When does every definable set have a definable member?
Joel David Hamkins
The City University of New York
Although the concept of `being definable’ is not generally expressible in the language of set theory, it turns out that the models of ${rm ZF}$ in which every definable nonempty set has a definable element are precisely the models of $V={rm HOD}$. Indeed, $V={rm HOD}$ is equivalent to the assertion merely that every $Pi_2$-definable set has an ordinal-definable element. Meanwhile, this is not true in the case of $Sigma_2$-definability, because every model of ZFC has a forcing extension satisfying $Vneq{rm HOD}$ in which every $Sigma_2$-definable set has an ordinal-definable element. This is joint work with François G. Dorais and Emil Jeřábek, growing out of some questions on MathOverflow, namely,
- Definable collections without definable members
- A question asked by Ashutosh five years ago, in which François and I gradually came upon the answer together.
- Is it consistent that every definable set has a definable member?
- A similar question asked last week by (anonymous) user38200
- Can $Vneq{rm HOD}$ if every $Sigma_2$-definable set has an ordinal-definable member?
- A question I had regarding the limits of an issue in my answer to the previous question.
In this talk, I shall present the answers to all these questions and place the results in the context of classical results on definability, including a review of basic concepts for graduate students.
Professor Hamkins (Ph.D. 1994 UC Berkeley) conducts research in mathematical and philosophical logic, particularly set theory, with a focus on the mathematics and philosophy of the infinite. He has been particularly interested in the interaction of forcing and large cardinals, two central themes of contemporary set-theoretic research. He has worked in the theory of infinitary computability, introducing (with A. Lewis and J. Kidder) the theory of infinite time Turing machines, as well as in the theory of infinitary utilitarianism and, more recently, infinite chess. His work on the automorphism tower problem lies at the intersection of group theory and set theory. Recently, he has been preoccupied with various mathematical and philosophical issues surrounding the set-theoretic multiverse, engaging with the emerging debate on pluralism in the philosophy of set theory, as well as the mathematical questions to which they lead, such as in his work on the modal logic of forcing and set-theoretic geology.