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2012 - 1202(1) - 1204(4) - 1206(1) - 1207(3) - 1209(2) - 1212(2)

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2017 - 1701(3) - 1702(1) - 1703(4) - 1704(4) - 1705(2) - 1706(1) - 1707(2) - 1708(1)

Any replacements are listed further down

[150] **viXra:1708.0156 [pdf]**
*submitted on 2017-08-14 10:23:58*

**Authors:** Philip Molyneux

**Comments:** 10 Pages.

This paper critically examines the Cantor Diagonal Argument (CDA) that is used in
set theory to draw a distinction between the cardinality of the natural numbers and
that of the real numbers. In the absence of a verified English translation of the original
1891 Cantor paper from which it is said to be derived, the CDA is discussed here
using a consensus from the forms found in a range of published sources (from
"popular" to "professional"). Some general comments are made on these sources. The
discussion then focusses on the CDA as applied to the correspondence between the set
of the natural numbers, and the set of real numbers in the open range (0,1) in their
expansion from decimal digits (0.123… etc.).
Four points critical of the CDA are raised: (1) The conventional presentation of the
CDA forms a putative new real number (X) from the "diagonal" of the chosen list of
real numbers and which is therefore not on this initial list; however, it omits to
consider that it may indeed be on the later part of the list, which is never exhausted
however far the "diagonal" list is extended. (2) This aspect, combined with the fact
that X is still composed of decimal digits, that is, it is a real number as defined,
indicates that it must be on the later part of the list, that is, it is not a "new" number at
all. (3) The conventional application of the CDA leads to one putative "new" real
number (X); however, the logical extension of this in its "exhaustive" application, that
is, by using all possible different methods of alteration of the decimal digits on the
"diagonal", and by reordering the list in all possible ways, leads to a list of putative
"new" real numbers that become orders of magnitude longer than the chosen
"diagonal" list. (4) The CDA is apparently considered to be a method that is
applicable generally; however, testing this applicability with the natural numbers
themselves leads to a contradiction.
Following on from this, it is found that it indeed is possible to set up a one-to-one
correspondence between the natural numbers and the real numbers in (0,1), that is, !
⇔ "; this takes the form: … a3 a2 a1 ⇔ 0. a1 a2 a3 …, where the right hand extension
of the natural number is intended to be a mirror image of the left hand extension of
the real number. It is also shown how this may be extended to real numbers outside
the range (0,1).
Additionally, a form of the CDA was presented by Wilfred Hodges in his 1998
critical review of "hopeless papers" dealing with the CDA; this form is also examined
from the same viewpoints, and to the same conclusions.
Finally, some comments are made on the concept of "infinity", pointing out that to
consider this as an entity is a category error, since it simply represents an absence, that
is, the absence of a termination to a process.

**Category:** Set Theory and Logic

[149] **viXra:1707.0336 [pdf]**
*submitted on 2017-07-26 03:15:00*

**Authors:** Carmine D'Errico

**Comments:** 16 Pages.

In this paper a new isomorphism is demonstred to exists who is demonstred to be a subset of boolean algebra.
Then all the properties of this new algebra are sketched out and the most important logic theorems are demonstred a new form.
Therefore a fully photonc logic gate is demonstred to be possible with a power consuption very near to zero.
Other analysis are offered.
The results of a prototype are showed to be compatible with theory

**Category:** Set Theory and Logic

[148] **viXra:1707.0220 [pdf]**
*submitted on 2017-07-16 10:14:11*

**Authors:** Thomas Limberg

**Comments:** 6 Pages. Language: German

We interpret 5 of the 10 axioms of ZFC (Zermelo-Fraenkel set theory with axiom of choice) as normal statements and proof them. So these 5 sentences don't need to be introduced as axioms, but can be used as proven statements.

**Category:** Set Theory and Logic

[147] **viXra:1706.0321 [pdf]**
*submitted on 2017-06-12 04:00:23*

**Authors:** Nikolay Dementev

**Comments:** 9 Pages.

An attempt of resolving Russell’s paradox with the help of Aristotle’s ideas is presented.

**Category:** Set Theory and Logic

[146] **viXra:1705.0226 [pdf]**
*submitted on 2017-05-14 17:15:50*

**Authors:** Farzad Didehvar

**Comments:** 9 Pages.

Throughout this paper, we are trying to show how and why our Mathematical frame-work seems inappropriate to solve problems in Theory of Computation. More exactly, the concept of turning back in time in paradoxes causes inconsistency in modeling of the concept of Time in some semantic situations. As we see in the first chapter, by introducing a version of “Unexpected Hanging Paradox”, first we attempt to open a new explanation for some paradoxes. In the second step, by applying this paradox, it is demonstrated that any formalized system for the Theory of Computation based on Classical Logic and Turing Model of Computation leads us to a contradiction. We conclude that our mathematical frame work is inappropriate for Theory of Computation. Furthermore, the result provides us a reason that many problems in Complexity Theory resist to be solved.
.

**Category:** Set Theory and Logic

[145] **viXra:1705.0173 [pdf]**
*submitted on 2017-05-10 12:07:14*

**Authors:** Ilija Barukčić

**Comments:** 23 Pages. Copyright © 2017 by Ilija Barukčić, Jever, Germany. Published by:

The division of zero by zero turns out to be a long lasting and not ending puzzle in mathematics and physics. An end of this long discussion is not in sight. In particular zero divided by zero is treated as indeterminate thus that a result cannot be found out. It is the purpose of this publication to solve the problem of the division of zero by zero while relying on the general validity of classical logic. According to classical logic, zero divided by zero is one.

**Category:** Set Theory and Logic

[144] **viXra:1704.0363 [pdf]**
*submitted on 2017-04-27 12:26:58*

**Authors:** Haroun Boutamani

**Comments:** 7 Pages.

I address the concept of selectivity using a main set and an auxiliary set, the auxiliary
serves the purpose of deciding which elements do or do not fit together within the output
sets chosen based on the elements of the main set.
This paper also presents an approach of comparing sets based on a defined relation and
the sets’ properties, and also the elementary properties being total compatibility and total
incompatibility.
The problem classes P and NP are two sets considered under the criteria of
distinguishability, then the possibility of comparison of the two sets is discussed. The
proposed comparison presents a distinction between P and NP.

**Category:** Set Theory and Logic

[143] **viXra:1704.0161 [pdf]**
*submitted on 2017-04-13 03:51:02*

**Authors:** Ken Seton

**Comments:** 6 Pages. Enjoy

In the world of transfinite cardinality, any talk of number density, dart-board hits, proportions or probability is just a pouring from the empty into the void. Transfinite cardinality is unrelated to any normal concept of proportion or density over an interval and no understanding of it can obtained from probabilistic analogies. This is demonstrated by comparing extremely sparse and extremely dense collections of reals which are normally understood as uncountable and countable respectively. Should this incline one to think that Cantor’s equivalence definition is inappropriate for identifying the “size” of an infinite collection ?

**Category:** Set Theory and Logic

[142] **viXra:1704.0115 [pdf]**
*submitted on 2017-04-10 01:13:52*

**Authors:** Ken Seton

**Comments:** 8 Pages. Enjoy

Many have suggested that the infinite set has a fundamental problem. The usual complaint rails against the actually infinite which (to critics of various finitist persuasions) unjustifiably goes beyond the finite. Here we observe the exact opposite. The problem of the infinite set defined to have an identity (content) that is specified and restricted to be forever finite .
Set theory is taken at its word. The existence of the infinite set and the representation of irrational reals as infinite sets of terms is accepted. In this context, it is shown that the standard definition of the infinite countable set is inconsistent with the existence of its own classic convergents of construction. If the set is infinite then it must be quite unlike that which set theory asserts it to be.
Set theory found itself into some trouble over a century ago trusting an unrestricted anthropic comprehension. But serious doubt is cast on the validity of infinite sets which have been defined by a comprehension which overly-restricts their content.

**Category:** Set Theory and Logic

[141] **viXra:1704.0008 [pdf]**
*submitted on 2017-04-01 18:47:38*

**Authors:** Igor Hrncic

**Comments:** 2 Pages.

This letter is the short continuation of the previous paper titled "The infinitesimal error", available for free at the internet address http://vixra.org/abs/1703.0280. This letter is written just to further clarify the subject of "The infinitesimal error".

**Category:** Set Theory and Logic

[140] **viXra:1703.0280 [pdf]**
*submitted on 2017-03-29 22:47:37*

**Authors:** Igor Hrncic

**Comments:** 6 Pages.

Unfortunately, Cantor was wrong. His notion of transfinite bijection is flawed. Cantor introduced this notion of transfinite bijection as the additional axiom, even though without even realising this. From this error, other errors sprung into the existence. He did all this in the heroic effort to justify the death of infinitesimals, even though he wasn't aware of this either. Cantor went bravely on to defend the established error in higher mathematics before his mentors and peers who banished infinitesimals. Instead, he demonstrated the error of it. He never realised this as well. This paper elucidates this link between Cantor's errors and infinitesimals.

**Category:** Set Theory and Logic

[139] **viXra:1703.0113 [pdf]**
*submitted on 2017-03-13 04:06:52*

**Authors:** Wolfgang Mückenheim

**Comments:** 3 Pages.

It is shown that Cantor's diagonal argument fails because either there is no actual infinity and hence no defined diagonal number or there is actual infinity but the diagonal number cannot be distinguished from all real numbers of the Cantor list. Further it is shown by another argument that there are not uncountably many paths in the complete infinite Binary Tree.

**Category:** Set Theory and Logic

[138] **viXra:1703.0112 [pdf]**
*submitted on 2017-03-13 04:15:07*

**Authors:** Wolfgang Mückenheim

**Comments:** 3 Pages.

Limits of sequences of sets required to define infinite bijections do not only raise paradoxes but cause self-contradictory results.

**Category:** Set Theory and Logic

[137] **viXra:1703.0032 [pdf]**
*submitted on 2017-03-03 16:07:51*

**Authors:** W. Mückenheim

**Comments:** 5 Pages.

Contrary to the assumptions of transfinite set theory, limit and union of infinite sequences of sets differ. We will show this for the set of natural numbers by the newly devised powerful tool of arithmogeometry.

**Category:** Set Theory and Logic

[136] **viXra:1702.0293 [pdf]**
*submitted on 2017-02-24 03:55:26*

**Authors:** W. Mückenheim

**Comments:** 2 Pages.

It is shown that the enumeration of rational numbers cannot be complete.

**Category:** Set Theory and Logic

[135] **viXra:1701.0564 [pdf]**
*submitted on 2017-01-22 04:25:16*

**Authors:** Amir Deljoo

**Comments:** 11 Pages.

This is a declaration. The identity of mathematics and number theory or arithmetics. I have defined a pattern here that shows consciousness is a pure unique entity that is present everywhere and whole the existence is a graphical manifestation that has been phenomenoned over to enclose it and I hermetically simplify my intuition to transfer it to curious ones. Since explaining the methodology requires in thousands of pages, the final concluded statements and equations are only declared here.

**Category:** Set Theory and Logic

[134] **viXra:1701.0563 [pdf]**
*submitted on 2017-01-22 04:47:06*

**Authors:** امیر دلجو

**Comments:** 12 Pages.

این سند یک منشور است برای بیان ماهیت ریاضیات و نظریه ی اعداد یا حساب. در اینجا من الگویی را تعریف کرده ام که نشان می دهد، خودآگاهی یک وجود واحده بسیط و همه جاحاضر بوده و هستی به مثابه یک کلیّت، یک تجلّی گرافیکی ست که در جهت افشای این خودآگاهی عارض شده و من هرمس وار، شهود خود بر یگانگی و آفرینش را برای انتقال به انسان های کنجکاو ساده سازی و تحریر کرده ام. از آنجا که تبیین روش شناختی این منشور مستلزم هزاران صفحه است، در اینجا تنها عبارات و معادلات منتج شده ی نهایی اعلان می شود.

**Category:** Set Theory and Logic

[133] **viXra:1701.0328 [pdf]**
*submitted on 2017-01-07 23:13:35*

**Authors:** Roger Granet

**Comments:** 3 Pages.

The Russell Paradox (1) considers the set, R, of all sets that are not members of themselves. On its surface, it seems like R belongs to itself only if it doesn't belong to itself. This is where the paradox come from. Here, a solution is proposed that is similar to Russell's method based on his theory of types (1,2) but is instead based on the definition of why things exist as described in previous work (3). In that work, it was proposed that a thing exists if it is a grouping defining what is contained within. A corollary is that a thing, such as a set, does not exist until what is contained within is defined. A second corollary is that after a grouping defining what is contained within is present, and the thing exists, if one then alters the definition of what is contained within, the first existent entity is destroyed and a different existent entity is created. Based on this, set R of the Russell Paradox does not even exist until after the list of the elements it contains (e.g. the list of all sets that aren't members of themselves) is defined. Once this list of elements is completely defined, R then springs into existence. Therefore, because it doesn't exist until after its list of elements is defined, R obviously can't be in this list of elements and, thus, cannot be a member of itself; so, the paradox is resolved. Additionally, one can't then put R back into its list of elements after the fact because if this were done, it would be a different list of elements, and it would no longer be the original set R, but some new set. This same type of reasoning is then applied to the Godel Incompleteness Theorem, which roughly states that there will always be some statements within a formal system of arithmetic (system P) that are true but that can't be proven to be true. Briefly, this reasoning suggests that arguments such as the Godel sentence and diagonalization arguments confuse references to future, not yet existent statements with a current and existent statement saying that the future statements are unprovable. Current and existent statements are different existent entities than future, not yet existent statements and should not be conflated. In conclusion, a new resolution of the Russell Paradox and some issues with the Godel Incompleteness Theorem are described.

