- 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 . Note that, although the definition is similar to one of the definitions of supercompact cardinals, the elementary embedding here only has to exist in , not in .
- 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, MR 1765054
- Gitman, Victoria (2016), Virtual large cardinals (PDF)
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