# Elementary epimorphisms between models of set theory

## Norman Perlmutter

### LaGuardia Community College, CUNY

This talk concerns joint work with Robert S. Lubarsky.

Elementary epimorphisms were introduced by Philipp Rothmaler. A surjective homomorphism f: M –> N between two model-theoretic structures is an elementary epimorphism if and only if every formula with parameters satisfied by N is satisfied in M using a preimage of those parameters.

Philipp asked me whether nontrivial elementary epimorphisms between models of set theory exist. We answer this question in the negative for fully elementary epimorphisms between models of ZFC, but in the positive under weaker assumptions.

In particular, we show that every Pi_1-elementary epimorphism between models of ZF is an isomorphism. On the other hand, nonisomorphic Sigma_1-elementary epimorphisms between models of ZF can be constructed, as can fully elementary epimorphisms between models of ZFC^-.

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.