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

*Saeed Salehi (University of Tabriz)*

**24-Nov-2022, 00:00-01:30 (23 months ago)**

**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 *un*provability 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

**New York City Category Theory Seminar **

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

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