# Kenneth Kunen

Kenneth Kunen

Herbert Kenneth Kunen (born August 2, 1943) is an emeritus professor of mathematics at the University of Wisconsin–Madison[1] who works in set theory and its applications to various areas of mathematics, such as set-theoretic topology and measure theory. He also works on non-associative algebraic systems, such as loops, and uses computer software, such as the Otter theorem prover, to derive theorems in these areas.

Kunen showed that if there exists a nontrivial elementary embedding j:LL of the constructible universe, then 0# exists. He proved the consistency of a normal, ${\displaystyle \aleph _{2}}$-saturated ideal on ${\displaystyle \aleph _{1}}$ from the consistency of the existence of a huge cardinal. He introduced the method of iterated ultrapowers, with which he proved that if ${\displaystyle \kappa }$ is a measurable cardinal with ${\displaystyle 2^{\kappa }>\kappa ^{+}}$ or ${\displaystyle \kappa }$ is a strongly compact cardinal then there is an inner model of set theory with ${\displaystyle \kappa }$ many measurable cardinals. He proved Kunen's inconsistency theorem showing the impossibility of a nontrivial elementary embedding ${\displaystyle V\to V}$, which had been suggested as a large cardinal assumption (a Reinhardt cardinal).

Away from the area of large cardinals, Kunen is known for intricate forcing and combinatorial constructions. He proved that it is consistent that the Martin Axiom first fails at a singular cardinal and constructed under CH a compact L-space supporting a nonseparable measure. He also showed that ${\displaystyle P(\omega )/Fin}$ has no increasing chain of length ${\displaystyle \omega _{2}}$ in the standard Cohen model where the continuum is ${\displaystyle \aleph _{2}}$. The concept of a Jech–Kunen tree is named after him and Thomas Jech.

Kunen completed his undergraduate degree at Caltech[2] and received his Ph.D. in 1968 from Stanford University,[3] where he was supervised by Dana Scott.

## Personal life

Kunen was born in New York in 1943. He lives in Madison, Wisconsin with his wife Anne. They have two sons, Isaac and Adam.[2]

## Bibliography

The journal Topology and its Applications has dedicated a special issue to "Ken" Kunen,[2] containing a biography by Arnold W. Miller, and surveys about Kunen research in various fields by Mary Ellen Rudin, Akihiro Kanamori, István Juhász, Jan van Mill, Dikran Dikranjan, and Michael Kinyon.

## References

1. ^ http://www.math.wisc.edu/~apache/emeriti.html
2. ^ a b c Hart, Joan, ed. (1 Dec 2011). "Special Issue: Ken Kunen". Topology and Its Applications. 158 (18): 2443–2564.
3. ^
4. ^ Henson, C. Ward (1984). "Review: Set theory, an introduction to independence proofs, by Kenneth Kunen" (PDF). Bull. Amer. Math. Soc. (N.S.). 10 (1): 129–131. doi:10.1090/s0273-0979-1984-15214-5.
5. ^ Baldwin, Stewart (December 1987). "Review: Handbook of set-theoretic topology edited by Kenneth Kunen and Jerry E. Vaughan". The Journal of Symbolic Logic. 52 (4): 1044–1045. JSTOR 2273837. doi:10.2307/2273837.