University of Birmingham > Talks@bham > Midlands Logic Seminar > Set theoretic principles required for categoricity II

Set theoretic principles required for categoricity II

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This is a continuation of the last talk in this series and based on a paper by Simpson and Yokoyama on the set theoretic principles required for categoricity of the axioms for the natural numbers. If time permits we will also look at a follow-up paper by Kolodziejczyk and Yokoyama.

DETAILS OF THE PREVIOUS TALK WILL NOT BE REQUIRED . All welcome.

Dedekind proved that two structures satisfying the second order axioms for natural numbers are canonically isomorphic. Obviously a key point in the argument is that, assuming two such structures are not isomorphic, an inductive set must be constructed in one structure that is not the whole thing. Thus set theoretic principles are required for categoricity results of this kind, and these arguments are sensitive to the set theoretic framework we are working in, and in particular what set existence axioms are available.

So the question is, which set existence axioms are actually needed? Simpson and Yokoyama investigated this question over the base theory that they call RCA0star. The second half of this paper looks at different formulations of the question where, instead of Dedekind’s system of (N,S,0) (a set, a 1—1 successor function and a number not in the image of S) they look at structures based on order relations. The answers are subtly different.

A question left open was whether or not there is some characterization of the natural numbers by a second order sentence which is provably categorical in RCA0star. This was answered negatively by Kolodziejczyk and Yokoyama.

We shall give a reading of these papers. This will be a useful introduction to some of the methods and results in so-called “reverse mathematics”.

This talk is part of the Midlands Logic Seminar series.

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