The Department of Logic and Philosophy of Science Colloquium Series presents
“How (not) to Prove Consistency”
with Bernd Buldt, Professor of Philosophy, Indiana-Purdue University, Fort Wayne
Friday, Feb 20, 2015
Social Science Tower, Room 777 (LPS Conference Room)
While Gödel’s first incompleteness theorem (= G1) remains valid under substitution of various provability predicates, Gödel’s second theorem (G2) does not. This is one reason to label G1 as “extensional”but to call G2 “intensional.” Although this asymmetry between G1 and G2 is known for long, no satisfying or generally agreed upon account of G2’s intensionality has been put forward. After briefly reviewing the previous discussion, the talk presents a new analysis of G2 based on, among others, two observations.
First, the underestimated role of provable closure under modus (or similar derivability conditions). Provable closure under modus ponens or similar requirements (e.g., as discussed by Bernays in Hilbert/Bernays, Grundlagen der Mathematik, vol. 2), do not get much attention since their proof isn’t perceived as much of challenge. Closer inspection shows, however, that the requirement of provable closure under modus ponens interacts differently with different formalized proof predicates and has thus a direct bearing on the discussion of G2’s intensionality.
Second, in order to establish the co-extensionality of various proof predicates and hence the extensionality of Gödel’s first incompleteness theorem, the role informal consistency assumption play goes largely unnoticed and unanalyzed. Closer inspection shows, however, that if such informal consistency assumptions are being made precise, the former distinction between G1 and G2 becomes blurry and G2’s intensionality can be made go away.
As a result Buldt will defend, among others, the thesis that the traditional extensional/intensional distinction is not as robust as the received view believed it to be.
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