**Category:** Set Theory and Logic

[132] **viXra:1612.0286 [pdf]**
*submitted on 2016-12-17 23:42:11*

**Authors:** Roger Granet

**Comments:** 4 Pages.

Here, the conclusion in set theory that the size of an infinite set is the same as the size of an infinite subset derived from it is questioned.This is done not to try and invalidate any mathematical results because mathematics is an abstract field and does not necessarily have to accurately describe the physical world but in order to prompt the reexamination of the use of this result in physics, which does have to accurately describe the real, physical world and the relationships between its components. The rationale is as follows. First, it is suggested that thought experiments are still experiments and should follow the rules for good experimental technique, which include the need to study a system in a setting as close as possible to the "natural setting" to try and avoid experimental artifacts. Now, starting with the single set of the positive integers, one wants to compare the total number of integers to the total number of even integers within the "natural setting" of the single original set. The traditional experimental processing method extracts the even integers, puts them into a separate subset and pairs off the subset's and set's members one-to-one with a function. After doing this, no elements are left over, and, therefore, the original set and the subset extracted from it are said to be the same size. However, extracting the evens and putting them into a separate subset dramatically alters the original single set system. This is analogous to a biologist extracting the nucleus from a cell, studying the nucleus and remaining parts of the cell in isolation and assuming that the results obtained are the same as in the original intact cell. They often are not. Does extracting the even integers out into a subset alter the results compared to those that would be obtained in the natural single set system? Yes. In the single set system, the positive integers march lockstep and in- phase with the odd integers from one to infinity, meaning that there is a built-in relationship in this system of one positive integer for every two total integers, which means that there are only one-half as many positive integers as total integers. This is a different result than that obtained after the subset extraction method, which means that the result produced by this method is an experimental artifact. This should be unacceptable in a well done experiment even if it is a thought experiment. It is suggested that this artifact may be related to some of the problems associated with infinities in physics.

**Category:** Set Theory and Logic

[131] **viXra:1611.0415 [pdf]**
*submitted on 2016-11-30 16:52:52*

**Authors:** Arthur Shevenyonov

**Comments:** 6 Pages. new foundations

The proposed extension of propositional logic exhibits a striking similarity to generalized games with a chance player while pointing to accidental applications in quantum computing.

**Category:** Set Theory and Logic

[130] **viXra:1611.0330 [pdf]**
*submitted on 2016-11-24 04:20:28*

**Authors:** Hewayda Elghawalby, A. A. Salama

**Comments:** 8 Pages.

In this paper we present a new neutrosophic crisp family generated from the three components’
neutrosophic crisp sets presented by Salama [4]. The idea behind Salam’s neutrosophic crisp set
was to classify the elements of a universe of discourse with respect to an event ”A” into three
classes: one class contains those elements that are fully supportive to A, another class contains
those elements that totally against A, and a third class for those elements that stand in a distance
from being with or against A. Our aim here is to study the elements of the universe of discourse
which their existence is beyond the three classes of the neutrosophic crisp set given by Salama. By adding more components we will get a four components’ neutrosophic crisp sets called the Ultra Neutrosophic Crisp Sets. Four types of set’s operations is defined and the properties of the new ultra neutrosophic crisp sets are studied. Moreover, a definition of the relation between two ultra neutrosophic crisp sets is given.

**Category:** Set Theory and Logic

[129] **viXra:1611.0281 [pdf]**
*submitted on 2016-11-19 11:31:44*

**Authors:** Damodar Rajbhandari

**Comments:** 3 Pages.

This paper will introduce a new notation named as F-notation. This notation will help us to prove the statement, "The cartesian product of natural numbers is countably infinite".

**Category:** Set Theory and Logic

[128] **viXra:1611.0079 [pdf]**
*submitted on 2016-11-06 06:48:50*

**Authors:** Max Null, Sergey Belov

**Comments:** 4 Pages.

This article is a mathematical experiment with the sets and the formulas.
We consider new elements which are called the electors. The elector has the
properties of the sets and the formulas.

**Category:** Set Theory and Logic

[127] **viXra:1608.0395 [pdf]**
*submitted on 2016-08-29 10:19:42*

**Authors:** Max Null, Sergey Belov

**Comments:** 12 Pages.

We define the topology atop(χ) on a complete upper semilattice χ = (M, ≤). The limit points are determined by the formula
Lim(D,X) = sup{a ∈ M| {x ∈ X| a ≤ x} ∈ D},
where X ⊆ M is an arbitrary set, D is an arbitrary non-principal ultrafilter on X. We investigate Lim(D,X) and topology atop(χ) properties. In particular, we prove the compactness of the topology atop(χ).

**Category:** Set Theory and Logic

[126] **viXra:1608.0057 [pdf]**
*submitted on 2016-08-05 07:48:37*

**Authors:** Adrian Chira

**Comments:** 4 Pages.

Curry's paradox is generally considered to be one of the hardest paradoxes to solve. However, it is shown here that the solution is however trivial and the paradox turns out to be no paradox at all. Reviewing the starting point of the paradox, it is concluded that it amounts to a false definition or assertion and therefore it is to be expected, as opposed to being paradoxical, to arrive to a false conclusion. Despite that fact that verifying the truth value of the first statement of the paradox is trivial, mathematicians and logicians have failed to do so and merely assumed that it is true. Taking this into consideration that it is false, the paradox is however dismissed. This conclusion puts to rest an important paradox that preoccupies logicians and points out the importance of verifying one's assumptions.

**Category:** Set Theory and Logic

[125] **viXra:1607.0421 [pdf]**
*submitted on 2016-07-22 08:40:32*

**Authors:** Robert A. Herrmann

**Comments:** 8 Pages.

The basic mathematical aspects of the GGU and GID models are discussed. As an illustration, the modified Robinson approach is used to give a more direct prediction as to the composition of ultra-propertons. Relative to logic-systems, the refined developmental paradigm is applied to the General Intelligence Design (GID) model and the basic GID statement are given.

**Category:** Set Theory and Logic

[124] **viXra:1607.0153 [pdf]**
*submitted on 2016-07-13 04:47:18*

**Authors:** S.Kalimuthu

**Comments:** 06 Pages. NA

According to James R. Meyer, In mathematics, a theorem is intended to be a term for a very precise and definite concept - a theorem is a statement that is proved, using rigorous mathematical reasoning, to follow according to a set of logical rules, from a set of initial statements. These initial statements are usually called axioms, and these are statements that are accepted without being proven. The set of logical rules which determine how one statement can follow from another are usually called the rules of inference . And basically, Gödel's incompleteness theorem is any statement that says that for every formal mathematical system, there are sentences that cannot be proved to be true or false in that system.

**Category:** Set Theory and Logic

[123] **viXra:1607.0124 [pdf]**
*submitted on 2016-07-11 02:37:20*

**Authors:** Florentin Smarandache

**Comments:** 170 Pages.

Neutrosophic Over-/Under-/Off-Set and -Logic were defined for the first time by Smarandache in 1995 and published in 2007. They are totally different from other sets/logics/probabilities.
He extended the neutrosophic set respectively to Neutrosophic Overset {when some neutrosophic component is > 1}, Neutrosophic Underset {when some neutrosophic component is < 0}, and to Neutrosophic Offset {when some neutrosophic components are off the interval [0, 1], i.e. some neutrosophic component > 1 and other neutrosophic component < 0}.
This is no surprise with respect to the classical fuzzy set/logic, intuitionistic fuzzy set/logic, or classical/imprecise probability, where the values are not allowed outside the interval [0, 1], since our real-world has numerous examples and applications of over-/under-/off-neutrosophic components.

**Category:** Set Theory and Logic

[122] **viXra:1606.0160 [pdf]**
*submitted on 2016-06-15 08:59:21*

**Authors:** Robert A. Herrmann

**Comments:** 52 Pages.

This part contains the Contents and Chapters 1, 2, 3 and 4, which include Alphabets, Words, Deduction, The Nonstandard Structure, as well as Adjective, Propositional, Predicate Reasoning, Reasoning from the Prefect and Order.

**Category:** Set Theory and Logic

[121] **viXra:1606.0159 [pdf]**
*submitted on 2016-06-15 08:59:39*

**Authors:** Robert A. Herrmann

**Comments:** 48 Pages.

This part contains Chapters 5, 6, 7, 8, 9, which include Consequence Operators (Operations) Perception, An Alternate Approach, Developmental Paradigms, Ultrawords, A Neutron Altering Process, The Extended Structure and General Paradigms.

**Category:** Set Theory and Logic

[120] **viXra:1606.0158 [pdf]**
*submitted on 2016-06-15 09:00:00*

**Authors:** Robert A. Herrmann

**Comments:** 38 Pages.

This part contains Chapters 10, 11, Symbols and the Index, which include Laws and Rules, Propertons (subparticles) and the MA-model.

**Category:** Set Theory and Logic

[119] **viXra:1606.0005 [pdf]**
*submitted on 2016-06-01 09:04:24*

**Authors:** Robert A. Herrmann

**Comments:** 7 Pages.

The major purpose of this article is to establish Theorem 9.3.1 for the ESG, with the modified Robinson approach, and to make other improvements in Section 9 of The Theory of Ultralogics.

**Category:** Set Theory and Logic

[118] **viXra:1605.0231 [pdf]**
*submitted on 2016-05-22 20:23:04*

**Authors:** Philip Druck

**Comments:** 27 Pages.

A totally ordered set is identified with cardinality strictly between natural (N) and real (R) numbers. This set, denoted DS, is essentially an experimental finding, identified in unrelated patented research on nonuniform data sampling and self-stabilizing computer arithmetic. Its theoretical validation here will provide concrete proof that the Continuum Hypothesis (CH) is false. Note that this is distinct from determining whether CH can or cannot be proven from current axioms of set theory, which is settled. Also note that the Generalized Continuum Hypothesis is not addressed. First, Cantor diagonalization is applied isomorphically to prove that DS has strictly more than Cardinality(N) points. Then three (3) distinct proofs are provided to show that DS contains strictly fewer than Cardinality(R) elements. Each proof relies on a distinct property of primes. It is surmised that the considerable research efforts to-date on CH missed this result due to over-generalization, by considering all Alephi sets, i=0.., ∞. Those efforts thereby missed the impact of primes specifically on Aleph0/Aleph1 sets.

**Category:** Set Theory and Logic

[117] **viXra:1605.0052 [pdf]**
*submitted on 2016-05-04 05:05:38*

**Authors:** J. Martina Jency, I. Arockiarani

**Comments:** 9 Pages.

In this paper, we design a model based on adjustable and mean potentiality approach to single valued neutrosophic level soft sets. Further, we introduce the notion of weighted single valued neutrosophic soft set and investigate its application in decision making.

**Category:** Set Theory and Logic

[116] **viXra:1605.0051 [pdf]**
*submitted on 2016-05-04 05:07:03*

**Authors:** Chunfang Liu, YueSheng Luo

**Comments:** 4 Pages.

Interval-valued neutrosophic set (INS) is a
generalization of fuzzy set (FS) that is designed for some practical situations in which each element has different truth membership function, indeterminacy membership function and falsity membership function and permits the membership degrees to be expressed by interval values.

**Category:** Set Theory and Logic

[115] **viXra:1605.0050 [pdf]**
*submitted on 2016-05-04 05:08:47*

**Authors:** Partha Pratim Dey, Surapati Pramanik, Bibhas C. Giri

**Comments:** 10 Pages.

This paper investigates an extended grey relational analysis method for multiple attribute decision making problems under interval neutrosophic uncertain linguistic environment. Interval neutrosophic uncertain
linguistic variables are hybridization of uncertain linguistic variables and interval neutrosophic sets and they can easily express the imprecise, indeterminate and inconsistent
information which normally exist in real life situations.

**Category:** Set Theory and Logic

[114] **viXra:1605.0049 [pdf]**
*submitted on 2016-05-04 05:09:58*

**Authors:** Florentin Smarandache

**Comments:** 3 Pages.

We have introduced for the first time the
degree of dependence (and consequently the degree of independence) between the components of the fuzzy set, and also between the components of the neutrosophic set in our 2006 book’s fifth edition. Now we extend it for the first time to the refined neutrosophic set considering the degree of dependence or independence of subcomponets.

**Category:** Set Theory and Logic

[113] **viXra:1605.0048 [pdf]**
*submitted on 2016-05-04 05:11:04*

**Authors:** Said Broumi, Assia Bakali, Mohamed Talea, Florentin Smarandache

**Comments:** 5 Pages.

Many results have been obtained on isolated
graphs and complete graphs. In this paper, a necessary and sufficient condition will be proved for a single valued neutrosophic graph to be an isolated single valued neutrosophic graph.

**Category:** Set Theory and Logic

[112] **viXra:1605.0047 [pdf]**
*submitted on 2016-05-04 05:12:42*

**Authors:** Gaurav, Megha Kumar, Kanika Bhutani, Swati Aggarwal

**Comments:** 12 Pages.

This paper employs a new soft computing based
methodology for identifying and analyzing the relationships among the causes and implications of the two challenging problems in India: unbalanced sex ratio and poverty.

**Category:** Set Theory and Logic

[111] **viXra:1605.0046 [pdf]**
*submitted on 2016-05-04 05:13:56*

**Authors:** Nouran Radwan, M. Badr Senousy, Alaa El Din M. Riad

**Comments:** 5 Pages.

This paper reviews some of the multivalued
logic models which are fuzzy set, intuitionistic
fuzzy set, and suggests a new approach which is neutrosophic set for handling uncertainty in expert systems to derive decisions. The paper highlights, compares and clarifies the differences of these models in terms of the application area of problem solving.

**Category:** Set Theory and Logic

[110] **viXra:1605.0045 [pdf]**
*submitted on 2016-05-04 05:15:14*

**Authors:** Madad Khan, Florentin Smarandache, Sania Afzal

**Comments:** 16 Pages.

In this paper we have defined neutrosophic ideals,
neutrosophic interior ideals, netrosophic quasi-ideals and neutrosophic bi-ideals (neutrosophic generalized bi-ideals) and proved some results related to them.

