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?