Saul A. Kripke’s research while affiliated with CUNY School of Law and other places

The following is a typeset copy of a letter sent by Saul Kripke to David Lewis on August 11, 1969 regarding the article “Counterpart Theory and Quantified Modal Logic” (Lewis, 1968). The original letter was typeset by Mimi Foster (indicated by the initials “mf” at the end of the letter) at Rockefeller University. In consultation with Saul Kripke, we corrected some typos, filled in blank formulas, and added three footnotes. Keywords have been added before the letter, references have been added after the letter, and the original pagination of the letter is reflected in bracketed parentheticals. (The Editors.)

This paper is an amended version of a talk Saul Kripke gave at the Eastern Division meeting of the APA in 1992 (an extended abstract was previously published as Kripke, 1992). It contains philosophical reflections and technical results concerning “Carnapian” quantified modal logic, that is, modal logic with quantification over individual concepts. The paper contains the fullest statement by the author available of (un)axiomatizability results he obtained in the 1970s. (The Editors.)

Under the influence of Quine’s famous manifesto, many philosophers have thought that logical theories are scientific theories that can be ‘adopted’ and tested as scientific theories. Here we argue that this idea is untenable. We discuss it with special reference to Putnam’s proposal to ‘adopt’ a particular non-classical logic to solve the foundational problems of quantum mechanics in his famous paper ‘Is Logic Empirical?’ (1968), which we argue was not really coherent.

Wittgenstein gave a clearly erroneous refutation of Russell’s logicist project. The errors were ably pointed out by Mark Steiner. Nevertheless, I was motivated by Wittgenstein and Steiner to consider various ideas about the natural numbers. I ask which notations for natural numbers are ‘buck-stoppers’. For us it is the decimal notation and the corresponding verbal system. Based on the idea that a proper notation should be ‘structurally revelatory’, I draw various conclusions about our own concept of the natural numbers.KeywordsWittgensteinRussellSteinerNatural numbersBuck-stoppersStructurally revelatoryDecimal notation

The Hilbert program was actually a specific approach for proving consistency, a kind of constructive model theory. Quantifiers were supposed to be replaced by ε-terms. ε xA ( x ) was supposed to denote a witness to $\exists xA(x)$ , or something 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 ( ${\Sigma}_1^0$ -correct). Here we show that if the result is supposed to be provable within S , a statement about all ${\Pi}_2^0$ 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.

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 Peano 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.

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 $\epsilon$-terms. $\epsilon{x}A(x)$ 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 $\epsilon$-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 ($\Sigma_{1}^{0}$-correct). Here we show that if the result is supposed to be provable within S, a statement about all $\Pi_{2}^{0}$ statements that subsumes itself within its own scope must be provable, yielding a contradiction. The result resembles G\"odel's but arises naturally out of the Hilbert program itself.

In his paper on the incompleteness theorems, G\"odel 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.