We give complete proofs of the two famous incompleteness theorems of Gödel. However, instead of the usual choice of Peano Arithmetic, we take the theory of hereditarily finite sets as the basis for the presentation.
Using the monotonicity theorem of L. van den Dries for RCF-definable real functions, and a further result of that author about RCF-definable equivalence relations on ℝ, we show that the theory of order with successors is not interpretable in the theory RCF. This confirms a conjecture by J. Mycielski, P. Pudlák and A. Stern.
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