Parameter-free schemes in second-order arithmetic
Victoria Gitman (CUNY Graduate Center)
Abstract: Second-order arithmetic has two types of objects: numbers and set of numbers, which we think of as the reals. Which sets (reals) have to exist in a model of second-order arithmetic is determined by the various set-existence axioms. These usually come in the form of schemes, of which the most common are the comprehension scheme, the choice scheme, and the collection scheme. The \emph{comprehension scheme} - asserts for a -formula , with a set parameter , that the collection it determines is a set. The \emph{choice scheme} - asserts for a -formula that if for every number there is a set such that holds, then there is a single set such that its slice is a witness for . The \emph{collection scheme} - asserts more generally that among the slices of , there is a witness for every . The full comprehension scheme for all second-order assertions is denoted by , the full choice scheme by , and the full collection scheme by . Although the theories + and are equiconsistent, Feferman and L\' evy showed that is independent of . It is also not difficult to see that implies over -, and hence that implies over -.
In this talk, I will explore how significant the inclusion of set parameters is in the second-order set-existence schemes. Let , , and denote the respective parameter-free schemes. H. Friedman showed that the theories and are equiconsistent and recently Kanovei and Lyubetsky showed that the theory can have extremely badly behaved models in which the sets aren't even closed under complement. They also constructed a more ``nice" model of in which - holds, but - fails. They asked whether one can construct a model of in which - holds, but there is an optimal failure of -. I will answer their question by constructing such a model. I will also construct a model of in which - holds, but fails, thus showing that does not imply even over -.
logic
Audience: researchers in the topic
Series comments: Description: Seminar on all areas of logic
Organizer: | Wesley Calvert* |
*contact for this listing |