Universality: new criterion for non-existence

Saharon Shelah

doi:10.1007/s40574-022-00327-7

Universality: new criterion for non-existence

Saharon Shelah^*

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We find new “reasons” for a class of models for not having a universal model in a cardinal λ. This work, though has consequences in model theory, is really in combinatorial (set theory). We concentrate on a prototypical class which is a simply defined class of models, of combinatorial character—models of T_ceq (essentially another representation of T_feq which was already considered but the proof with T_ceq is more transparent). Models of T_ceq consist essentially of an equivalence relation on one set and a family of choice functions for it. This class is not simple (in the model theoretic sense) but seems to be very low among the non-simple (first order complete countable) ones. We give sufficient conditions for the non-existence of a universal model for it in λ. This work may be continued in Shelah et al. (Tba, In preparation. Preliminary number: Sh:F2150).

Original language	English
Pages (from-to)	43-64
Number of pages	22
Journal	Bolletino dell Unione Matematica Italiana
Volume	16
Issue number	1
DOIs	https://doi.org/10.1007/s40574-022-00327-7
State	Published - Mar 2023

Bibliographical note

Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Unione Matematica Italiana.

Keywords

Classification theory
Combinatorial set theory
Non-simple theories
Universality

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10.1007/s40574-022-00327-7

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Universality: new criterion for non-existence

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