In
mathematics, a
cardinal number
In mathematics, cardinal numbers, or cardinals for short, are a generalization of the natural numbers used to measure the cardinality (size) of sets. The cardinality of a finite set is a natural number: the number of elements in the set. T ...
κ is called superstrong
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bi ...
there exists an
elementary embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one ofte ...
''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with
critical point κ and
⊆ ''M''.
Similarly, a cardinal κ is n-superstrong if and only if there exists an
elementary embedding In model theory, a branch of mathematical logic, two structures ''M'' and ''N'' of the same signature ''σ'' are called elementarily equivalent if they satisfy the same first-order ''σ''-sentences.
If ''N'' is a substructure of ''M'', one ofte ...
''j'' : ''V'' → ''M'' from ''V'' into a transitive inner model ''M'' with
critical point κ and
⊆ ''M''.
Akihiro Kanamori has shown that the consistency strength of an n+1-superstrong cardinal exceeds that of an
n-huge cardinal for each n > 0.
References
*
Set theory
Large cardinals
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