In
set theory
Set theory is the branch of mathematical logic that studies Set (mathematics), sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory – as a branch of mathema ...
, a branch of mathematics, Kunen's inconsistency theorem, proved by , shows that several plausible
large cardinal
In the mathematical field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the name suggests, generally very "large" (for example, bigger than the least ...
axioms are
inconsistent
In deductive logic, a consistent theory is one that does not lead to a logical contradiction. A theory T is consistent if there is no formula \varphi such that both \varphi and its negation \lnot\varphi are elements of the set of consequences o ...
with the
axiom of choice
In mathematics, the axiom of choice, abbreviated AC or AoC, is an axiom of set theory. Informally put, the axiom of choice says that given any collection of non-empty sets, it is possible to construct a new set by choosing one element from e ...
.
Some consequences of Kunen's theorem (or its proof) are:
*There is no non-trivial
elementary embedding
In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one oft ...
of the universe ''V'' into itself. In other words, there is no
Reinhardt cardinal
In set theory, a Reinhardt cardinal is a kind of large cardinal. Reinhardt cardinals are considered under ZF (Zermelo–Fraenkel set theory without the Axiom of Choice), because they are inconsistent with ZFC (ZF with the Axiom of Choice). They wer ...
.
*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 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''
λ+2 into itself.
It is not known if Kunen's theorem still holds in ZF (ZFC without the axiom of choice), though 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'':
Kunen used
Morse–Kelley set theory
In the foundations of mathematics, Morse–Kelley set theory (MK), Kelley–Morse set theory (KM), Morse–Tarski set theory (MT), Quine–Morse set theory (QM) or the system of Quine and Morse is a first-order axiomatic set theory that is closely ...
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.
It is possible to have models of set theory that have elementary embeddings into themselves, at least if one assumes some mild large cardinal axioms. For example, if
0# exists then there is an elementary embedding from the
constructible universe
In mathematics, in set theory, the constructible universe (or Gödel's constructible universe), denoted by L, is a particular Class (set theory), class of Set (mathematics), sets that can be described entirely in terms of simpler sets. L is the un ...
''L'' into itself. This does not contradict Kunen's theorem because if 0
# exists then ''L'' cannot be the whole universe of sets.
See also
*
Rank-into-rank
In set theory, a branch of mathematics, a rank-into-rank embedding is a large cardinal property defined by one of the following four axioms given in order of increasing consistency strength. (A set of rank < \lambda is one of the elements o ...
References
*
*
*
*
Large cardinals
Theorems in the foundations of mathematics
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