**Category:** Set Theory and Logic

[109] **viXra:1605.0044 [pdf]**
*submitted on 2016-05-04 05:16:41*

**Authors:** Nasir Shah, Asim Hussain

**Comments:** 14 Pages.

The aim of this paper is to propose a new type of
graph called neutrosophic soft graphs. We have established a link between graphs and neutrosophic soft sets. Basic operations of neutrosophic soft graphs such as union, intersection and complement are defined here. The concept of strong neutrosophic soft graphs is also discussed in this paper.

**Category:** Set Theory and Logic

[108] **viXra:1605.0043 [pdf]**
*submitted on 2016-05-04 05:18:07*

**Authors:** Partha Pratim Dey, Surapati Pramanik, Bibhas C. Giri

**Comments:** 9 Pages.

The present paper proposes neutrosophic soft
multi-attribute decision making based on grey relational projection method. Neutrosophic soft sets is a combination of neutrosophic sets and soft sets and it is a new mathematical apparatus to deal with realistic problems in the fields of medical sciences, economics, engineering, etc.

**Category:** Set Theory and Logic

[107] **viXra:1605.0041 [pdf]**
*submitted on 2016-05-04 05:21:11*

**Authors:** Vasile Patrascu

**Comments:** 10 Pages.

In this article, starting from primary representation of neutrosophic information, namely the triplet (μ, ω, ν) made up of the degree of truth μ, degree of indeterminacy
ω and degree of falsity ν, we define a refined representation in a penta-valued fuzzy space, described by the index of truth t, index of falsity f, index of ignorance u, index of contradiction c and index of hesitation h. In
the proposed penta-valued refined representation the indeterminacy was split into three sub-indeterminacies such as ignorance, contradiction and hesitation.

**Category:** Set Theory and Logic

[106] **viXra:1605.0040 [pdf]**
*submitted on 2016-05-04 05:22:55*

**Authors:** Huda E. Khalid

**Comments:** 5 Pages.

This article sheds light on the possibility of finding the minimum solution set of neutrosophic relational geometric programming with (max, min) composition. This work examines the privacy enjoyed by both neutrosophic logic and geometric programming, and how it affects the minimum solutions. It is the first attempt to solve this type of problems.

**Category:** Set Theory and Logic

[105] **viXra:1605.0020 [pdf]**
*submitted on 2016-05-03 01:19:33*

**Authors:** Florentin Smarandache - Editor-in-Chief

**Comments:** 113 Pages.

“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics
that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.

**Category:** Set Theory and Logic

[104] **viXra:1604.0392 [pdf]**
*submitted on 2016-04-30 17:18:06*

**Authors:** István Aggott Hönsch

**Comments:** 19 Pages.

A terminological framework is proposed for the mathematical examination and analysis of the Mandelbrot set's correlative ectocopial set. The Apeiropolis and anthropobrot multisets are defined and explained to be the mathematical entities underlying the well-known Buddhabrot visualization.
The definitions are presented as tools conducive to finding novel approaches and generating discoveries that might otherwise be missed via a primarily programmatic approach.
The anthropobrot multisets are introduced as a new, infinite repository of unique pareidolic figures as richly diverse as the Julia sets.

**Category:** Set Theory and Logic

[103] **viXra:1604.0340 [pdf]**
*submitted on 2016-04-25 07:04:43*

**Authors:** Vasile Pătraşcu

**Comments:** 10 Pages. Neutrosophic Sets and Systems, Volume 11, pp. 57-66, 2016

In this article, starting from primary representation of neutrosophic information, namely the triplet (μ, ω, ν) made up of the degree of truth μ, degree of indeterminacy ω and degree of falsity ν, we define a refined representation in a penta-valued fuzzy space, described by the index of truth t, index of falsity f, index of ignorance u, index of contradiction c and index of hesitation h. In the proposed penta-valued refined representation the indeterminacy was split into three sub-indeterminacies such as ignorance, contradiction and hesitancy. The set of the proposed five indexes represent the similarities of the neutrosophic information (μ, ω, ν) with these particular values: T=(1,0,0), F=(0,0,1), U=(0,0,0), C=(1,0,1) and H=(0.5,1,0.5). This representation can be useful when the neutrosophic information is obtained from bipolar information which is defined by the degree of truth and the degree of falsity to which is added the third parameter, its cumulative degree of imprecision.

**Category:** Set Theory and Logic

[102] **viXra:1604.0118 [pdf]**
*submitted on 2016-04-06 06:46:52*

**Authors:** Yakov A. Iosilevskii

**Comments:** 1134 Pages.

In contrast to Church, who proved in 1936, based on papers by Gödel, that a dual decision problem for the conventional axiomatic first-order predicate calculus is unsolvable, I have solved a trial decision problem algebraically (and hence analytically, not tabularily) for a properly designed axiomatic first-order algebraico-predicate calculus, called briefly the trial logic (TL), and have successfully applied the pertinent algebraic decision procedures to all conceivable logical relations of academic or practical interest, including the 19 categorical syllogisms. The structure of the TL is a synthesis of the structure of a conventional axiomatic first-order predicate calculus (briefly CAPC) and of the structure of an abstract integral domain. Accordingly, the TL contains as its autonomous parts the so-called Predicate-Free Relational Trial Logic (PFRTL), which is parallel to a conventional axiomatic sentential calculus (CASC), and the so-called Binder-Free Predicate Trial Logic (BFPTL), which is parallel to the predicate-free part of a pure CAPC. This treatise, presenting some of my findings, is alternatively called “the Theory of Trial Logic” (“the TTL”) or “the Trial Logic Theory” (“the TLT”). The treatise reopens the entire topic of symbolic logic that is called “decision problem” and that Church actually closed by the fact of synecdochically calling the specific dual decision problem, the insolvability of which he had proved, by the generic name “decision problem”, without the qualifier “dual”. Any additional axiom that is incompatible with the algebraic decision method of the trial logic and that is therefore detrimental for that method is regarded as one belonging to either to another logistic system or to mathematics.

**Category:** Set Theory and Logic

[101] **viXra:1603.0226 [pdf]**
*submitted on 2016-03-16 03:11:25*

**Authors:** Vasile Pătraşcu

**Comments:** 12 Pages.

Starting from the primary representation of neutrosophic information, namely the degree of truth, degree of indeterminacy and degree of falsity, we define a nuanced representation in a penta valued fuzzy space, described by the index of truth, index of falsity, index of ignorance, index of contradiction and index of hesitance. Also, it was constructed an associated penta valued logic and then using this logic, it was defined for the proposed penta valued structure the following operators: union, intersection, negation, complement and dual. Then, the penta valued representation is extended to a hexa valued one, adding the sixth component, namely the index of ambiguity.

**Category:** Set Theory and Logic

[100] **viXra:1602.0198 [pdf]**
*submitted on 2016-02-16 21:38:55*

**Authors:** Florentin Smarandache

**Comments:** 107 Pages.

Welcome into my scientific lab!
My lab[oratory] is a virtual facility with non-controlled conditions in which I mostly perform scientific chats.
I called the jottings herein scilogs (truncations of the words scientific, and gr. Λόγος – appealing rather to its original meanings "ground", "opinion", "expectation"), combining the welly of both science and informal (via internet) talks.
In this book, one may find new and old questions and ideas, some of them already put at work, others dead or waiting, referring to various fields of research (e.g. from neutrosophic algebraic structures to Zhang's degree of intersection, or from Heisenberg uncertainty principle to neutrosophic statistics) – email messages to research colleagues, or replies, notes about authors, articles or books, so on.
Feel free to budge in the lab or use the scilogs as open source for your own ideas.

**Category:** Set Theory and Logic

[99] **viXra:1601.0193 [pdf]**
*submitted on 2016-01-17 18:49:05*

**Authors:** Nikolaj Roerich

**Comments:** 3 Pages.

We show that Riemann Hypothesis is actually an Axiom.
Prooving it would mean knowing how to build the Universe.
That is the future that people evolution will lead to.
To build the Universee one needs to know the details about the corresponding Hilbert Space since Universe = Hilbert Space,
and Riemann Hypothesis solution is equivalent to the knowing the linear operator in that Hilbert Space that is called "L" which has the eigenvalues equal to the zeros of Riemann Function.

**Category:** Set Theory and Logic

[98] **viXra:1601.0023 [pdf]**
*submitted on 2016-01-04 06:29:47*

**Authors:** Janis Belov

**Comments:** 2 Pages.

We solve P vs NP Millenium problem.

**Category:** Set Theory and Logic

[97] **viXra:1512.0357 [pdf]**
*submitted on 2015-12-17 20:25:15*

**Authors:** Bolonkin A.A.

**Comments:** 8 Pages.

Предлагается принципиально новый метод оптимизации. В отличие от классической постановки задачи:
а) Дан функционал – найти его минималь.
Рассматриваются также задачи:
б) найти более «узкое» подмножество, содержащее абсолютную минималь;
в) найти подмножество решений лучших, чем данное;
г) найти оценки снизу данного функционала.
В настоящее время большинство исследователей, работающих в области оптимизации заняты решение задачи в классической постановке – отысканием точной минимали. Инженера же, как правило, в реальных задачах интересует подмножество квази-оптимальных решений, выбирая из которого, он заранее уверен, что получит значение функционала не хуже заданной величины (задача в) и оценка снизу, показывающая насколько он далек от точного оптимального решения (задача г). Кроме того у него есть много дополнительных соображений, которые нельзя учесть в математической модели или которые бы ее сильно усложнили. Постановка задачи в форме «в» дает ему определенную свободу выбора.
This method, called the “Method of Deformation of Functional (Extreme)”, solves for a total minimum and finds a solution set near the optimum. Solutions found by this method can be exact or approximate. Most other methods solve only for a unique local minimum. The ability to create a set of solutions rather than a unique solution has important practical ramifications in many designs, economic and scientific problems because a unique solution usually is difficult to realize in practice.
This method has the additional virtue of a simple proof, one that is useful for studying other methods of optimization, since most other methods can be delivered from the Method of Deformation.

**Category:** Set Theory and Logic

[96] **viXra:1511.0160 [pdf]**
*submitted on 2015-11-18 12:28:07*

**Authors:** Alex Patterson

**Comments:** 7 Pages.

Will be look at (data) type inference for the four major arithmetic types to search for symmetry-checks and factorization in the Lie algebra, using the multiplicative decomposition by such searches in the Lie Algebra to Poincare Group, Poincare Group important only for the theory check.

**Category:** Set Theory and Logic

[95] **viXra:1510.0133 [pdf]**
*submitted on 2015-10-16 03:30:37*

**Authors:** Takis Tsoukalas, Panagiotatos Mitropolitis Thessalonikis Anthimos Roussas

**Comments:** 8 Pages.

Assume Z ≤ D. Recent developments in applied singular oper- ator theory [11] have raised the question of whether t < π. We show that Borel’s criterion applies. In [11, 11, 28], the main result was the derivation of multiplicative, linearly tangential paths. In [11], the authors extended lines.

**Category:** Set Theory and Logic

[94] **viXra:1510.0041 [pdf]**
*submitted on 2015-10-05 06:44:34*

**Authors:** editors Florentin Smarandache, Mumtaz Ali

**Comments:** Pages. 98

This volume is a collection of fourteen papers, written by different authors and co-authors (listed in the order of the papers): F. Yuhua, K. Mandal, K. Basu, S. Pramanik, K. Mondal, S. Alkhazaleh, J. Nescolarde-Selva, J. L. Usó-Doménech, A. Betancourt-Vázquez, K. Pérez-Teruel, M. Leyva-Vázquez, A. Aydoğdu, I. Arockiarani, C. A. C. Sweety, F. Smarandache, L. Zhengda, S. Kar, S. Mukherjee, P. Das, and T. K. Kumar.

**Category:** Set Theory and Logic

[93] **viXra:1508.0309 [pdf]**
*submitted on 2015-08-30 19:26:33*

**Authors:** Minseong Kim

**Comments:** 2 Pages.

In computer science, a character set $\Sigma$ is often defined. Then, Kleene plus and Kleene star for formal language are defined. Then, $\Sigma^{+} = \Sigma^{*}\Sigma$ is proved, which means every string (set) in $\Sigma^{+}$ can be represented as a concatenation of a set in $\Sigma^{*}$ and a set in $\Sigma$. However, if one forms a set that cannot be defined by a formula but what people would believe as existing, then while the proof itself does not break down, it may be possible that state of matter is inconsistent. This paper explores this possibility.

**Category:** Set Theory and Logic

[92] **viXra:1508.0299 [pdf]**
*submitted on 2015-08-29 08:36:40*

**Authors:** Samuel Amok

**Comments:** 1 Page.

In this paper, I answer to a question that has been raised in http://www.les-mathematiques.net/phorum/read.php?16,1137927,1137947#msg-1137947

**Category:** Set Theory and Logic

[91] **viXra:1508.0284 [pdf]**
*submitted on 2015-08-27 01:08:39*

**Authors:** Shawkat Alkhazaleh, Emad Marei

**Comments:** 112 Pages.

In 1995 Smarandache introduced the concept of
neutrosophic set which is a mathematical tool for handling problems involving imprecise, indeterminacy and inconsistent data. In 2013 Maji introduced the concept of neutrosophic soft
set theory as a general mathematical tool for dealing with uncertainty.

**Category:** Set Theory and Logic

[90] **viXra:1508.0161 [pdf]**
*submitted on 2015-08-20 09:19:58*

**Authors:** Alex Patterson

**Comments:** 12 Pages. Special thanks to Michael J. Burns

This paper uses its own peculiar lettering system for each paragraph.
This paper proposes an overall solution to Godel’s incompleteness theorem and the Gödel sentence. Both are handled as one, by using Gödel numbers as the exemplary objects of incompleteness.
New terms and tools are introduced for quantification that creates a more synthetic (logical, reasonable, coherent) intervention and inter-weaving into these now classical problems of the assumptions in the Gödel material and literature.
Asymptotes are used within vertical and horizontal graphs to justify a future that need not be seen as a future in the sense of grammatical future-tense, but as a potential part such systems themselves that we deal with respect to incompleteness.
The thesis is that we can approach incompleteness by using theoretical reasoning and available tools that are allowed in theoretical reasoning to critique the very theory of incompleteness itself. That is the essential Abstract Thesis. It will be seen that a real attempt is attempted.

