In
set theory, Berkeley cardinals are certain
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 Î ...
s suggested by
Hugh Woodin
William Hugh Woodin (born April 23, 1955) is an American mathematician and set theorist at Harvard University. He has made many notable contributions to the theory of inner models and determinacy. A type of large cardinals, the Woodin cardinals ...
in a seminar at the
University of California, Berkeley in about 1992.
A Berkeley cardinal is a cardinal ''κ'' in a model of
Zermelo–Fraenkel set theory with the property that for every
transitive set ''M'' that includes ''κ'' and α < κ, there is a nontrivial
elementary embedding of ''M'' into ''M'' with α <
critical point < ''κ''.
Berkeley cardinals are a strictly stronger cardinal axiom than
Reinhardt cardinal In set theory, a branch of mathematics, 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 A ...
s, implying that they are not compatible with the
axiom of choice.
A weakening of being a Berkeley cardinal is that for every binary relation ''R'' on ''V''
''κ'', there is a nontrivial elementary embedding of (''V''
''κ'', ''R'') into itself. This implies that we have elementary
: ''j''
1, ''j''
2, ''j''
3, ...
: ''j''
1: (''V''
''κ'', ∈) → (''V''
''κ'', ∈),
: ''j''
2: (''V''
''κ'', ∈, ''j''
1) → (''V''
''κ'', ∈, ''j''
1),
: ''j''
3: (''V''
''κ'', ∈, ''j''
1, ''j''
2) → (''V''
''κ'', ∈, ''j''
1, ''j''
2),
and so on. This can be continued any finite number of times, and to the extent that the model has dependent choice, transfinitely. Thus, plausibly, this notion can be strengthened simply by asserting more dependent choice.
While all these notions are incompatible with Zermelo–Fraenkel set theory (ZFC), their
consequences do not appear to be false. There is no known inconsistency with ZFC in asserting that, for example:
For every ordinal λ, there is a transitive model of ZF + Berkeley cardinal that is closed under λ sequences.
See also
*
List of large cardinal properties
This page includes a list of cardinals with large cardinal properties. It is arranged roughly in order of the consistency strength of the axiom asserting the existence of cardinals with the given property. Existence of a cardinal number κ of a g ...
References
Sources
*
*{{citation, url=http://logic.harvard.edu/blog/wp-content/uploads/2014/11/Deep_Inconsistency.pdf, first=Peter , last=Koellner, title=The Search for Deep Inconsistency, year=2014
Large cardinals