Reinhardt cardinal
Encyclopedia
In set theory
, a branch of mathematics
, a Reinhardt cardinal is a large cardinal κ, suggested by , that is the critical point
of a non-trivial elementary embedding j of V into itself.
A minor technical problem is that this property cannot be formulated in the usual set theory ZFC: the embedding j is a class of the form
for some set a and formula φ, and in the language of set theory it is not possible to quantify over all classes (or formulas). There are several ways to get round this. One way is to add a new function symbol j to the language of ZFC, together with axioms stating that j is an elementary embedding of V (and of course adding separation and replacement axioms for formulas involving j). Another way is to use a class theory
such as NBG or KM. A third way is to treat Kunen's theorem as a countable infinite collection of theorems, one for each formula φ. (It is possible to have elementary embeddings of models of ZFC into themselves assuming a mild large cardinal hypothesis, but these elementary embeddings are not classes of the model.)
proved Kunen's inconsistency theorem
showing that the existence of such an embedding contradicts NBG with the axiom of choice (and ZFC extended by j), but it is consistent with weaker class theories
. His proof uses the axiom of choice, and it is still an open question as to whether such an embedding is consistent with NBG without the axiom of choice (or with ZF plus the extra symbol j and its attendant axioms).
Reinhardt cardinals are essentially the largest ones that have been defined (as of 2006) that are not known to be inconsistent in ZF-set theory.
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, a branch of mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a Reinhardt cardinal is a large cardinal κ, suggested by , that is the critical point
Critical point (set theory)
In set theory, the critical point of an elementary embedding of a transitive class into another transitive class is the smallest ordinal which is not mapped to itself....
of a non-trivial elementary embedding j of V into itself.
A minor technical problem is that this property cannot be formulated in the usual set theory ZFC: the embedding j is a class of the form
for some set a and formula φ, and in the language of set theory it is not possible to quantify over all classes (or formulas). There are several ways to get round this. One way is to add a new function symbol j to the language of ZFC, together with axioms stating that j is an elementary embedding of V (and of course adding separation and replacement axioms for formulas involving j). Another way is to use a class theoryClass (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
such as NBG or KM. A third way is to treat Kunen's theorem as a countable infinite collection of theorems, one for each formula φ. (It is possible to have elementary embeddings of models of ZFC into themselves assuming a mild large cardinal hypothesis, but these elementary embeddings are not classes of the model.)
proved Kunen's inconsistency theorem
Kunen's inconsistency theorem
In set theory, a branch of mathematics, Kunen's inconsistency theorem, proved by , shows that several plausible large cardinal axioms are inconsistent with the axiom of choice.Some consequences of Kunen's theorem are:...
showing that the existence of such an embedding contradicts NBG with the axiom of choice (and ZFC extended by j), but it is consistent with weaker class theories
Class (set theory)
In set theory and its applications throughout mathematics, a class is a collection of sets which can be unambiguously defined by a property that all its members share. The precise definition of "class" depends on foundational context...
. His proof uses the axiom of choice, and it is still an open question as to whether such an embedding is consistent with NBG without the axiom of choice (or with ZF plus the extra symbol j and its attendant axioms).
Reinhardt cardinals are essentially the largest ones that have been defined (as of 2006) that are not known to be inconsistent in ZF-set theory.