As a part of the Spring 2012 Mid-Atlanatic Mathematical Logic Seminar, to be held March 9-10, 2012 at the CUNY Graduate Center, I shall participate in the following panel discussion.

Panel discussion: The unity and diversity of logic

**Abstract.** The field of mathematical logic sometimes seems to be fracturing into ever-finer subdisciplines, with little connection between them, and many logicians now identify themselves by their specific subdiscipline. On the other hand, certain new themes have appeared which tend to unify the diverse discoveries of the many subdisciplines. This discussion will address these trends and ask whether one is likely to dominate the other in the long term. Will logic remain a single field, or will it split into many unrelated branches?

The panelists will be Prof. Gregory Cherlin, myself, Prof. Rohit Parikh, and Prof. Jouko Väänänen, with the discussion moderated by Prof. Russell Miller. Questions and participation from the audience are encouraged.

As preparation for this panel discussion, please suggest points or topics that might brought up at the panel discussion, by posting suitable comments below. Perhaps we’ll proceed with our own pre-discussion discussion here!

One salient feature that ties together the modern practice of mathematical logic is that all of its areas seem to make more use of (transfinite) ordinal numbers than do other branches of math. Model theory also seems to tie together mathematical logic, as seen by its application in set theory (inner models, use of elementary substructures) and in computability theory, where computable model theory has become a major area of study.

I agree, Norman, that well-foundedness is prominent in logic, but it is emphasized far less in some areas, such as computabilty theory, than in others, such as set theory. Adrian Mathias has emphasized that set theory is the study of well-foundedness, an idea with the ordinals surely at its core. As for model theory, one sometimes hears arguments that the recent exciting developments in model theory, which have led to increasing connections in other parts of mathematics, are also the least connected to the rest of logic. Indeed, Angus MacIntyre has argued that the rest of logic could learn from these applications that a more unified integration with the rest of mathematics is possible.

If one holds that the multiverse view of set theory is valid, where does this leave the view that set theory is ‘the’ foundation of mathematics?

Is there a way to reinterpret the universal quantifier in the definition of the power set axiom (‘there is some set B such that for every set C ( if C is a subset of A then C is a member of B)’) as ‘for every possible set C’ where one takes as the power set the forcing extension which adds the largest number of ‘new’ subsets

C of A short of inconsistency? Is there a way to make this notion mathematically precise?