**Category:** Set Theory and Logic

[89] **viXra:1508.0089 [pdf]**
*submitted on 2015-08-11 16:25:07*

**Authors:** Peiman Ghasemi

**Comments:** 6 Pages.

Until the current moment, mankind is not realized that there is a diverse population of intelligent civilizations living in our universe. In the current article we will deduce the occurrence/existence of extraterrestrial life by mathematical proof. I would show you that even inside our galaxy, the Milky Way, a sufficient number of alien creatures are living. It's a mathematical proof for the extraterrestrial life debate, for the first time in mankind's history.

**Category:** Set Theory and Logic

[88] **viXra:1506.0165 [pdf]**
*submitted on 2015-06-22 20:30:53*

**Authors:** Takahiro Kato

**Comments:** 14 Pages.

This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a more direct connection between propositional logic and Boolean algebras.

**Category:** Set Theory and Logic

[87] **viXra:1506.0147 [pdf]**
*submitted on 2015-06-19 08:23:22*

**Authors:** Thomas Colignatus

**Comments:** 12 Pages.

The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) axioms for set theory appear to be inconsistent. A step in developing this proof is the observation that ZFC would be deductively incomplete if it were consistent. Both points are proven by means of the singleton. The axioms are still too lax on the notion of a 'well-defined set'.

**Category:** Set Theory and Logic

[86] **viXra:1505.0173 [pdf]**
*submitted on 2015-05-24 14:58:20*

**Authors:** A.D Albert

**Comments:** Pages.

Traditionally Fuzzy Logic is defined in the unit interval of [0,1]. These notes explore what will happen if this interval was larger than the unit interval (where normalization of truth values will violate the definition of a particular fuzzy set). These notes also derive suitable logical operators applicable to this extended interval utilizing first principles.

**Category:** Set Theory and Logic

[85] **viXra:1505.0122 [pdf]**
*submitted on 2015-05-16 12:35:47*

**Authors:** Misha Mikhaylov

**Comments:** 24 Pages.

It seems that statements determining features of some algebraic structures behavior are based on just intuitive assumptions or empiric observations and for sake of convenience (simplest example is the phrase: “let’s consider 0! =1”… perhaps, just because sir I. Newton entrusted, so, why not 2, 5, 7.65 – choose any). So, without logical explanation these are looking a little mysterious or sometimes even magic. This article is a humble attempt to get it straight rather formally. Some troubles may appear on the way – e.g. as it was shown earlier (in the ref. [2], for example), there are at least two binary relations hav-ing properties of idempotent equivalences – algebra’s elements that may aspire to be an identity. Apparently, probable obtaining of some well-known results in the text is not an attempt of their re-discovering, but it is rather “check-points” that confirm theory validity, more by token that it was made by using of the only exceptionally formal way, while usually they are obtained rather intuitively. Usually the notion of tensor product is determined for each kind of algebraic structure – especially for modulus (in group theory it is often called direct product – but this is a matter of semantics, so, it’s rather negligible). Here it is shown that tensor product may be introduced without defining of concrete algebraic structure. Without such introduction defining of algebraic operation is strongly complicated.

**Category:** Set Theory and Logic

[84] **viXra:1505.0079 [pdf]**
*submitted on 2015-05-10 13:10:05*

**Authors:** Alexander S. Nudelman

**Comments:** 5 Pages.

In this paper we define an arithmetic theory PAM, which is an extension of Peano arithmetic PA, and prove that theory PAM has only one (up to isomorphism) model, which is the standard PA–model.

**Category:** Set Theory and Logic

[83] **viXra:1504.0086 [pdf]**
*submitted on 2015-04-11 03:11:45*

**Authors:** Vadim V Nazarenko

**Comments:** 1 Page.

As much as 0 is the opposite of प, every number of the Poorna series has it's opposite number from the Shoonya series.

**Category:** Set Theory and Logic

[82] **viXra:1503.0115 [pdf]**
*submitted on 2015-03-14 14:58:16*

**Authors:** Florentin Smarandache

**Comments:** 11 Pages.

In this paper we introduce for the first time a new type of structures, called (T, I, F)-Neutrosophic Structures, presented from a neutrosophic logic perspective, and we show particular cases of such structures in geometry and in algebra.
In any field of knowledge, each structure is composed from two parts: a space, and a set of axioms (or laws) acting (governing) on it. If the space, or at least one of its axioms (laws), has some indeterminacy, that structure is a (T, I, F)-Neutrosophic Structure.
The (T, I, F)-Neutrosophic Structures [based on the components T=truth, I=indeterminacy, F=falsehood] are different from the Neutrosophic Algebraic Structures [based on neutrosophic numbers of the form a+bI, where I=indeterminacy and I^n = I], that we rename as Neutrosophic I-Algebraic Structures (meaning algebraic structures based on indeterminacy “I” only). But we can combine both and obtain the (T, I, F)-Neutrosophic I-Algebraic Structures, i.e. algebraic structures based on neutrosophic numbers of the form a+bI, but also having indeterminacy related to the structure space (elements which only partially belong to the space, or elements we know nothing if they belong to the space or not) or indeterminacy related to at least one axiom (or law) acting on the structure space. Then we extend them to Refined (T, I, F)-Neutrosophic Refined I-Algebraic Structures.

**Category:** Set Theory and Logic

[81] **viXra:1503.0085 [pdf]**
*submitted on 2015-03-12 05:25:13*

**Authors:** Takahiro Kato

**Comments:** 323 Pages

Modules (also known as profunctors or distributors) and morphisms among them subsume categories and functors and provide more general and abstract framework to explore the theory of structures. In this book we generalize and redevelop the basic notions and results of category theory using this framework of modules.

**Category:** Set Theory and Logic

[80] **viXra:1502.0045 [pdf]**
*submitted on 2015-02-06 03:21:31*

**Authors:** Vasile Patrascu

**Comments:** 7 Pages. Neutrosophic Sets and Systems, pp. 40-46, Vol. 7, 2015.

This paper presents two variants of penta-valued representation for neutrosophic entropy. The first is an extension of Kaufmann's formula and the second is an extension of Kosko's formula.
Based on the primary three-valued information represented by the degree of truth, degree of falsity and degree of neutrality there are built some penta-valued representations that better highlights some specific features of neutrosophic entropy. Thus, we highlight five features of neutrosophic uncertainty such as ambiguity, ignorance, contradiction, neutrality and saturation. These five features are supplemented until a seven partition of unity by adding two features of neutrosophic certainty such as truth and falsity.
The paper also presents the particular forms of neutrosophic entropy obtained in the case of bifuzzy representations, intuitionistic fuzzy representations, paraconsistent fuzzy representations and finally the case of fuzzy representations.

**Category:** Set Theory and Logic

[79] **viXra:1501.0107 [pdf]**
*submitted on 2015-01-09 03:49:37*

**Authors:** Shrikrishna Jayraj Kalgaonkar

**Comments:** 33 Pages.

Almost everywhere “brotherhood relation” as non-reflexive, non-symmetric but transitive. I couldn’t agree with this. The reasons for this disagreement are explained in the article.
Brotherhood concept is discussed as a binary relation, types of binary relations, equivalence relations and its effect.
Finally suggesting that the brotherhood relation should be considered as a equivalence relation.

**Category:** Set Theory and Logic

[78] **viXra:1412.0269 [pdf]**
*submitted on 2014-12-29 20:01:52*

**Authors:** Jaykov Foukzon

**Comments:** 29 Pages.

In this paper paraconsistent first-order logic
LP^# with infinite hierarchy levels of contradiction is proposed. Corresponding paraconsistent set theory KSth^# is
proposed. Axiomatical system HST^#,as inconsistent generalization of Hrbacek set
theory HST is considered.

**Category:** Set Theory and Logic

[77] **viXra:1412.0235 [pdf]**
*submitted on 2014-12-25 05:46:46*

**Authors:** Thomas Colignatus

**Comments:** Short version of the argument (6 pages)

Paul of Venice (1369-1429) provides a consistency enhancer that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem on the power set. It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a "well-defined set". The transfinites of ZFC may be a mirage of still imperfect axiomatics w.r.t. the proper foundations for set theory.

**Category:** Set Theory and Logic

[76] **viXra:1412.0234 [pdf]**
*submitted on 2014-12-25 05:54:03*

**Authors:** Thomas Colignatus

**Comments:** 36 Pages. March 2012, written in Mathematica

> Context \[Bullet] In the philosophy of mathematics there is the distinction between platonism (realism), formalism, and constructivism. There seems to be no distinguishing experiment however to determine which approach is best according to which criteria. Philosophy finds a sounding board in the didactics of mathematics, as the relevant empirical context, instead of mathematics itself. Mathematicians are trained for abstract thought but in class they meet with real world students. Traditional mathematicians resolve their cognitive dissonance by relying on tradition. That tradition however is not targetted at didactic clarity and empirical relevance with respect to psychology. The mathematical curriculum is a mess. Philosophers can go astray when they don't realize the distinction between mathematics and the didactics of mathematics. > Problem \[Bullet] Aristotle distinguished between potential and actual infinite, Cantor proposed the transfinites, and Occam would want to reject those transfinites if they aren't really necessary. My book "A Logic of Exceptions" already refuted 'the' general proof of Cantor's Theorem on the power set, so that the latter holds only for finite sets but not for 'any' set. There still remains Cantor's diagonal argument on the real numbers. > Results \[Bullet] There is a 'bijection by abstraction' between \[DoubleStruckCapitalN] and \[DoubleStruckCapitalR]. Potential and actual infinity are two faces of the same coin. Potential infinity associates with counting, actual infinity with the continuum, but they would be 'equally large'. The notion of a limit in \[DoubleStruckCapitalR] cannot be defined independently from the construction of \[DoubleStruckCapitalR] itself. Occam's razor eliminates Cantor's transfinites. > Constructivist content \[Bullet] Constructive steps Subscript[S, 1], ..., Subscript[S, 5] are identified, where Subscript[S, 3] gives potential infinity and Subscript[S, 4] actual infinity. The latter is taken as "proper constructivism" and it contains abstraction. The confusions about Subscript[S, 6], nonconstructivism and the transfinites, derive rather from logic than from infinity.

**Category:** Set Theory and Logic

[75] **viXra:1412.0233 [pdf]**
*submitted on 2014-12-25 05:58:23*

**Authors:** Thomas Colignatus

**Comments:** 10 Pages. Paper of 2007, written in Mathematica

Adding some reasonable properties to the Gödelian system of Peano Arithmetic creates a new system for which Gödel's completeness theorems collapse and the Gödeliar becomes the Liar paradox again. Rejection of those properties is difficult since they are reasonable. Three-valued logic is a better option to deal with the Liar and its variants.

**Category:** Set Theory and Logic

[74] **viXra:1412.0201 [pdf]**
*submitted on 2014-12-19 02:03:14*

**Authors:** Karan Doshi

**Comments:** 14 Pages.

In this paper the author submits a proof using the Power Set relation for the existence of a transfinite cardinal strictly smaller than Aleph Zero, the cardinality of the Naturals. Further, it can be established taking these arguments to their logical conclusion that even smaller transfinite cardinals exist. In addition, as a lemma using these new found and revolutionary concepts, the author conjectures that some outstanding unresolved problems in number theory can be brought to heel. Specifically, a proof of the twin prime conjecture is given.

**Category:** Set Theory and Logic

[73] **viXra:1412.0155 [pdf]**
*submitted on 2014-12-09 20:21:48*

**Authors:** Florentin Smarandache

**Comments:** 500 Pages.

Neutrosophic Theory means Neutrosophy applied in many fields in order to solve problems related to indeterminacy.
Neutrosophy considers every entity <A> together with its opposite or negation <antiA>, and with their spectrum of neutralities <neutA> in between them (i.e. entities supporting neither nor <antiA>). Where

**Category:**

[72] **viXra:1412.0130 [pdf]**
*submitted on 2014-12-06 15:09:42*

**Authors:** Jaykov Foukzon

**Comments:** 17 Pages.

In recent years there has been a revitalised interest in non-classical solutions to the
semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei's paradox without rejection any contraction postulate is proposed.

**Category:** Set Theory and Logic

[71] **viXra:1412.0006 [pdf]**
*submitted on 2014-12-01 09:42:44*

**Authors:** Vasile Patrascu

**Comments:** 10 Pages.

The paper presents some steps for multi-valued representation of neutrosophic information. These steps are provided in the framework of multi-valued logics using the following logical value: true, false, neutral, unknown and saturated. Also, this approach provides some calculus formulae for the following neutrosophic features: truth, falsity, neutrality, ignorance, under-definedness, over-definedness, saturation and entropy. In addition, it was defined net truth, definedness and neutrosophic score.

**Category:** Set Theory and Logic

[70] **viXra:1411.0529 [pdf]**
*submitted on 2014-11-21 07:00:54*

**Authors:** W. B. Vasantha Kandasamy, Florentin Smarandache, Ilanthenral K.

**Comments:** 242 Pages.

In this book the authors for the first time have ventured to study, analyse and investigate fuzzy and neutrosophic models and the experts opinion. To make such a study, innovative techniques and defined and developed. Several important conclusions about these models are derived using these new techniques. Open problems are suggested in this book.

**Category:** Set Theory and Logic

[69] **viXra:1411.0528 [pdf]**
*submitted on 2014-11-21 07:02:15*

**Authors:** Vasantha Kandasamy, Florentin Smarandache, Ilanthenral K.

**Comments:** 275 Pages.

In this book the authors for the first time have merged vertices and edges of lattices to get a new structure which may or may not be a lattice but is always a graph. This merging is done for graph too which will be used in the merging of fuzzy models. Further merging of graphs leads to the merging of matrices; both these concepts play a vital role in merging the fuzzy and neutrosophic models.
Several open conjectures are suggested.

