Abstract

We use the Boyer-Moore Prover, Nqthm, to verify the Paris-Harrington version of Ramsey's Theorem. The proof we verify is a modification of the one given by Ketonen and Solovay. The Theorem is not provable in Peano Arithmetic, and one key step in the proof requires ffl 0 induction. x0. Introduction. The most well-known formalizations of finite mathematics are PA (Peano Arithmetic) and PRA (Primitive Recursive Arithmetic). In both, the "intended" domain of discourse is the set of natural numbers. PA is formalized in standard first-order logic, and contains the induction schema, which can apply to arbitrary first-order formulas. The logic of PRA allows only quantifier-free formulas, which are thought of as being universally quantified, and PRA has the induction scheme for quantifier-free formulas, expressed as a proof rule. Also, for each primitive recursive function f , PRA contains a function symbol for f and has the recursive definition of f as an axiom. Clearly, PRA is much weaker tha...

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