arXiv subjects
Publications and source records attributed to Saul A. Kripke.
In his paper on the incompleteness theorems, Gödel seemed to say that a direct way of constructing a formula that says of itself that it is unprovable might involve a faulty circularity. In this note, it is proved that 'direct' self-reference can actually be used to prove his result.
In the Handbook of Mathematical Logic, the Paris-Harrington variant of Ramsey's theorem is celebrated as the first result of a long 'search' for a purely mathematical incompleteness result in first-order arithmetic. This paper questions the existence of any such search and the status of the Paris-Harrington result as the first mathematical incompleteness result. In fact, I argue that Gentzen gave the first such result, and that it was restated by Goodstein in a number-theoretic form.
The Hilbert program was actually a specific approach for proving consistency. Quantifiers were supposed to be replaced by $ε$-terms. $ε{x}A(x)$ was supposed to denote a witness to $\exists{x}A(x)$, arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system $S$, each $ε$-term can be replaced by a numeral, making each line provable and true. This implies that $S$ must not only be consistent, but also 1-consistent ($Σ_{1}^{0}$-correct). Here we show that if the result is supposed to be provable within $S$, a statement about all $Π_{2}^{0}$ statements that subsumes itself within its own scope must be provable, yielding a contradiction. The result resembles Gödel's but arises naturally out of the Hilbert program itself.