## A Simple Proof That Finite Mathematics Is More Fundamental Than Classical One

**Authors:** Felix M. Lev

Classical mathematics (involving such notions as infinitely small/large and continuity) is usually treated as fundamental while finite mathematics is treated as inferior which is used only in special applications. In our previous publications we argue that the situation is the opposite: classical mathematics is only a special degenerate case of finite one in the formal limit when the characteristic of the ring or field in finite mathematics goes to infinity.
In the present paper we give a simple and rigorous proof of this fundamental fact. In general, introducing infinity automatically implies transition to a degenerate theory because in that case all operations modulo a number are lost. So, {\it even from the pure mathematical point of view}, the very notion of infinity cannot be fundamental, and theories involving infinities can be only approximations to more general theories. We also prove that standard quantum theory based on classical mathematics is a special degenerate case of
quantum theory based on finite mathematics. Motivation and implications are discussed.

**Comments:** 9 Pages. Statement 2 in Sec. 3 has been considerably elaborated

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### Submission history

[v1] 2018-11-03 23:46:55

[v2] 2018-11-15 23:29:07

[v3] 2018-11-23 18:29:06

[v4] 2018-12-13 23:09:36

[v5] 2019-01-12 12:43:10

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