en>Headbomb: Various citation cleanup and WP:AWB general fixes using AWB

2011-09-17T03:12:01Z

Various citation cleanup and WP:AWB general fixes using AWB

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In [[set theory]], a branch of mathematics, '''Kunen's inconsistency theorem''', proved by {{harvs|txt|first=Kenneth|authorlink=Kenneth Kunen|last=Kunen|year=1971}}, shows that several plausible [[large cardinal]] axioms are [[Consistency|inconsistent]] with the [[axiom of choice]].

Some consequences of Kunen's theorem are:
*There is no non-trivial [[Elementary equivalence|elementary embedding]] of the universe ''V'' into itself. In other words, there is no [[Reinhardt cardinal]].
*If ''j'' is an elementary embedding of the universe ''V'' into an inner model ''M'', and λ is the smallest fixed point of ''j'' above the [[Critical point (set theory)|critical point]] κ of ''j'', then ''M'' does not contain the set ''j'' "λ (the image of ''j'' restricted to λ).
*There is no [[ω-huge cardinal]].
*There is no non-trivial elementary embedding of ''V''<sub>λ+2</sub> into itself.

It is not known if Kunen's theorem still holds in ZF (ZFC without the axiom of choice), though {{harvtxt|Suzuki|1999}} showed that there is no definable elementary embedding from ''V'' into ''V''. That is there is no formula ''J'' in the language of set theory such that for some parameter ''p''∈''V'' for all sets ''x''∈''V'' and ''y''∈''V'': <math>j(x)=y \leftrightarrow J(x,y,p) \,.</math>

Kunen used [[Morse–Kelley set theory]] in his proof. If the proof is re-written to use ZFC, then one must add the assumption that replacement holds for formulas involving ''j''. Otherwise one could not even show that ''j'' "λ exists as a set. The forbidden set ''j'' "λ is crucial to the proof. The proof first shows that it cannot be in ''M''. The other parts of the theorem are derived from that.

==See also==
*[[Rank-into-rank]]

==References==
*{{Citation | last1=Kanamori | first1=Akihiro | title=The Higher Infinite : Large Cardinals in Set Theory from Their Beginnings | publisher=[[Springer-Verlag]] | location=Berlin, New York | edition=2nd | isbn=978-3-540-00384-7 | doi=10.1007/978-3-540-88867-3 | year=2003}}
*{{citation
|last=Kunen|first= Kenneth
|title=Elementary embeddings and infinitary combinatorics
|journal=J. Symbolic Logic |volume=36 |year=1971|pages= 407–413
|doi=10.2307/2269948|jstor=2269948
|mr=0311478
|issue=3}}
*{{Citation | last1=Suzuki | first1=Akira | title=No elementary embedding from V into V is definable from parameters | id={{MathSciNet | id = 1780073}} | year=1999 | journal=The Journal of Symbolic Logic | issn=0022-4812 | volume=64 | issue=4 | pages=1591–1594 | doi=10.2307/2586799}}
*{{Citation | last1=Zapletal | first1=Jindřich | title=A new proof of Kunen's inconsistency | id={{MathSciNet | id = 1317054}} | year=1996 | journal=[[Proceedings of the American Mathematical Society]] | issn=0002-9939 | volume=124 | issue=7 | pages=2203–2204 | doi=10.1090/S0002-9939-96-03281-9}}

[[Category:Large cardinals]]

{{settheory-stub}}

Chang's conjecture - Revision history

en>Headbomb: Various citation cleanup and WP:AWB general fixes using AWB