# Definability and the Math Tea argument: must there be numbers we cannot describe or define? University of Warsaw, 22 January 2021

This will be a talk for a new mathematical logic seminar at the University of Warsaw in the Department of Hhilosophy, entitled Epistemic and Semantic Commitments of Foundational Theories, devoted to formal truth theories and implicit commitments of foundational theories as well as their conceptual surroundings.

My talk will be held 22 January 2021, 8 pm CET (7 pm UK), online via Zoom https://us02web.zoom.us/j/83366049995.

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.

Pointwise definable models of set theory

• J. D. Hamkins, D. Linetsky, and J. Reitz, “Pointwise definable models of set theory,” Journal of Symbolic Logic, vol. 78, iss. 1, p. 139–156, 2013.
[Bibtex]
@article {HamkinsLinetskyReitz2013:PointwiseDefinableModelsOfSetTheory,
AUTHOR = {Hamkins, Joel David and Linetsky, David and Reitz, Jonas},
TITLE = {Pointwise definable models of set theory},
JOURNAL = {Journal of Symbolic Logic},
FJOURNAL = {Journal of Symbolic Logic},
VOLUME = {78},
YEAR = {2013},
NUMBER = {1},
PAGES = {139--156},
ISSN = {0022-4812},
MRCLASS = {03E55},
MRNUMBER = {3087066},
MRREVIEWER = {Bernhard A. König},
DOI = {10.2178/jsl.7801090},
URL = {http://jdh.hamkins.org/pointwisedefinablemodelsofsettheory/},
eprint = "1105.4597",
archivePrefix = {arXiv},
primaryClass = {math.LO},
}