**Category:** Set Theory and Logic

[68] **viXra:1411.0051 [pdf]**
*submitted on 2014-11-07 05:57:21*

**Authors:** Ricardo Alvira

**Comments:** 5 Pages.

It reviewes the difference between cocnepts involving Certainty/Uncertainty.

**Category:** Set Theory and Logic

[67] **viXra:1411.0009 [pdf]**
*submitted on 2014-11-01 23:51:59*

**Authors:** Karan Doshi

**Comments:** 8 Pages.

Well-ordering of the Reals@@ presents a major challenge in Set theory. Under the standard Zermelo Fraenkel Set theory (ZF) with the Axiom of Choice (ZFC), a well-ordering of the Reals is indeed possible. However the Axiom of Choice (AC) had to be introduced to the original ZF theory which is then shown equivalent to the well-ordering theorem. Despite the result however, no way has still been found of actually constructing a well-ordered Set of Reals. In this paper the author attempts to generate a well ordered Set of Reals without using the AC i.e. under ZF theory itself using the Axiom of the Power Set as the guiding principle.

**Category:** Set Theory and Logic

[66] **viXra:1410.0131 [pdf]**
*submitted on 2014-10-22 17:37:47*

**Authors:** A. A. Salama, Said Broumi

**Comments:** 5 Pages.

Since the world is full of indeterminacy, the neutrosophics found their place into
contemporary research. In this paper we define rough neutrosophic sets and study their
properties. Some propositions in this notion are proved. Possible application to computer
sciences is touched upon.

**Category:** Set Theory and Logic

[65] **viXra:1410.0022 [pdf]**
*submitted on 2014-10-04 20:57:05*

**Authors:** Yilun Shang

**Comments:** 6 Pages.

Let $U^{(n)}$ denote the maximal length arithmetic
progression in a non-uniform random subset of $\{0,1\}^n$, where $1$
appears with probability $p_n$. By using dependency graph and
Stein-Chen method, we show that $U^{(n)}-c_n\ln n$ converges in law
to an extreme type distribution with $\ln p_n=-2/c_n$. Similar
result holds for $W^{(n)}$, the maximal length aperiodic arithmetic
progression (mod $n$).

**Category:** Set Theory and Logic

[64] **viXra:1409.0174 [pdf]**
*submitted on 2014-09-25 14:54:50*

**Authors:** Misha Mikhaylov

**Comments:** 7 Pages.

Thoughts expressed in previous paper [3] were developed. There was shown formally that collection of sets’ properties may not appear chaotically and independently on each other. Presence or absence of one leads to rise or drop of another.

**Category:** Set Theory and Logic

[63] **viXra:1409.0056 [pdf]**
*submitted on 2014-09-08 16:49:27*

**Authors:** Misha Mikhaylov

**Comments:** 11 Pages.

There was an attempt to formalize an appearance of main relation properties in contrast to the usual one. There were paid an attention only for natural relations. Surely, such relations as “better than” are not observed here since the notion of “good” itself is still not defined.

**Category:** Set Theory and Logic

[62] **viXra:1409.0041 [pdf]**
*submitted on 2014-09-06 09:58:42*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we redefine the infinitesimals by using axiomatic method.

**Category:** Set Theory and Logic

[61] **viXra:1409.0006 [pdf]**
*submitted on 2014-09-02 01:36:05*

**Authors:** Felix M. Lev

**Comments:** 5 Pages.

Standard mathematics involves such notions as infinitely small/large, continuity and standard division. However, some of these notions are treated differently in traditional and constructive versions. This mathematics is usually treated as fundamental while finite mathematics is treated as inferior. Standard mathematics has foundational problems (as follows, for example, from G\"{o}del's incompleteness theorems) but people usually believe that this is less important than the fact that it describes many experimental data with high accuracy. We argue that the situation is the opposite: standard mathematics is only a special case of finite one in the
formal limit when the characteristic of the ring or field used in finite mathematics goes to infinity. Therefore foundational problems
in standard mathematics are not fundamental.

**Category:** Set Theory and Logic

[60] **viXra:1408.0211 [pdf]**
*submitted on 2014-08-29 10:11:03*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

Our result is the explicit form of the infinitesimals.

**Category:** Set Theory and Logic

[59] **viXra:1408.0039 [pdf]**
*submitted on 2014-08-07 22:42:05*

**Authors:** Oh Jung Uk

**Comments:** 8 Pages.

If B(P) is the truth value of proposition P then connectives could be translated to arithmetic operation in the congruent expression of 2 as described below.
B(~p)≡1+B(p)( mod 2 ),B(p∧q)≡B(p)B(q)( mod 2 )
B(p∨q)≡B(p)+B(q)+B(p)B(q)( mod 2 )
B(p⟶q)≡1+B(p)+B(p)B(q)( mod 2 )
B(p⟷q)≡1+B(p)+B(q)(mod 2)
By using this, logical laws could be proved, compound proposition could be simplified, and logical equation could be solved.

**Category:** Set Theory and Logic

[58] **viXra:1407.0033 [pdf]**
*submitted on 2014-07-04 06:53:06*

**Authors:** Subhajit Ganguly

**Comments:** 19 Pages.

Our aim is to help build a machine that can reduce the possibility of mishaps in navigation to zero. For that devise a new system of numbers, in which the real numbers are represented on the y-axis and complex numbers on the x-axis. Inside such a system, we incorporate the equivalent Ideal Fuzzy Logic that can be used by the machine to predict and avoid mishaps.

**Category:** Set Theory and Logic

[57] **viXra:1406.0115 [pdf]**
*submitted on 2014-06-18 12:02:18*

**Authors:** Mustafa A. Khan

**Comments:** 9 Pages.

This paper introduces a new type of set called the Sohraab-Hyder set or the SH set which is quite different from the well known and studied set of the Set Theory. This new set theory is logically consistent and leads to some fascinating results such as, (1) the necessary existence of an ultimate SH set that created itself in pre-eternity and will continue to exist to post-eternity, (2) the ultimate SH set is self sufficient, (2) the ultimate SH set must give rise to infinite number of other SH sets which represent logically consistent algebras and geometries, (3) the ultimate SH set must give rise to the existence of infinite universes, including our own, (4) the ultimate SH set must give rise to mathematical cells, organs and organisms which undergo must undergo evolution, (5) some of these mathematical cells, organs and organisms are same as the organic cells, organs and organisms which leads to the conclusion that the evolution of organisms is a mathematical fact and finally, (6) it is shown that every living and non-living entity in the universe, with the universe included of course, is a matrix of 0's and 1's.

**Category:** Set Theory and Logic

[56] **viXra:1405.0360 [pdf]**
*submitted on 2014-05-30 12:59:54*

**Authors:** Ricardo Alvira

**Comments:** 42 Pages.

Presentation based on book expected to be published in 1-2 months: "A Unified Theory of Complexity". ISBN: 978-1499335859. 250 pages aprox.

**Category:** Set Theory and Logic

[55] **viXra:1405.0227 [pdf]**
*submitted on 2014-05-14 03:52:18*

**Authors:** Leszek W. Guła

**Comments:** 2 Pages. no

**Category:** Set Theory and Logic

[54] **viXra:1405.0225 [pdf]**
*submitted on 2014-05-13 07:06:04*

**Authors:** Jailton C. Ferreira

**Comments:** 3 Pages.

A counterexample to Cantor's Diagonal Argument is revisited.

**Category:** Set Theory and Logic

[53] **viXra:1405.0202 [pdf]**
*submitted on 2014-05-10 01:05:38*

**Authors:** Bertrand Wong

**Comments:** 3 Pages.

This article raises some important points about logic, e.g., mathematical logic.

**Category:** Set Theory and Logic

[52] **viXra:1401.0149 [pdf]**
*submitted on 2014-01-22 05:43:50*

**Authors:** Antonio Leon

**Comments:** 14 Pages.

It is proved in this paper the undecidable formula involved in Gödel's first incompleteness theorem would be inconsistent if the formal system where it is defined were complete. So, before proving the formula is undecidable it is necessary to assume the system is not complete in order to ensure the formula is not inconsistent. Consequently, Gödel proof does not prove the formal system is incomplete but that, once assumed it is incomplete, it is possible to define an undecidable formula within the system. This conclusion makes Gödel's incompleteness theorems devoid of substance.

**Category:** Set Theory and Logic

[51] **viXra:1310.0221 [pdf]**
*submitted on 2013-10-24 16:37:16*

**Authors:** A. A. Salama, Mohamed eisa, Florentin Smarandache

**Comments:** 1 Page.

The purpose of this paper is to introduce and study the characteristic
function of a neutrosophic set. After given the fundamental definitions of
neutrosophic set operations generated by, we obtain several properties, and discussed
the relationship between neutrosophic sets generated by Ng and others. Finally, we
introduce the neutrosophic topological spaces generated by Ng . Possible application
to GIS topology rules are touched upon.

**Category:** Set Theory and Logic

[50] **viXra:1309.0028 [pdf]**
*submitted on 2013-09-05 20:36:23*

**Authors:** Florentin Smarandache

**Comments:** 9 Pages.

In this paper we present a short history of logics: from particular cases of 2-symbol or numerical
valued logic to the general case of n-symbol or numerical valued logic. We show generalizations
of 2-valued Boolean logic to fuzzy logic, also from the Kleene’s and Lukasiewicz’ 3-symbol
valued logics or Belnap’s 4-symbol valued logic to the most general n-symbol or numerical
valued refined neutrosophic logic. Two classes of neutrosophic norm (n-norm) and neutrosophic
conorm (n-conorm) are defined. Examples of applications of neutrosophic logic to physics are
listed in the last section.
Similar generalizations can be done for n-Valued Refined Neutrosophic Set, and respectively n-
Valued Refined Neutrosopjhic Probability.

**Category:** Set Theory and Logic

[49] **viXra:1309.0013 [pdf]**
*submitted on 2013-09-04 07:32:54*

**Authors:** Robert A. Herrmann

**Comments:** 19 Pages.

Relative to universal logic, it is demonstrated how useful it is to utilize general logic-systems to investigate finite consequence operators (operations). Among many other examples relative to the lattice of finite consequence operators, a general characterization for the lattice-theoretic supremum for a nonempty collection of finite consequence operators is given. Further, it is shown that for any denumerable language there is rather simple collection of finite consequence operators and for a propositional language, three simple modifications to the finitary rules of inference that demonstrate that the lattice of finite consequence operators is not meet-complete.

**Category:** Set Theory and Logic

[48] **viXra:1307.0124 [pdf]**
*submitted on 2013-07-23 18:54:31*

**Authors:** editor Linfan Mao

**Comments:** 135 Pages.

The First International Conference on Smarandache Multispace and Multistructure was organized by Prof. Linfan Mao, and it was held in the Beijing University of Civil Engineering and Architecture of P. R. China on June 28-30, 2013. There were 46 researchers from China, India, Iran, Nigeria, and USA that have taken part in this conference with 14 papers on Smarandache multispace and geometry, birings, neutrosophy,neutrosophic groups, regular maps and topological graphs with applications to non-solvable equation systems.
Definition.
In any domain of knowledge, a Smarandache multispace (or S-multispace) with its multistructure is a finite or infinite (countable or uncountable) union of many spaces that have various
structures. The spaces may overlap. The notions of multispace (also spelt multi-space) and multi-
structure (also spelt multi-structure) were introduced by Smarandache in 1969 under his idea of hybrid science: combining different fields into a unifying field, which is closer to our real life world since we live in a heterogeneous space. Today, this idea is widely accepted by the world of sciences. S-multispace is a qualitative notion, since it is too large and includes both metric and non-metric spaces. It is believed that the smarandache multispace with its multistructure is the best candidate for 21st century Theory of Everything in any domain. It unifies many knowledge fields.
Applications.
A such multispace can be used for example in physics for the Unified Field Theory that tries to unite the gravitational, electromagnetic, weak and strong interactions. Or in the parallel quantum computing and in the mu-bit theory, in multi-entangled states or particles and up to multi-entangles objects. We also mention: the algebraic multispaces (multi-groups, multi-rings, multi-vector spaces, multi-operation systems and multi-manifolds, also multi-voltage graphs, multi-embedding of a graph in an n-manifold, etc.), geometric multispaces (combinations of Euclidean and Non-Euclidean geometries into one space as in Smarandache geometries), theoretical physics, including the relativity theory, the M-theory and the cosmology, then multi-space models for p-branes and cosmology, etc.
- The multispace and multistructure were first used in the Smarandache geometries (1969), which are combinations of different geometric spaces such that at least one geometric axiom behaves differently in each such space.
- In paradoxism (1980), which is a vanguard in literature, arts, and science, based on finding
common things to opposite ideas [i.e. combination of contradictory fields].
- In neutrosophy (1995), which is a generalization of dialectics in philosophy, and takes into con-
sideration not only an entity < A > and its opposite < antiA > as dialectics does, but also the
neutralities < neutA > in between. Neutrosophy combines all these three < A >,< antiA > and
< neutA > together. Neutrosophy is a metaphilosophy.
- Then in neutrosophic logic (1995), neutrosophic set (1995), and neutrosophic probability (1995), which have, behind the classical values of truth and falsehood, a third component called indeterminacy (or neutrality, which is neither true nor false, or is both true and false simultaneously - again a combination of opposites: true and false in indeterminacy).
- Also used in Smarandache algebraic structures (1998), where some algebraic structures are in-
cluded in other algebraic structures.

**Category:** Set Theory and Logic

[47] **viXra:1306.0209 [pdf]**
*submitted on 2013-06-25 11:06:36*

**Authors:** Andrew Nassif

**Comments:** 15 Pages. May need formatting, but not sure because they are notes.

A 15 page collection of my notes on mathematical articles I have created.

**Category:** Set Theory and Logic

[111] **viXra:1705.0173 [pdf]**
*replaced on 2017-05-12 13:18:20*

**Authors:** Ilija Barukčić

**Comments:** 24 pages. Copyright © 2017 by Ilija Barukčić, Jever, Germany. Published by:

The division of zero by zero turns out to be a long lasting and not ending puzzle in mathematics and physics. An end of this long discussion is not in sight. In particular zero divided by zero is treated as indeterminate thus that a result cannot be found out. It is the purpose of this publication to solve the problem of the division of zero by zero while relying on the general validity of classical logic. According to classical logic, zero divided by zero is one.

