In mathematics, an Erdős cardinal, also called a partition cardinal is a certain kind of large cardinal number introduced by Paul Erdős and András Hajnal (1958).
, there is a set of order type
In the notation of the partition calculus,
-Erdős if Under this definition, any cardinal larger than the least
The existence of zero sharp implies that the constructible universe
satisfies "for every countable ordinal
In fact, for every indiscernible
satisfies "for every ordinal
-Erdős cardinal in
( ω , α )
" (the Lévy collapse to make
-Erdős cardinal implies existence of zero sharp.
is the satisfaction relation for
(using ordinal parameters), then the existence of zero sharp is equivalent to there being an
-Erdős ordinal with respect to
-Erdős cardinal implies that the axiom of constructibility is false.
-Erdős cardinal is not weakly compact,[1]p. 39. nor is the least
-Erdős cardinal.[1]p.
-Erdős in every transitive model satisfying "
For a limit ordinal
, a cardinal
-Erdős if for every closed unbounded
An equivalent definition is that
of order-indiscernibles for the structure
such that: The least cardinal
to satisfy the partition relation
κ → ( α
-Erdős cardinal is an inaccessible limit of ineffable cardinals.
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