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 New York City Category Theory Seminar Department of Computer Science Department of Mathematics The Graduate Center of The City University of New York

Some talks will be in-person (with a Zoom connection) and some talks will only be on Zoom. Time: Wednesdays 02:00-3:00 PM Eastern Time (US and Canada) NOTICE NEW TIME!!!

IN-PERSON INFORMATION: 365 Fifth Avenue (at 34th Street) map (Diagonally across from the Empire State Building) New York, NY 10016-4309 Room 4214.03 NOTICE NEW ROOM!!! THere will be a Zoom link to join us on in-person talks. This means that from now on, in-person talks will be hybrid.

ZOOM INFORMATION: brooklyn-cuny-edu.zoom.us/j/84134080992?pwd=MNvAvOfYemk2bh7qCtI9I5zQTRkFdR.1 Meeting ID: 841 3408 0992 Passcode: NYCCTS

Seminar web page. Videoed talks. Previous semesters. List of previous speakers. Researchseminars.org page.

Co-organizers: Emilio Minichiello and Noson S. Yanofsky.

Contact E. Minichiello or N. Yanofsky to schedule a speaker or to add a name to the seminar mailing list.

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