<div class="csl-bib-body">
<div class="csl-entry">Schlutzenberg, F. S. (2025, November 7). <i>Large cardinals beyond the bounds of the Axiom of Choice</i> [Conference Presentation]. Australasian Association for Logic 2025, Brisbane, Australia. http://hdl.handle.net/20.500.12708/224923</div>
</div>
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dc.identifier.uri
http://hdl.handle.net/20.500.12708/224923
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dc.description.abstract
The axioms of set theory – the Zermelo Fraenkel axioms with the Axiom of Choice
(ZFC) – are highly incomplete, with many basic questions known to be independent.
Large cardinal axioms, a major discovery of the last century, form a natural and com-
pelling hierarchy of extensions of ZFC, which resolve many of the unanswered ques-
tions. Their investigation is a central program in the foundations of mathematics.
In the 1970s, Kenneth Kunen demonstrated, with what is now known as the Kunen
inconsistency, that the large cardinal hierarchy has a rather surprising hard upper
bound, with the non-existence of elementary embeddings of the form j : V_{λ+2} → V_{λ+2}.
This dispelled hopes for potentially much larger cardinals, such as Reinhardt cardinals.
However, his proof made strong use of the Axiom of Choice, leaving a long-standing
open question, as to whether his result could be proved without it. I will discuss some
recent work demonstrating that, assuming the consistency of the large cardinal axiom
I0, Choice is in fact necessary. This result is part of a recent body of work, due to
various researchers, exploring significant structure in these large cardinals “beyond”
the Axiom of Choice, such as Reinhardt cardinals. This structure suggests that these
principles are coherent, and could in fact play a foundational role. But their direct
conflict with Choice leads to significant questions as to what that role could be, and
how they might actually feature in the set theoretic universe.
en
dc.description.sponsorship
FWF - Österr. Wissenschaftsfonds
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dc.language.iso
en
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dc.subject
Large cardinal
en
dc.subject
Elementary embedding
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dc.subject
Kunen inconsistency
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dc.subject
Axiom of Choice
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dc.subject
Reinhardt cardinal
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dc.title
Large cardinals beyond the bounds of the Axiom of Choice
en
dc.type
Presentation
en
dc.type
Vortrag
de
dc.relation.grantno
Y1498
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dc.type.category
Conference Presentation
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tuw.project.title
Determiniertheit und Woodin Limes von Woodin Kardinalzahlen
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tuw.researchTopic.id
I1
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tuw.researchTopic.name
Logic and Computation
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tuw.researchTopic.value
100
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tuw.publication.orgunit
E104-08 - Forschungsbereich Mengenlehre
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tuw.event.name
Australasian Association for Logic 2025
en
dc.description.sponsorshipexternal
Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)
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dc.relation.grantnoexternal
EXC 2044-390685587
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tuw.event.startdate
03-11-2025
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tuw.event.enddate
07-11-2025
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tuw.event.online
Hybrid
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tuw.event.type
Event for scientific audience
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tuw.event.place
Brisbane
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tuw.event.country
AU
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tuw.event.institution
Australasian Association for Logic / University of Queensland
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tuw.event.presenter
Schlutzenberg, Farmer
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wb.sciencebranch
Mathematik
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wb.sciencebranch.oefos
1010
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wb.sciencebranch.value
100
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item.openairecristype
http://purl.org/coar/resource_type/c_18cp
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item.cerifentitytype
Publications
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item.openairetype
conference paper not in proceedings
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item.languageiso639-1
en
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item.grantfulltext
none
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item.fulltext
no Fulltext
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crisitem.author.dept
E104-08 - Forschungsbereich Mengenlehre
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crisitem.author.parentorg
E104 - Institut für Diskrete Mathematik und Geometrie
