Joel David Hamkins: The Math Tea argument—must there be numbers we cannot describe or define?

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.

[NikolajKuntner]

Thanks for this.
I find myself coming back to that MO post on the definability definability every now and again.
:)

[davidtaffs]

I suspect (infinite) definitions are uncountable, you can map them one-to-one with real numbers. Of course finite definitions are finite.

[NoNTrvaL]

Absolutely amazing video! Could you please suggest books on mathematical logic specifically in the work of Saharon Shelah and Hrushovski?

[pmcate]

@12:50 you say that A not being periodic implies Leibnizian. But doesn't the inclusion of < with the usual interpretation already suffice? Also, how is definability affected by infinitary logic? For whatever reason my intuition is that languages allowing countably long sentences could have point-wise definability for models with cardinality of the reals, but for models of greater size there would still be undefinable elements. Also, @21:32 by V do you mean the von Neumann universe?