Remarkable cardinal explained
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 → Hθ 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 = HρN, i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than ρ"
Equivalently,
is remarkable if and only if for every
there is
such that in some
forcing extension
, there is an elementary embedding
satisfying
j(\operatorname{crit}(j))=\kappa
. Although the definition is similar to one of the definitions of
supercompact cardinals, the elementary embedding here only has to exist in
, not in
.
See also