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Very large cardinals without choice

   School of Mathematics

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  Dr David Aspero  No more applications being accepted  Self-Funded PhD Students Only

About the Project

Large cardinal axioms are usually stated as assertions concerning the existence of nontrivial elementary embeddings from the universe V into some subclass M; the more M resembles V the stronger the axiom is. Kenneth Kunen famously proved the nonexistence of any nontrivial elementary embedding from V into itself, assuming ZFC, thus establishing a natural impassable barrier for all consistent large cardinals. The Axiom of Choice figures prominently in all known proofs of this result; in fact it is not known whether the use of Choice can be dispensed with. As it turns out, there is a long hierarchy of large cardinal axioms in the ZF context subsuming, and extending, the hierarchy of ZFC large cardinal axioms. This hierarchy comprises Reinhardt cardinals, Berkeley cardinals, etc. The theory of these cardinals remains largely unexplored, although it has seen some important developments in recent years. In this project we aim to explore structural and combinatorial consequences of axioms in the higher reaches of the large cardinal hierarchy in the absence of Choice. One of our tools will be the use of forcing over models with the relevant large cardinals.   

Funding Notes

This PhD project is offered on a self-funding basis. It is open to applicants with funding or those applying to funding sources. Details of tuition fees can be found at
A bench fee is also payable on top of the tuition fee to cover specialist equipment or laboratory costs required for the research. Applicants should contact the primary supervisor for further information about the fee associated with the project.


i) A. Kanamori, The higher infinite, Springer, 1994
ii) J. Bagaria, P. Koellner, H. Woodin, Large cardinals beyond choice, Bull. of Symbolic Logic, vol. 25, 283-318 (2019)
iii) Periodicity in the cumulative hierarchy, G. Goldberg and F. Schlutzenberg, J. European Math. Soc, 2021,
iv) G. Goldberg, Measurable cardinals and choiceless axioms,
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