The Math Tea argument: must there be numbers we can neither describe nor define? Barcelona March 2023

This will be a talk 15 March 2023 for the Mathematics Department of the University of Barcelona, organized jointly with the Set Theory Seminar.

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, including every real number, every
function, every set. We shall discuss the relevance for the math tea argument.

4 thoughts on “The Math Tea argument: must there be numbers we can neither describe nor define? Barcelona March 2023

  1. Very interesting. May I suggest modifying your statement about the absence of a formula df(x). As written it reads as if you are saying for all models M of ZFC there is no formula df(x) holding of all the defineable elements. I presume you mean there is no formula df(x) s.t. in all models M df(x) holds only of the pointwise defineable elements?

    Though, why do you use ZFC there at all since presumably the claim holds for any theory T having infinite models. Right?

    • Thanks for these comments. Indeed, I had meant that there is no formula df(x) that picks out the definable elements in every model M. In some models, the definable elements do form a definable class, such as any pointwise definable model. And indeed, the argument generalizes to other theories, not just ZFC, provided that there are pointwise definable models. (Not every theory with infinite models has a pointwise definable model.)

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