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

  1. π : MHθ is an elementary embedding
  2. M is countable and transitive
  3. π(λ) = κ
  4. σ : MN is an elementary embedding with critical point λ
  5. N is countable and transitive
  6. ρ = MOrd is a regular cardinal in N
  7. σ(λ) > ρ
  8. M = HρN, i.e., MN 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

λ>\kappa

there is

\barλ<\kappa

such that in some forcing extension

V[G]

, there is an elementary embedding
V
j:V
\barλ
V
V
λ
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

V[G]

, not in

V

.

See also