**Category:** Set Theory and Logic

[110] **viXra:1704.0363 [pdf]**
*replaced on 2017-05-12 02:48:50*

**Authors:** Haroun Boutamani

**Comments:** 7 Pages.

This paper addresses the concept of selectivity using a main set and an auxiliary set, the auxiliary serves the purpose of deciding which elements do or do not fit together within the output sets chosen based on the elements of the main set.
This paper also presents an approach of comparing sets based on a defined relation, the sets’ properties, and the elementary properties being total compatibility and total incompatibility.
The problem classes P and NP are two sets considered under the criteria of distinguishability, then the possibility of comparison of the two sets is discussed. The proposed comparison presents a distinction between P and NP.

**Category:** Set Theory and Logic

[109] **viXra:1704.0363 [pdf]**
*replaced on 2017-05-05 07:29:39*

**Authors:** Haroun Boutamani

**Comments:** 7 Pages.

I address the concept of selectivity using a main set and an auxiliary set, the auxiliary
serves the purpose of deciding which elements do or do not fit together within the output
sets chosen based on the elements of the main set.
This paper also presents an approach of comparing sets based on a defined relation and
the sets’ properties, and also the elementary properties being total compatibility and total
incompatibility.
The problem classes P and NP are two sets considered under the criteria of
distinguishability, then the possibility of comparison of the two sets is discussed. The
proposed comparison presents a distinction between P and NP.

**Category:** Set Theory and Logic

[108] **viXra:1704.0363 [pdf]**
*replaced on 2017-05-04 09:43:56*

**Authors:** Haroun Boutamani

**Comments:** 7 Pages.

I address the concept of selectivity using a main set and an auxiliary set, the auxiliary
serves the purpose of deciding which elements do or do not fit together within the output
sets chosen based on the elements of the main set.
This paper also presents an approach of comparing sets based on a defined relation and
the sets’ properties, and also the elementary properties being total compatibility and total
incompatibility.
The problem classes P and NP are two sets considered under the criteria of
distinguishability, then the possibility of comparison of the two sets is discussed. The
proposed comparison presents a distinction between P and NP.

**Category:** Set Theory and Logic

[107] **viXra:1704.0363 [pdf]**
*replaced on 2017-05-02 09:51:47*

**Authors:** Haroun Boutamani

**Comments:** 7 Pages.

**Category:** Set Theory and Logic

[106] **viXra:1704.0363 [pdf]**
*replaced on 2017-04-30 07:25:19*

**Authors:** Haroun Boutamani

**Comments:** 7 Pages.

**Category:** Set Theory and Logic

[105] **viXra:1704.0363 [pdf]**
*replaced on 2017-04-28 12:39:10*

**Authors:** Haroun Boutamani

**Comments:** 7 Pages.

**Category:** Set Theory and Logic

[104] **viXra:1704.0363 [pdf]**
*replaced on 2017-04-28 05:12:42*

**Authors:** Haroun Boutamani

**Comments:** 7 Pages.

**Category:** Set Theory and Logic

[103] **viXra:1704.0115 [pdf]**
*replaced on 2017-05-16 21:36:12*

**Authors:** Ken Seton

**Comments:** 9 Pages. Enjoy

Many have suggested that the infinite set has a fundamental problem. The usual complaint rails against the actually infinite which (say critics of various finitist persuasions) unjustifiably goes beyond the finite. Here we identify the exact opposite. The problem of the infinite set defined to have an identity (content) that is specified and restricted to be forever finite .
Set theory is taken at its word. The existence of the infinite set and the representation of irrational reals as infinite sets of terms is accepted. In this context, it is shown that the standard definition of the infinite countable set is inconsistent with the existence of its own classic convergents of construction. If the set is infinite then it must be quite unlike that which set theory asserts it to be.
Set theory found itself in some trouble over a century ago trusting an unrestricted anthropic comprehension. But serious doubt is cast on the validity of infinite sets which have been defined by a comprehension which overly-restricts their content.

**Category:** Set Theory and Logic

[102] **viXra:1704.0115 [pdf]**
*replaced on 2017-05-11 01:29:26*

**Authors:** Ken Seton

**Comments:** 9 Pages. Enjoy

Many have suggested that the infinite set has a fundamental problem. The usual complaint rails against the actually infinite which (say critics of various finitist persuasions) unjustifiably goes beyond the finite. Here we identify the exact opposite. The problem of the infinite set defined to have an identity (content) that is specified and restricted to be forever finite .
Set theory is taken at its word. The existence of the infinite set and the representation of irrational reals as infinite sets of terms is accepted. In this context, it is shown that the standard definition of the infinite countable set is inconsistent with the existence of its own classic convergents of construction. If the set is infinite then it must be quite unlike that which set theory asserts it to be.
Set theory found itself in some trouble over a century ago trusting an unrestricted anthropic comprehension. But serious doubt is cast on the validity of infinite sets which have been defined by a comprehension which overly-restricts their content.

**Category:** Set Theory and Logic

[101] **viXra:1703.0032 [pdf]**
*replaced on 2017-03-06 04:56:10*

**Authors:** Wolfgang Mückenheim

**Comments:** 5 Pages.

Contrary to the assumptions of transfinite set theory, limit and union of infinite sequences of sets differ.

**Category:** Set Theory and Logic

[100] **viXra:1702.0293 [pdf]**
*replaced on 2017-02-26 04:27:32*

**Authors:** W. Mückenheim

**Comments:** 2 Pages.

It is shown that the enumeration of rational numbers cannot be complete.

**Category:** Set Theory and Logic

[99] **viXra:1702.0293 [pdf]**
*replaced on 2017-02-24 05:08:54*

**Authors:** W. Mückenheim

**Comments:** 2 Pages.

It is shown that the enumeration of rational numbers cannot be complete.

**Category:** Set Theory and Logic

[98] **viXra:1701.0563 [pdf]**
*replaced on 2017-01-31 11:38:56*

**Authors:** امیر دلجو

**Comments:** 12 Pages.

این سند یک منشور است برای بیان ماهیت ریاضیات و نظریه ی اعداد یا حساب. در اینجا من الگویی را تعریف کرده ام که نشان می دهد، خودآگاهی یک وجود واحده بسیط و همه جاحاضر بوده و هستی به مثابه یک کلیّت، یک تجلّی گرافیکی ست که در جهت افشای این خودآگاهی عارض شده و من هرمس وار، شهود خود بر یگانگی و آفرینش را برای انتقال به انسان های کنجکاو ساده سازی و تحریر کرده ام. از آنجا که تبیین روش شناختی این منشور مستلزم هزاران صفحه است، در اینجا تنها عبارات و معادلات منتج شده ی نهایی اعلان می شود.

**Category:** Set Theory and Logic

[97] **viXra:1608.0395 [pdf]**
*replaced on 2016-11-03 03:06:10*

**Authors:** Max Null, Sergey Belov

**Comments:** 23 Pages.

We define the topology atop(χ) on a complete upper semilattice χ = (M, ≤).
The limit points are determined by the formula
lim (X) = sup{a ∈ M | {x ∈ X| a ≤ x} ∈ D},
D
where X ⊆ M is an arbitrary set, D is an arbitrary non-principal ultrafilter
on X. We investigate lim (X) and topology atop(χ) properties. In particular,
D
we prove the compactness of the topology atop(χ).

**Category:** Set Theory and Logic

[96] **viXra:1608.0395 [pdf]**
*replaced on 2016-09-14 08:54:47*

**Authors:** Max Null, Sergey Belov

**Comments:** 17 Pages.

We define the topology atop(χ) on a complete upper semilattice χ = (M, ≤). The limit points are determined by the formula Lim(D,X) = sup{a ∈ M| {x ∈ X| a ≤ x} ∈ D}, where X ⊆ M is an arbitrary set, D is an arbitrary non-principal ultrafilter on X. We investigate Lim(D,X) and topology atop(χ) properties. In particular, we prove the compactness of the topology atop(χ).

**Category:** Set Theory and Logic

[95] **viXra:1608.0358 [pdf]**
*replaced on 2017-06-04 02:59:26*

**Authors:** Vatolin Dm.

**Comments:** 7 Pages. Russian

Here are definitions of «completeness» and «incompleteness» for mathematical theories. These definitions are different from those that gave Godel. Сontradictions of the Godel's arguments have been eliminated. Found are theo-rems that put everything in its place.

**Category:** Set Theory and Logic

[94] **viXra:1608.0358 [pdf]**
*replaced on 2017-03-23 21:58:40*

**Authors:** Vatolin Dm.

**Comments:** 7 Pages. Rassian

Here are definitions of «completeness» and «incompleteness» for mathematical theories. These definitions are different from those that gave Godel. Сontradictions of the Godel's arguments have been eliminated. Found are theo-rems that put everything in its place.

**Category:** Set Theory and Logic

[93] **viXra:1608.0057 [pdf]**
*replaced on 2016-09-07 19:32:34*

**Authors:** Adrian Chira

**Comments:** 7 Pages.

Curry's paradox is generally considered to be one of the hardest paradoxes to solve. It is shown here that the paradox can be arrived in fewer steps and also for a different term of the original biconditional. Further, using different approaches, it is also shown that the conclusion of the paradox must always be false and this is not paradoxical but it is expected to be so. One of the approaches points out that the starting biconditional of the paradox amounts to a false definition or assertion which consequently leads to a false conclusion. Therefore, the solution is trivial and the paradox turns out to be no paradox at all. Despite that fact that verifying the truth value of the first biconditional of the paradox is trivial, mathematicians and logicians have failed to do so and merely assumed that it is true. Taking this into consideration that it is false, the paradox is however dismissed. This conclusion puts to rest an important paradox that preoccupies logicians and points out the importance of verifying one's assumptions.

**Category:** Set Theory and Logic

[92] **viXra:1607.0421 [pdf]**
*replaced on 2016-12-07 08:10:32*

**Authors:** Robert A. Herrmann

**Comments:** 8 Pages.

The basic mathematical aspects of the GGU and GID models are discussed. As an illustration, the modified Robinson approach is used to give a more direct prediction as to the composition of ultra-propertons. Relative to logic-systems, the refined developmental paradigm is applied to the General Intelligent Design (GID) model and the basic GID statements are given.

**Category:** Set Theory and Logic

[91] **viXra:1607.0421 [pdf]**
*replaced on 2016-08-11 09:56:27*

**Authors:** Robert A. Herrmann

**Comments:** 9 Pages.

The basic mathematical aspects of the GGU and GID models are discussed. As an illustration, the modified Robinson approach is used to give a more direct prediction as to the composition of ultra-propertons. Relative to logic-systems, the refined developmental paradigm is applied to the General Intelligence Design (GID) model and basic GID statements are given.

**Category:** Set Theory and Logic

[90] **viXra:1606.0159 [pdf]**
*replaced on 2016-12-01 08:07:04*

**Authors:** Robert A. Herrmann

**Comments:** 48 Pages.

This part contains Chapters 5, 6, 7, 8, 9, which include Consequence Operators (Operations), Perception, An Alternate Approach, Developmental Paradigms, Ultrawords, A Neutron Altering Process, The Extended Structure and General Paradigms.

**Category:** Set Theory and Logic

[89] **viXra:1606.0005 [pdf]**
*replaced on 2016-12-16 09:20:24*

**Authors:** Robert A. Herrmann

**Comments:** 8 Pages.

The major purpose for this article is to reestablish Theorem 9.3.1, for the EGS, with the modified Robinson approach and make other improvements in Section 9 of The Theory of Ultralogics (Herrmann, (1978-93, 1999)). Additionally, what constitutes a saturated enlargement is now fixed as of this date.

**Category:** Set Theory and Logic

[88] **viXra:1606.0005 [pdf]**
*replaced on 2016-11-18 08:38:56*

**Authors:** Robert A. Herrmann

**Comments:** 7 Pages.

The major purpose for this article is to reestablish Theorem 9.3.1, for the EGS, with the modified Robinson approach and make other improvements in Section 9 of The Theory of Ultralogics (Herrmann, (1978-93)). Additionally, in this version, certain notational conventions are discussed.

**Category:** Set Theory and Logic

[87] **viXra:1606.0005 [pdf]**
*replaced on 2016-08-09 09:58:46*

**Authors:** Robert A. Herrmann

**Comments:** 7 Pages.

The major purpose for this article is to reestablish Theorem 9.3.1 for the EGS, with the modified Robinson approach, and make other improvements in Section 9 of The Theory of Ultralogics. Further, an important improvement is made in the (2013) article on Nonstandard Ultra-logic-systems.

**Category:** Set Theory and Logic

[86] **viXra:1606.0005 [pdf]**
*replaced on 2016-06-02 09:45:38*

**Authors:** Robert A. Herrmann

**Comments:** 7 Pages.

The major purpose for this article is to reestablish Theorem 9.3.1 for the EGS, with the modified Robinson approach, and make other improvements in Section 9 of The Theory of Ultralogics.

**Category:** Set Theory and Logic

[85] **viXra:1604.0104 [pdf]**
*replaced on 2016-04-10 18:42:42*

**Authors:** Allen D Allen

**Comments:** Abstract contains 200 words, ms runs 6 pages

By proving that his “last theorem” (FLT) is true for the integral exponent n = 3, Fermat took the first step in a standard method of proving there exists no greatest lower bound on n for which FLT is true, thus proving the theorem. Unfortunately, there are two reasons why the standard method of proof is not available for FLT. First, transitive inequality lies at the heart of that method. Secondly, FLT admits to a change from > to < rendering their transitive natures unavailable. A related, self evident symmetry illustrates another problem that would have plagued Fermat and centuries of successors. FLT asserts such a narrow proposition, it is difficult to find an antecedent while easy to find a non equivalent consequence. For example, if FLT asserted that the exponent n is even, then FLT would be equivalent to the proposition that Fermat’s equation has two solutions, one for positive bases and one for their negative counterparts. This could be addressed with conservative transformations. The example provided by FLT motivates the use of an early paper by the author to prove a theorem on theorems. The theorem on theorems demonstrates there are infinitely many theorems as difficult to prove as FLT.

