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

1. π : M → H_θ is an elementary embedding
2. M is countable and transitive
3. π(λ) = κ
4. σ : M → N is an elementary embedding with critical point λ
5. N is countable and transitive
6. ρ = M ∩ Ord is a regular cardinal in N
7. σ(λ) > ρ
8. M = H_ρ^N, i.e., M ∈ N and N ⊨ "M is the set of all sets that are hereditarily smaller than ρ"

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

See also

• Hereditarily countable set

References

• Schindler, Ralf (2000), "Proper forcing and remarkable cardinals", The Bulletin of Symbolic Logic 6 (2): 176–184, doi:10.2307/421205, ISSN 1079-8986, https://www.math.ucla.edu/~asl/bsl/0602/0602-003.ps 
• Gitman, Victoria (2016), Virtual large cardinals, http://nylogic.org/wp-content/uploads/virtualLargeCardinals.pdf