Herbert Kenneth Kunen (August 2, 1943 – August 14, 2020 ) was a professor of mathematics at the University of Wisconsin–Madison who worked in set theory and its applications to various areas of mathematics, such as set-theoretic topology and measure theory . He also worked on non-associative algebraic systems, such as loops , and used computer software, such as the Otter theorem prover , to derive theorems in these areas.
12-630: Kunen may refer to: Kenneth Kunen , American mathematician former name of Acharkut , Armenia Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Kunen . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Kunen&oldid=942394274 " Categories : Disambiguation pages Disambiguation pages with surname-holder lists Hidden categories: Short description
24-417: A strongly compact cardinal is a certain kind of large cardinal . An uncountable cardinal κ is strongly compact if and only if every κ-complete filter can be extended to a κ-complete ultrafilter. Strongly compact cardinals were originally defined in terms of infinitary logic , where logical operators are allowed to take infinitely many operands. The logic on a regular cardinal κ is defined by requiring
36-440: Is a strongly compact cardinal then there is an inner model of set theory with κ {\displaystyle \kappa } many measurable cardinals. He proved Kunen's inconsistency theorem showing the impossibility of a nontrivial elementary embedding V → V {\displaystyle V\to V} , which had been suggested as a large cardinal assumption (a Reinhardt cardinal ). Away from
48-607: Is different from Wikidata All article disambiguation pages All disambiguation pages Kenneth Kunen Kunen was born in New York City in 1943 and died in 2020. He lived in Madison, Wisconsin , with his wife Anne, with whom he had two sons, Isaac and Adam. Kunen completed his undergraduate degree at the California Institute of Technology and received his Ph.D. in 1968 from Stanford University , where he
60-402: Is equiconsistent with that of a supercompact cardinal. However, a proof is unlikely until a canonical inner model theory for supercompact cardinals is developed. Jech obtained a variant of the tree property which holds for an inaccessible cardinal if and only if it is strongly compact. Extendibility is a second-order analog of strong compactness. This set theory -related article
72-406: Is strongly compact, or that the first strongly compact cardinal is supercompact; these cannot both be true, however. A measurable limit of strongly compact cardinals is strongly compact, but the least such limit is not supercompact. The consistency strength of strong compactness is strictly above that of a Woodin cardinal . Some set theorists conjecture that existence of a strongly compact cardinal
84-533: The area of large cardinals, Kunen is known for intricate forcing and combinatorial constructions. He proved that it is consistent that Martin's axiom first fails at a singular cardinal and constructed under the continuum hypothesis a compact L-space supporting a nonseparable measure. He also showed that P ( ω ) / F i n {\displaystyle P(\omega )/Fin} has no increasing chain of length ω 2 {\displaystyle \omega _{2}} in
96-408: The consistency of the existence of a huge cardinal . He introduced the method of iterated ultrapowers , with which he proved that if κ {\displaystyle \kappa } is a measurable cardinal with 2 κ > κ + {\displaystyle 2^{\kappa }>\kappa ^{+}} or κ {\displaystyle \kappa }
108-434: The number of operands for each operator to be less than κ; then κ is strongly compact if its logic satisfies an analog of the compactness property of finitary logic. Specifically, a statement which follows from some other collection of statements should also follow from some subcollection having cardinality less than κ. The property of strong compactness may be weakened by only requiring this compactness property to hold when
120-425: The original collection of statements has cardinality below a certain cardinal λ; we may then refer to λ-compactness. A cardinal κ is weakly compact if and only if it is κ-compact; this was the original definition of that concept. Strong compactness implies measurability , and is implied by supercompactness . Given that the relevant cardinals exist, it is consistent with ZFC either that the first measurable cardinal
132-568: The standard Cohen model where the continuum is ℵ 2 {\displaystyle \aleph _{2}} . The concept of a Jech–Kunen tree is named after him and Thomas Jech . The journal Topology and its Applications has dedicated a special issue to "Ken" Kunen, containing a biography by Arnold W. Miller , and surveys about Kunen's research in various fields by Mary Ellen Rudin , Akihiro Kanamori , István Juhász , Jan van Mill , Dikran Dikranjan , and Michael Kinyon . Strongly compact cardinal In set theory ,
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#1732855059473144-408: Was supervised by Dana Scott . Kunen showed that if there exists a nontrivial elementary embedding j : L → L of the constructible universe , then 0 exists. He proved the consistency of a normal, ℵ 2 {\displaystyle \aleph _{2}} -saturated ideal on ℵ 1 {\displaystyle \aleph _{1}} from
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