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Topic Archive: Kunen inconsistency

Set theory seminarFriday, September 20, 201310:00 amGC

Joel David Hamkins

The role of the axiom of foundation in the Kunen inconsistency

The City University of New York

The axiom of foundation plays an interesting role in the Kunen inconsistency, the assertion that there is no nontrivial elementary embedding of the set-theoretic universe to itself, for the truth or falsity of the Kunen assertion depends on one’s specific anti-foundational stance.  The fact of the matter is that different anti-foundational theories come to different conclusions about this assertion.  On the one hand, it is relatively consistent with ZFC without foundation that the Kunen assertion fails, for there are models of  ZFC-F  in which there are definable nontrivial elementary embeddings $j:Vto V$. Indeed, in Boffa’s anti-foundational theory BAFA, the Kunen assertion is outright refutable, and in this theory there are numerous nontrivial elementary embeddings of the universe to itself. Meanwhile, on the other hand, Aczel’s anti-foundational theory GBC-F+AFA, as well as Scott’s theory GBC-F+SAFA and other anti-foundational theories, continue to prove the Kunen assertion, ruling out the existence of a nontrivial elementary embedding $j:Vto V$.

This is very recent joint work with Emil Jeřábek, Ali Sadegh Daghighi and Mohammad Golshani, based on an interaction growing out of Ali’s question on MathOverflow.  Our paper will be completed soon.

Norman Perlmutter
LaGuardia Community College, CUNY
Norman Lewis Perlmutter grew up in Toledo, Ohio, earned his bachelor’s degree in mathematics at Grinnell College in Grinnell, Iowa, in 2007, and earned his Ph.D. in mathematics at the CUNY Graduate Center in 2013 under the supervision of Joel David Hamkins. After a year as a visiting assistant professor at Florida Atlantic University, he returned to New York City and to CUNY, taking a position as an assistant professor of mathematics at LaGuardia Community College in 2014. Besides mathematics, his interests include theater, board games, food, travel, and science fiction.