This will be a talk for the Logic Seminar at the IUSS, Scuola Universitaria Superiore Pavia, 28 September 2022.

(Note: This seminar will be held the day before the related conference Philosophy of Mathematics: Foundations, Definitions and Axioms, Italian Network for the Philosophy of Mathematics, 29 September to 1 October 2022. I shall be speaking at that conference on the topic, Fregean abstraction in set theory, a deflationary account.)

**Abstract.** According to the math tea argument, perhaps heard at a good afternoon tea, there must be some real numbers that we can neither describe nor define, since there are uncountably many real numbers, but only countably many definitions. Is it correct? In this talk, I shall discuss the phenomenon of pointwise definable structures in mathematics, structures in which every object has a property that only it exhibits. A mathematical structure is Leibnizian, in contrast, if any pair of distinct objects in it exhibit different properties. Is there a Leibnizian structure with no definable elements? We shall discuss many interesting elementary examples, eventually working up to the proof that every countable model of set theory has a pointwise definable extension, in which every mathematical object is definable.

How small can a model of set theory be and not have a Leibnizian extension? Does every model of continuum size have one?

This is an interesting question! I don’t know the answer straight away, but want to think about it.

As far as I know, definitive answers to the questions posed by Joseph Shipman are not known, but there are a number of related partial results (and similar open questions) in my 2004 JSL-paper “Leibnizian models of set theory”.

Thanks for your reply, Ali.

I can now prove that every model of PA of size continuum has a Leibnizian extension, and I can make Sigma_k elementary for any desired k. The argument seems to generalize to set theory to show that every model of ZF of size continuum has a Leibnizian extension, and if the original model has V=HOD, then one can arrange it to be Sigma_k elementary. Victoria Gitman and I are currently writing this up.