Logic Seminar - Zachiri McKenzie
| Dates: | 21 October 2026 |
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| Times: | 15:00 - 15:00 |
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| What is it: | Seminar |
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| Organiser: | Department of Mathematics |
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| Who is it for: | University staff, External researchers, Current University students |
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Speaker: Zachiri McKenzie (University of Chester)
Title: Transitive Containment in subsystems of set theory
Abstract: Transitive Containment (TCo) says that every set is contained in a transitive set. Students usually encounter an elementary proof of this fact in Zermelo-Fraenkel Set Theory (ZF) early on in an undergraduate course in set theory. Despite this, it has long been known that there are very strong fragments of ZF that fail to prove TCo. For example, TCo is not provable the theory obtained by strengthening Zermelo Set Theory with full set induction. Nor is it provable in theory obtained from ZF by replacing the Axiom of Infinity with its negation. I will survey some of these results before presenting a model of a significant fragment of ZF that includes the Axiom of Infinity and Bounded Collection in which TCo fails. I will end by discussing some open questions.
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Frank Adams 1
Alan Turing Building
Manchester
