Erdős cardinal
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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).
A cardinal is called
-Erdős if for every function
, there is a set of order type
that is homogeneous for
. In the notation of the partition calculus,
is
-Erdős if
.
The existence of zero sharp implies that the constructible universe satisfies "for every countable ordinal
, there is an
-Erdős cardinal". In fact, for every indiscernible
,
satisfies "for every ordinal
, there is an
-Erdős cardinal in
" (the Lévy collapse to make
countable).
However, the existence of an -Erdős cardinal implies existence of zero sharp. If
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
. Thus, the existence of an
-Erdős cardinal implies that the axiom of constructibility is false.
The least -Erdős cardinal is not weakly compact,[1]p. 39. nor is the least
-Erdős cardinal.[1]p. 39
If is
-Erdős, then it is
-Erdős in every transitive model satisfying "
is countable."