**Category:** Set Theory and Logic

[84] **viXra:1603.0226 [pdf]**
*replaced on 2016-03-17 02:40:06*

**Authors:** Vasile Pătraşcu

**Comments:** 12 Pages.

Starting from the primary representation of neutrosophic information, namely the degree of truth, degree of indeterminacy and degree of falsity, we define a nuanced representation in a penta valued fuzzy space, described by the index of truth, index of falsity, index of ignorance, index of contradiction and index of hesitance. Also, it was constructed an associated penta valued logic and then using this logic, it was defined for the proposed penta valued structure the following operators: union, intersection, negation, complement and dual. Then, the penta valued representation is extended to a hexa valued one, adding the sixth component, namely the index of ambiguity.

**Category:** Set Theory and Logic

[83] **viXra:1603.0226 [pdf]**
*replaced on 2016-03-17 02:40:06*

**Authors:** Vasile Pătraşcu

**Comments:** 12 Pages.

Starting from the primary representation of neutrosophic information, namely the degree of truth, degree of indeterminacy and degree of falsity, we define a nuanced representation in a penta valued fuzzy space, described by the index of truth, index of falsity, index of ignorance, index of contradiction and index of hesitance. Also, it was constructed an associated penta valued logic and then using this logic, it was defined for the proposed penta valued structure the following operators: union, intersection, negation, complement and dual. Then, the penta valued representation is extended to a hexa valued one, adding the sixth component, namely the index of ambiguity.

**Category:** Set Theory and Logic

[82] **viXra:1602.0006 [pdf]**
*replaced on 2016-08-29 12:45:44*

**Authors:** Andrew Banks

**Comments:** 20 Pages.

Cantor’s infinity (CI) depends on the capability of completing an infinite collection of successive steps (ICSS). Otherwise, CI does not solve Zeno’s dichotomy paradox and all natural numbers cannot be upward constructed using the successor function. In spite of the fact that no one has ever scientifically witnessed an ICSS nor has anyone ever furnished a direct proof demonstrating its theoretical existence, the assumed capability of completing an ICSS rests at the foundations of theoretical mathematics. As such, this article will introduce a very general type of accelerated Turing machine that can execute instructions in any given time including zero seconds. This device will demonstrate that the assumption that an ICSS can be completed contradicts the assumption that the natural numbers are unbounded (NNU). Hence, CI is not consistent since it contains both. Furthermore, using this computing device and other machinery, it will be proven that space and time are only finitely divisible.

**Category:** Set Theory and Logic

[81] **viXra:1508.0089 [pdf]**
*replaced on 2016-02-13 08:42:50*

**Authors:** Peiman Ghasemi

**Comments:** 7 Pages.

Until the current moment, mankind is not realized that there is a diverse population of intelligent civilizations living in our universe. In the current article we will deduce the occurrence/existence of extraterrestrial life by mathematical proof. I would show you that even inside our galaxy, the Milky Way, a sufficient number of alien creatures are living. The first section includes an algebraic probabilistic proof when the event of life is not highly biased and the second section includes a proof by contradiction that describes the event fundamentally. It's a mathematical proof for the extraterrestrial life debate, for the first time in mankind's history.

**Category:** Set Theory and Logic

[80] **viXra:1508.0089 [pdf]**
*replaced on 2016-02-13 03:43:30*

**Authors:** Peiman Ghasemi

**Comments:** 7 Pages.

Until the current moment, mankind is not realized that there is a diverse population of intelligent civilizations living in our universe. In the current article we will deduce the occurrence/existence of extraterrestrial life by mathematical proof. I would show you that even inside our galaxy, the Milky Way, a sufficient number of alien creatures are living. The first section includes an algebraic probabilistic proof when the event of life is not highly biased and the second section includes a proof by contradiction that describes the event fundamentally. It's a mathematical proof for the extraterrestrial life debate, for the first time in mankind's history.

**Category:** Set Theory and Logic

[79] **viXra:1508.0089 [pdf]**
*replaced on 2015-10-13 11:16:10*

**Authors:** Peiman Ghasemi

**Comments:** 6 Pages.

Until the current moment, mankind is not realized that there is a diverse population of intelligent civilizations living in our universe. In the current article we will deduce the occurrence/existence of extraterrestrial life by mathematical proof. I would show you that even inside our galaxy, the Milky Way, a sufficient number of alien creatures are living. It's a mathematical proof for the extraterrestrial life debate, for the first time in mankind's history.

**Category:** Set Theory and Logic

[78] **viXra:1506.0165 [pdf]**
*replaced on 2015-09-10 05:47:49*

**Authors:** Takahiro Kato

**Comments:** 16 Pages.

This article provides yet another characterization of Boolean algebras and, using this characterization, establishes a more direct connection between propositional logic and Boolean algebras.

**Category:** Set Theory and Logic

[77] **viXra:1506.0147 [pdf]**
*replaced on 2015-07-26 23:12:31*

**Authors:** Thomas Colignatus

**Comments:** 13 Pages.

The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) system of axioms for set theory appears to be inconsistent. A step in developing this proof is the observation that ZFC would be deductively incomplete if it were consistent. Both points are proven by means of the singleton. The axioms are still too lax on the notion of a well-defined set.

**Category:** Set Theory and Logic

[76] **viXra:1506.0147 [pdf]**
*replaced on 2015-06-27 03:05:22*

**Authors:** Thomas Colignatus

**Comments:** 11 Pages.

The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) system of axioms for set theory
appears to be inconsistent. A step in developing this proof is the observation
that ZFC would be deductively incomplete if it were consistent. Both points are
proven by means of the singleton. The axioms are still too lax on the notion of
a 'well-defined set'.

**Category:** Set Theory and Logic

[75] **viXra:1506.0147 [pdf]**
*replaced on 2015-06-24 10:54:12*

**Authors:** Thomas Colignatus

**Comments:** 11 Pages.

The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) system of axioms for set theory appears to be inconsistent. A step in developing this proof is the observation that ZFC would be deductively incomplete if it were consistent. Both points are proven by means of the singleton. The axioms are still too lax on the notion of a 'well-defined set'.

**Category:** Set Theory and Logic

[74] **viXra:1506.0145 [pdf]**
*replaced on 2017-02-15 03:38:49*

**Authors:** Jaykov Foukzon

**Comments:** 84 Pages.

In 1980 F. Wattenberg constructed the Dedekind completion∗d of the Robinson
non-archimedean field ∗ and established basic algebraic properties of ∗d [6]. In
1985 H. Gonshor established further fundamental properties of ∗d [7].In [4]
important construction of summation of countable sequence of Wattenberg numbers
was proposed and corresponding basic properties of such summation were
considered. In this paper the important applications of the Dedekind completion∗d in
transcendental number theory were considered. We dealing using set theory
ZFC ∃(-model of ZFC).Given an class of analytic functions of one complex
variable f ∈ z, we investigate the arithmetic nature of the values of fz at
transcendental points en,n ∈ ℕ. Main results are: (i) the both numbers e and e
are irrational, (ii) number ee is transcendental. Nontrivial generalization of the
Lindemann- Weierstrass theorem is obtained.

**Category:** Set Theory and Logic

[73] **viXra:1506.0145 [pdf]**
*replaced on 2016-05-14 14:11:23*

**Authors:** Jaykov Foukzon

**Comments:** 61 Pages. Advances in Pure Mathematics Vol.5 No.10, Pub. Date: August 19, 2015

In this paper the important applications of the Dedekind completion *R_d in transcendental number theory is considered. We dealing using set theory ZFC+~∃(omega-model of ZFC).Given an class of analytic functions of one complex variable f ∈Q[[z]],we investigate the arithmetic nature of the values of f(z) at transcendental points e^n. Main results are: (i) the both numbers e+pi and e-pi are irrational, (ii) number e^e are transcendental. Nontrivial generalization of the Lindemann-Weierstrass theorem is obtained.

**Category:** Set Theory and Logic

[72] **viXra:1505.0122 [pdf]**
*replaced on 2016-03-02 13:16:32*

**Authors:** Misha Mikhaylov

**Comments:** 24 Pages.

It seems that statements determining features of some algebraic structures behavior are based on just intuitive assumptions or empiric observations and for sake of convenience (simplest example is the phrase: “let’s consider 0! =1”… perhaps, just because Sir Isaac Newton entrusted, so, why not choose any: e.g. 2, 5, or 7.65). So, without logical explanation these are looking a little mysterious or sometimes even magic. This article is a humble attempt to get it straight rather formally. Some troubles may appear on the way – e.g. as it was shown earlier (in the ref. [2], for example), there are at least two binary relations having properties of idempotent equivalences – algebra’s elements that may aspire to be an identity. Apparently, probable obtaining of some well-known results in the text is not an attempt of their re-discovering, but it is rather “check-points” that confirm theory validity, more by token that it was made by using of the only exceptionally formal way, while usually they are obtained rather intuitively. Usually the notion of tensor product is determined for each kind of algebraic structure – especially for modulus (in group theory it is often called direct product – but this is a matter of semantics, so, it’s rather negligible). Here it is shown that tensor product may be introduced without defining of concrete algebraic structure. Without such introduction defining of algebraic operation is strongly complicated.

**Category:** Set Theory and Logic

[71] **viXra:1504.0086 [pdf]**
*replaced on 2015-04-12 00:44:59*

**Authors:** Vadim V Nazarenko

**Comments:** 1 Page.

As much as 0 is the opposite of प, every number of the Poorna series has it's opposite number from the Shoonya series.

**Category:** Set Theory and Logic

[70] **viXra:1503.0085 [pdf]**
*replaced on 2017-05-25 01:10:03*

**Authors:** Takahiro Kato

**Comments:** 377 Pages.

Modules (also known as profunctors or distributors) and morphisms among them subsume categories and functors and provide more general and abstract framework to explore the theory of structures, in particular, the notion of universal property. In this book we generalize and redevelop the basic notions and results of various universal constructions in category theory using this framework of modules.

**Category:** Set Theory and Logic

[69] **viXra:1412.0235 [pdf]**
*replaced on 2015-07-28 04:51:49*

**Authors:** Thomas Colignatus

**Comments:** 2 Pages. The paper refers to the book FMNAI that supersedes the paper

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) axioms for set theory appear to be inconsistent. They are still too lax on the notion of a well-defined set. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC for the foundations of set theory. For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).

**Category:** Set Theory and Logic

[68] **viXra:1412.0235 [pdf]**
*replaced on 2015-06-27 03:07:27*

**Authors:** Thomas Colignatus

**Comments:** 30 Pages.

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) axioms for set theory appear to be inconsistent. They are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC for the foundations of set theory. For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory). Theorems 2.5 & 2.7 show for the singleton that ZFC is inconsistent. Lemma 3.2 shows that a Cantorian reading of ZFC implies the possibility of the weaker Pauline reading. Theorem 3.3 gives a constructive proof of the existence of a Pauline set. Appendix D deproves Cantor's Theorem.

**Category:** Set Theory and Logic

[67] **viXra:1412.0235 [pdf]**
*replaced on 2015-06-17 11:59:17*

**Authors:** Thomas Colignatus

**Comments:** 30 Pages.

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). The Zermelo-Fraenkel-Axiom-of-Choice (ZFC) axioms for set theory appear to be inconsistent. They are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC for the foundations of set theory.
For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).
Theorems 2.5 & 2.7 show for the singleton that ZFC is inconsistent. Lemma 3.2 shows that a Cantorian reading of ZFC implies the possibility of the weaker Pauline reading. Theorem 3.3 gives a constructive proof of the existence of a Pauline set. Appendix D deproves Cantor's Theorem.

**Category:** Set Theory and Logic

[66] **viXra:1412.0235 [pdf]**
*replaced on 2015-06-12 16:10:45*

**Authors:** Thomas Colignatus

**Comments:** 40 Pages.

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC for the foundations for set theory.
For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).
Lemma 3.2 shows that a Cantorian reading of ZFC implies the possibility of the weaker Pauline reading. Theorem 3.3 gives the existence of a Pauline set. It has a fundamental constructive proof and a compact non-constructive proof. Theorem 1.1.6 shows that ZFC has an inconsistency.

**Category:** Set Theory and Logic

[65] **viXra:1412.0235 [pdf]**
*replaced on 2015-06-05 02:16:20*

**Authors:** Thomas Colignatus

**Comments:** 38 Pages.

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC for the foundations for set theory.
For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).
Lemma 3.2 shows that a Cantorian reading of ZFC implies the possibility of the weaker Pauline reading. Theorem 3.3 gives the existence of a Pauline set. It has a fundamental constructive proof and a compact non-constructive proof. Theorem 1.1.6 shows that ZFC has an anomaly. Corollary 3.3 turns that anomaly into an inconsistency.

**Category:** Set Theory and Logic

[64] **viXra:1412.0235 [pdf]**
*replaced on 2015-05-20 11:16:57*

**Authors:** Thomas Colignatus

**Comments:** 18 Pages.

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC w.r.t. the proper foundations for set theory. For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).

**Category:** Set Theory and Logic

[63] **viXra:1412.0235 [pdf]**
*replaced on 2015-05-01 04:33:25*

**Authors:** Thomas Colignatus

**Comments:** 13 Pages. Corrects a wrong bracket in Notation on p2 of the April 30 version

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC w.r.t. the proper foundations for set theory. For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).

**Category:** Set Theory and Logic

[62] **viXra:1412.0235 [pdf]**
*replaced on 2015-04-30 10:33:05*

**Authors:** Thomas Colignatus

**Comments:** 13 Pages.

Paul of Venice (1369-1429) provides a consistency condition that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem (in Russell's form, for the power set). It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC w.r.t. the proper foundations for set theory. For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).

