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, $\kappa$ is remarkable if and only if for every $\lambda >\kappa$ there is ${\bar {\lambda }}<\kappa$ such that in some forcing extension $V[G]$, there is an elementary embedding $j:V_{\bar {\lambda }}^{V}\rightarrow V_{\lambda }^{V}$ 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

• Hereditarily countable set

References

• Schindler, Ralf (2000), "Proper forcing and remarkable cardinals", The Bulletin of Symbolic Logic, 6 (2): 176–184, CiteSeerX 10.1.1.297.9314, doi:10.2307/421205, ISSN 1079-8986, JSTOR 421205, MR 1765054, S2CID 1733698
• Gitman, Victoria (2016), Virtual large cardinals (PDF)