Self-Reference and Diagonalization: their difference and a short history.

Abstract: What is now called the Diagonal (or the Self-Reference) Lemma, is the statement that for every formula F(x), with the only free variable x, there exists a sentence σ such that σ is equivalent to the F of the Gödel code of σ, i.e., σ ≡ F(#σ); and this equivalence is provable in certain weak arithmetics. This lemma is credited to Gödel (1931), in the special case when F is the unprovability predicate, and to Carnap (1934) in the more general case.

In this talk, we will argue that Gödel-Carnap's original Diagonal Lemma is not the modern formulation and was more similar to, but not exactly identical with, the Strong Diagonal (or Direct Self-Reference) Lemma. This lemma, so-called recently, says that for every formula F(x), in a sufficiently expressive language, there exists a sentence σ such that σ is equal to the F of the Gödel code of σ, i.e., σ = F(#σ); and this equality is provable in sufficiently strong theories. We will attempt at tracking down the first appearance of the modern formulation of the Diagonal Lemma in the equivalent form, also in the strong direct form of equality.

Computer scienceMathematics

Audience: researchers in the topic

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New York City Category Theory Seminar

Series comments: The seminar Web page: www.sci.brooklyn.cuny.edu/~noson/Seminar/index.html

THIS SEMESTER, SOME TALKS WILL BE IN-PERSON AND SOME WILL BE ON ZOOM. Time: Wednesdays 07:00 PM Eastern Time (US and Canada)

IN-PERSON INFORMATION: 365 Fifth Avenue (at 34th Street) map (Diagonally across from the Empire State Building) New York, NY 10016-4309 Room 6417 The videos of the lectures will be put up on YouTube a few hours after the lecture.

ZOOM INFORMATION: brooklyn-cuny-edu.zoom.us/j/83243451066?pwd=V3BkMCtxTnM3WTQ0QlN3K3NRRHNSQT09 Meeting ID: 832 4345 1066 Passcode: NYCCTS1

Most talks are uploaded to www.youtube.com/channel/UCNOfhimbNwZwJO2ltv1AZOw/videos

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