ⓘ Reinhardt cardinal. In mathematical set theory, a Reinhardt cardinal is a large cardinal κ in a model of ZF, Zermelo–Fraenkel set theory without the axiom of ch ..


ⓘ Reinhardt cardinal

In mathematical set theory, a Reinhardt cardinal is a large cardinal κ in a model of ZF, Zermelo–Fraenkel set theory without the axiom of choice. They were suggested by American mathematician William Nelson Reinhardt.


1. Definition

A Reinhardt cardinal 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, which in ZFC means something of the form { x | ϕ x, a } {\displaystyle \{x|\phi x,a\}} for some set a and formula φ, but in the language of set theory it is not possible to quantify over all classes or define the truth of 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 would be to treat Kunens theorem as a countable infinite collection of theorems, one for each formula φ, but that would trivialize the theorem. It is possible to have nontrivial elementary embeddings of transitive models of ZFC into themselves assuming a mild large cardinal hypothesis, but these elementary embeddings are not classes of the model.


2. Kunens theorem

Kunen 1971 proved Kunens 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.


3. Stronger cardinal axioms

There are some variations of Reinhardt cardinals. In ZF, there is a hierarchy of hypotheses asserting existence of elementary embeddings V→V J3: There is a nontrivial elementary embedding j: V→V J2: There is a nontrivial elementary embedding j: V→V, and DC λ holds, where λ is the least fixed-point above the critical point. J1: There is a cardinal κ such that for every α, there is an elementary embedding j: V→V with jκ> α and cpj = κ.

J2 implies J3, and J1 implies J3 and also implies consistency of J2. By adding a generic well-ordering of V to a model of J1, one gets ZFC plus a nontrivial elementary embedding of HOD into itself.

Berkeley cardinals are a stronger large cardinals suggested by Woodin.

  • the axiom of choice Reinhardt cardinal Berkeley cardinal Drake, F. R. 1974 Set Theory: An Introduction to Large Cardinals Studies in Logic and
  • extendible cardinals are large cardinals introduced by Reinhardt 1974 who was partly motivated by reflection principles. Intuitively, such a cardinal represents
  • essentially the strongest known large cardinal axioms not known to be inconsistent in ZFC the axiom for Reinhardt cardinals is stronger, but is not consistent
  • 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
  • with critical point below κ. Berkeley cardinals are a strictly stronger cardinal axiom than Reinhardt cardinals implying that they are not compatible
  • 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
  • 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
  • independently considered by H. Jerome Keisler, and was written up by Solovay, Reinhardt Kanamori 1978 According to Pudlak 2013, p. 204 Vopenka s principle
  • Lecture Notes in Mathematics, 669, Springer, 99 275. R. M. Solovay, W. N. Reinhardt A. Kanamori: Strong axioms of infinity and elementary embeddings, Annals
  • Replacement Axiom for j - formulas Thus, the wholeness axiom differs from Reinhardt cardinals another way of providing elementary embeddings from V to itself
  • Spirit for the life of individuals and of the church. He was created a cardinal of the Catholic Church in 1994. Congar was born in Sedan in northeast France