Remarkable cardinal

In mathematics, a remarkable cardinal is a certain kind of large cardinal number.

Formally, a cardinal κ is called remarkable iff for all regular cardinals θ > κ, there exist π, "M", λ, σ, "N" and ρ such that

# π : "M" → H_θ 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" = "H"_ρ^"N", i.e., "M" ∈ "N" and "N" |= "M is the set of all sets that are hereditarily smaller than ρ"

ee also

*Hereditarily countable set

References

*Schindler, Ralf: "Proper forcing and remarkable cardinals", Bulletin of Symbolic Logic 6, 2000, pp. 176-184 [http://www.math.ucla.edu/~asl/bsl/0602/0602-003.ps]