Two New(ish) Surveys on Gödel’s Incompleteness Theorems

Gödel's incompleteness theorems have many variants: semantic vs. syntactic versions, which specific theory is taken as basic, what model of computability is used, which logical system is assumed to underlie the provability relation, how syntax is arithmetized, what hypotheses the theorem itself uses (soundness, consistency, $latex \omega$-consistency, etc.). These result in trade-offs regarding simplicity of … Continue reading Two New(ish) Surveys on Gödel’s Incompleteness Theorems

Possible Postdoc on Genesis of Mathematical Knowledge

Via the APMP list: Expressions of interest are invited for a postdoc grant (financed by Junta de Andalucia) associated with the following research project:  “THE GENESIS OF MATHEMATICAL KNOWLEDGE: COGNITION, HISTORY, PRACTICES” (P12-HUM-1216). IP: Jose Ferreiros Contact: The grant consists in a 2-year research contract to be held at the University of Sevilla. Salary … Continue reading Possible Postdoc on Genesis of Mathematical Knowledge

Kalmár’s Compleness Proof

Dana Scott's proof reminded commenter "fbou" of Kalmár's 1935 completeness proof. (Original paper in German on the Hungarian Kalmár site.) Mendelsohn's Introduction to Mathematical Logic also uses this to prove completeness of propositional logic. Here it is (slightly corrected): We need the following lemma: Let $latex v$ be a truth-value assignment to the propositional variables … Continue reading Kalmár’s Compleness Proof

Dana Scott’s Favorite Completeness Proof

Last week I gave my decision problem talk at Berkeley. I briefly mentioned the 1917/18 Hilbert/Bernays completeness proof for propositional logic. It (as well as Post's 1921 completeness proof) made essential use of provable equivalence of a formula with its conjunctive normal form. Dana Scott asked who first gave (something like) the following simple completeness … Continue reading Dana Scott’s Favorite Completeness Proof

Lectures on the Epsilon Calculus

Back in 2009, I taught a short course on the epsilon calculus at the Vienna University of Technology.  I wrote up some of the material, intending to turn them into something longer.  I haven't had time to do that, but someone might find what I did helpful. So I put it up on arXiv: