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With only the additional axiom that $0\neq 1$, I think I have been able to formally construct the a subset $n$ of the ring $(R,+,*,0,1)$ using only a subset axiom (specification in ZF).

Informally, $n=\{1, 1+1, 1+1+1, ...\}$

I have shown the that equivalent of Peano's Axioms (including induction) holds on this subset. Can this be true? Would this not imply that all rings with $0\neq 1$ are infinite?

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    Maybe you could show the construction?2012-12-02

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As you pointed out, this cannot be true, because there exist finite rings. The Peano Axioms that is not satisfied in this case is:

For every natural number n, S(n) = 0 is false. That is, there is no natural number whose successor is 0.

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    Is it then correct to say that finite rings must "loop around" like modular arithmetic?2012-12-01