**Category:** Set Theory and Logic

[61] **viXra:1412.0235 [pdf]**
*replaced on 2014-12-31 02:45:55*

**Authors:** Thomas Colignatus

**Comments:** 8 Pages.

Paul of Venice (1369-1429) provides a consistency enhancer that resolves Russell's Paradox in naive set theory without using a theory of types. It allows a set of all sets. It also blocks the (diagonal) general proof of Cantor's Theorem on the power set. It is not unlikely that the Zermelo-Fraenkel (ZFC) axioms for set theory are still too lax on the notion of a 'well-defined set'. The transfinites of ZFC may be a mirage, and a consequence of still imperfect axiomatics in ZFC w.r.t. the proper foundations for set theory. For amendment of ZFC two alternatives are mentioned: ZFC-PV (amendment of de Axiom of Separation) or BST (Basic Set Theory).

**Category:** Set Theory and Logic

[60] **viXra:1412.0234 [pdf]**
*replaced on 2015-07-28 05:05:15*

**Authors:** Thomas Colignatus

**Comments:** 2 Pages. The paper refers to the book FMNAI that supersedes the paper

> Context • In the philosophy of mathematics there is the distinction between platonism (realism), formalism, and constructivism. There seems to be no distinguishing or decisive experiment to determine which approach is best according to non-trivial and self-evident criteria. As an alternative approach it is suggested here that philosophy finds a sounding board in the didactics of mathematics rather than mathematics itself. Philosophers can go astray when they don’t realise the distinction between mathematics (possibly pure modeling) and the didactics of mathematics (an empirical science). The approach also requires that the didactics of mathematics is cleansed of its current errors. Mathematicians are trained for abstract thought but in class they meet with real world students. Traditional mathematicians resolve their cognitive dissonance by relying on tradition. That tradition however is not targetted at didactic clarity and empirical relevance with respect to psychology. The mathematical curriculum is a mess. Mathematical education requires a (constructivist) re-engineering. Better mathematical concepts will also be crucial in other areas, such as e.g. brain research. > Problem • Aristotle distinguished between potential and actual infinite, Cantor proposed the transfinites, and Occam would want to reject those transfinites if they aren’t really necessary. My book “A Logic of Exceptions” already refuted ‘the’ general proof of Cantor's Conjecture on the power set, so that the latter holds only for finite sets but not for ‘any’ set. There still remains Cantor’s diagonal argument on the real numbers. > Results • There is a bijection by abstraction between N and R. Potential and actual infinity are two faces of the same coin. Potential infinity associates with counting, actual infinity with the continuum, but they would be ‘equally large’. The notion of a limit in R cannot be defined independently from the construction of R itself. Occam’s razor eliminates Cantor’s transfinites. > Constructivist content • Constructive steps S1, ..., S5 are identified while S6 gives non-constructivism (possibly the transfinites). Here S3 gives potential infinity and S4 actual infinity. The latter is taken as ‘proper constructivism with abstraction'. The confusions about S6 derive rather from logic than from infinity.

**Category:** Set Theory and Logic

[59] **viXra:1412.0201 [pdf]**
*replaced on 2015-01-27 21:14:44*

**Authors:** Karan Doshi

**Comments:** 11 Pages.

In this paper the author submits a proof using the Power Set relation for the existence of a transfinite cardinal strictly smaller than Aleph Zero, the cardinality of the Naturals. Further, it can be established taking these arguments to their logical conclusion that even smaller transfinite cardinals exist. In addition, as a lemma using these new found and revolutionary concepts, the author conjectures that some outstanding unresolved problems in number theory can be brought to heel. Specifically, a proof of the twin prime conjecture is given.

**Category:** Set Theory and Logic

[58] **viXra:1412.0155 [pdf]**
*replaced on 2015-01-30 13:43:13*

**Authors:** Florentin Smarandache

**Comments:** 480 Pages.

Neutrosophic Theory means Neutrosophy applied in many fields in order to solve problems related to indeterminacy.
Neutrosophy considers every entity <A> together with its opposite or negation <antiA>, and with their spectrum of neutralities <neutA> in between them (i.e. entities supporting neither nor <antiA>). Where

**Category:**

[57] **viXra:1412.0130 [pdf]**
*replaced on 2015-02-27 12:54:44*

**Authors:** Jaykov Foukzon

**Comments:** 8 Pages. DOI: 10.11648/j.pamj.s.2015040101.12

In 1942 Haskell B. Curry presented what is now called Curry's paradox which can be found in a logic independently of its stand on negation. In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes. In this article the non-classical resolution of Curry’s Paradox and Shaw-Kwei's paradox without rejection any contraction postulate is proposed.

**Category:** Set Theory and Logic

[56] **viXra:1409.0174 [pdf]**
*replaced on 2016-02-27 04:59:59*

**Authors:** Misha Mikhaylov

**Comments:** 7 Pages.

It was shown here that just transitive relation may be considered as closure and its presence is necessary and sufficient to order a set linearly, and it is not possible to do this by using other relation’s property – neither reflexivity nor symmetry. By interaction, it occurs due to ambiguity of their definition – it was shown earlier (ref. [3]) they have various appearances. Among them just the only transitivity is determined uniquely. At the same time the last one doesn’t exist separately from any others. Circumstances of their joint existence are clarifying in this article.

**Category:** Set Theory and Logic

[55] **viXra:1409.0174 [pdf]**
*replaced on 2016-02-25 11:57:16*

**Authors:** Misha Mikhaylov

**Comments:** 7 Pages.

It was shown here that just transitive relation may be considered as closure and its presence is necessary and sufficient to order a set linearly. It is not impossible to do this by using other relation’s property – neither reflexivity nor symmetry. By interaction, it occurs due to ambiguity of their definition – it was shown earlier (ref. [3]) they have various appearances. Among them just the only transitivity is determined uniquely. At the same time the last one doesn’t exist separately from any others. Circumstances of their joint existence are clarifying in this article.

**Category:** Set Theory and Logic

[54] **viXra:1409.0174 [pdf]**
*replaced on 2014-09-28 13:34:49*

**Authors:** Misha Mikhaylov

**Comments:** 7 Pages.

Thoughts expressed in previous paper [3] were developed. There was shown formally that collection of sets’ properties may not appear chaotically and independently on each other. Presence or absence of one leads to rise or drop of another.

**Category:** Set Theory and Logic

[53] **viXra:1409.0056 [pdf]**
*replaced on 2016-02-23 12:30:09*

**Authors:** Misha Mikhaylov

**Comments:** 11 Pages.

Usually itemizing relations’ properties those of them are always pointed out – some appearance of reflexivity, symmetry and transitivity. Also it is not so clear whether they are introduced artificially – i.e. axiomatically, rather for the sake of convenience or it may be done due to inartificial reasons. At the same time an origin of them is not so clear – whether they appear chaotically and independently on each other or there should be rigorous association between them. It is shown here that request of relation’s reversibility leads to these properties’ presence or absence. Often symmetry appearances are defined by using of ambiguous way. In fact, anti-symmetry is not direct negation for symmetry – there is also something that may be called as asymmetry or it may be something else. To avoid it here there was found the unified method of their definition. The same thing may be told about reflexivity and it was shown that just the only intransitivity may be represented as direct negation of transitivity.

**Category:** Set Theory and Logic

[52] **viXra:1409.0056 [pdf]**
*replaced on 2014-09-14 04:03:45*

**Authors:** Misha Mikhaylov

**Comments:** 11 Pages.

Usually itemizing relations’ properties those of them are always pointed out – some appearance of reflexivity, symmetry and transitivity. Also it is not so clear whether they are introduced artificially – i.e. axiomatically, rather for the sake of convenience or it may be done due to inartificial reasons. At the same time an origin of them is not so clear – whether they appear chaotically and independently on each other or there should be rigorous association between them. It is shown here that request of relation’s reversibility leads to these properties’ presence or absence. Often symmetry appearances are defined by using of ambiguous way. In fact, anti-symmetry is not direct negation for symmetry – there is also something that may be called as asymmetry or it may be something else. To avoid it here there was found the unified method of their definition. The same thing may be told about reflexivity and it was shown that just the only intransitivity may be represented as direct negation of transitivity.

**Category:** Set Theory and Logic

[51] **viXra:1409.0041 [pdf]**
*replaced on 2015-06-01 08:54:57*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we redefine the infinitesimals by using axiomatic method.

**Category:** Set Theory and Logic

[50] **viXra:1409.0041 [pdf]**
*replaced on 2015-05-31 18:06:55*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

In this paper, we redefine the infinitesimals by using axiomatic method.

**Category:** Set Theory and Logic

[49] **viXra:1409.0041 [pdf]**
*replaced on 2014-11-06 09:08:11*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 1 Page.

**Category:** Set Theory and Logic

[48] **viXra:1409.0006 [pdf]**
*replaced on 2014-10-20 00:20:20*

**Authors:** Felix M. Lev

**Comments:** 7 Pages. A figure added

Standard mathematics involves such notions as infinitely small/large, continuity and standard division. This mathematics is usually treated as fundamental while finite mathematics is treated as inferior. Standard mathematics has foundational problems (as follows, for example, from G\"{o}del's incompleteness theorems) but it is usually believed that this is less important than the fact that it describes many experimental data with high accuracy. We argue that the situation is the opposite: standard mathematics is only a degenerate case of finite one in the formal limit when the characteristic of the ring or field used in finite mathematics goes to infinity. Therefore foundational problems in standard mathematics are not fundamental.

**Category:** Set Theory and Logic

[47] **viXra:1408.0211 [pdf]**
*replaced on 2014-08-29 22:58:49*

**Authors:** Ihsan Raja Muda Nasution

**Comments:** 2 Pages.

Our result is the explicit form of the infinitesimals.

**Category:** Set Theory and Logic

[46] **viXra:1406.0189 [pdf]**
*replaced on 2017-04-21 12:36:58*

**Authors:** Daniel Cordero Grau

**Comments:** 2 Pages.

In this paper I lay down the Quantum Continuum Theory

**Category:** Set Theory and Logic

[45] **viXra:1405.0360 [pdf]**
*replaced on 2014-11-16 10:51:17*

**Authors:** Ricardo Alvira

**Comments:** 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Complexity Theory'. ISBN: 978-1499335859. 274 pages

**Category:** Set Theory and Logic

[44] **viXra:1405.0360 [pdf]**
*replaced on 2014-07-23 14:52:39*

**Authors:** Ricardo Alvira

**Comments:** 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Complexity Theory'. ISBN: 978-1499335859. 274 pages.

**Category:** Set Theory and Logic

[43] **viXra:1405.0360 [pdf]**
*replaced on 2014-06-30 05:41:35*

**Authors:** Ricardo Alvira

**Comments:** 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Theory of Complexity'. ISBN: 978-1499335859. 270 pages aprox. The book is currently available in spanish as 'Una Teoria Unificada de la Complejidad'

**Category:** Set Theory and Logic

[42] **viXra:1405.0360 [pdf]**
*replaced on 2014-06-23 06:15:21*

**Authors:** Ricardo Alvira

**Comments:** 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Theory of Complexity'. ISBN: 978-1499335859. 270 pages aprox. The book is currently available in spanish as 'Una Teoria Unificada de la Complejidad'

**Category:** Set Theory and Logic

[41] **viXra:1405.0360 [pdf]**
*replaced on 2014-05-30 17:14:51*

**Authors:** Ricardo Alvira

**Comments:** 42 Pages.

Presentation based on Chapter 2.4 of the book 'A Unified Theory of Complexity'. ISBN: 978-1499335859. 250 pages aprox. The book is expected to be published soon.

**Category:** Set Theory and Logic

[40] **viXra:1405.0202 [pdf]**
*replaced on 2014-05-13 09:26:34*

**Authors:** Bertrand Wong

**Comments:** 3 Pages.

This article raises some important points about logic, e.g., mathematical logic.

**Category:** Set Theory and Logic

[39] **viXra:1310.0075 [pdf]**
*replaced on 2014-04-18 11:11:16*

**Authors:** Daniel Cordero Grau

**Comments:** 3 Pages.

In this paper we prove the continuum hypothesis by categorical logic, proving that the theory of initial ordinals and the theory of cardinals are isomorphic. To prove that the theorems of the theory of cardinals are theorems of the theory of initial ordinals, and conversely, the theorems of the theory of initial ordinals are theorems of the theory of cardinals, and so, since isomorphic structures are isomorphic theories by the fundamental theorem of mathematical logic, cardinals and initial ordinals are isomorphic structures, we use the definition of a theory, the definition of an isomorphism of structures in its equivalent form, the definition of an isomorphism of categories, the definition of a structure, the definition of a formal language, the definition of a functor, the definition of a category, the axioms of mathematical logic and the axioms of the theory of categories, which include the Gödel-Bernays-von Neumann axioms, so as to apply both the theorem on the comparablity of ordinals to the theory of cardinals and the fundamental theorem of cardinal arithmetic to the theory of ordinals.

**Category:** Set Theory and Logic

[38] **viXra:1310.0006 [pdf]**
*replaced on 2017-04-04 09:59:39*

**Authors:** Daniel Cordero Grau

**Comments:** 2 Pages.

In this paper I prove the Space Time Intractability of Quantum Computation and the Fundamental Theorem of Computational Complexity

**Category:** Set Theory and Logic

[37] **viXra:1309.0013 [pdf]**
*replaced on 2015-11-30 06:58:58*

**Authors:** Robert A. Herrmann

**Comments:** 19 Pages.

Relative to universal logic, it is demonstrated how useful it is to utilize general logic-systems to investigate finite consequences operators (operations). Among many other examples relative to the lattice of finite consequence operators, a general characterization for the lattice-theoretic supremum for a nonempty collection of finite consequence operators is given. Further, it is shown that for any denumerable language there is a rather simple collection of finite consequence operators and for a propositional language, three simple modifications to the finitary rule of inference that demonstrate that the lattice of finite consequence operators is not meet-complete.

**Category:** Set Theory and Logic