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Cardinal Characteristics above the Continuum (ASPERODU20SF)

  • Full or part time
  • Application Deadline
    Sunday, May 31, 2020
  • Self-Funded PhD Students Only
    Self-Funded PhD Students Only

Project Description

The study of cardinal characteristics of the continuum constitutes a vast area in set theory. A typical question in this area asks whether a given pair of such characteristics can be consistently separated, i.e., it asks a question of the following form: “Is it consistent with the standard axioms for mathematics to have that the smallest set of reals with property P_0 is smaller than the smallest set of reals with property P_1?’’. Here, P_0 and P_1 are usually natural combinatorial properties. The method of iterated forcing has provided us with an extremely powerful toolbox for resolving this type of questions and, in the other direction, these questions have historically been one of the main driving forces in the development of iterated forcing techniques. Our understanding of the corresponding characteristics for high analogues of the reals (e.g., the space of all functions from a given uncountable cardinal into itself) is at the moment much more limited. Nevertheless, there are natural questions in this realm that seem to be amenable to recent methods involving iterated forcing with side conditions.

This project aims at developing this area of set theory, focusing both on what can be proved within ZFC, and on the use and development of iteration techniques for the independence direction.

For more information on the supervisor for this project, please go here:

This is a PhD programme.

The start date of the project is 1 October 2020.

The mode of study is full-time. The studentship length is 3 years.

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 View Website.

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.

Entry requirements

Acceptable first degree in Mathematics.

The standard minimum entry requirement is 2:1.


i) Aspero, Adding many Baumgartner clubs}. Archive for Mathematical Logic, vol. 56 (7) (2017), no. 7-8, 797-810.

ii) Cummings and S. Shelah, Cardinal invariants above the continuum, Annals of Pure and Applied Logic, vol\ 75 (1995), 251-268.

iii) Jech, Set Theory, third millennium edition (2003).

iv) K. Kunen, Set theory. An introduction to independence proofs (1980).

Related Subjects

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