Remarkable cardinal

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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, 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 .

See also[edit]


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