Remarkable Cardinal

In mathematics, a remarkable cardinal is a certain kind of large cardinal number.

A cardinal ''κ'' is called remarkable if for all regular cardinals ''θ'' > ''κ'', there exist ''π'', ''M'', ''λ'', ''σ'', ''N'' and ''ρ'' such that

# ''π'' : ''M'' → H_''θ'' is an elementary embedding
# ''M'' is countable and transitive
# ''π''(''λ'') = ''κ''
# ''σ'' : ''M'' → ''N'' is an elementary embedding with critical point ''λ''
# ''N'' is countable and transitive
# ''ρ'' = ''M'' ∩ Ord is a regular cardinal in ''N''
# ''σ''(''λ'') > ''ρ''
# ''M'' = ''H''_''ρ''^''N'', i.e., ''M'' ∈ ''N'' and ''N'' ⊨ "''M is the set of all sets that are hereditarily smaller than ρ''"

Equivalently, $\kappa$ is remarkable if and only if for every $\lambda>\kappa$ there is $\bar\lambda<\kappa$ such that in some forcing extension V /math>, there is an elementary embedding $j:V_^V\rightarrow V_\lambda^V$ satisfying $j(\operatorname(j))=\kappa$. Although the definition is similar to one of the definitions of supercompact cardinals, the elementary embedding here only has to exist in V /math>, not in $V$.

See also

*Hereditarily countable set

References

*
*

Large cardinals

{{settheory-stub