ⓘ 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
 π λ = κ
 M = H ρ N, i.e., M ∈ N and N ⊨ M is the set of all sets that are hereditarily smaller than ρ
 N is countable and transitive
 σ λ > ρ
 π: M → H θ is an elementary embedding
 M is countable and transitive
 ρ = M ∩ Ord is a regular cardinal in N
 σ: M → N is an elementary embedding with critical point λ
Equivalently, κ {\displaystyle \kappa } is remarkable if and only if for every λ > κ {\displaystyle \lambda > \kappa } there is λ ¯ < κ {\displaystyle {\bar {\lambda }}
 totally ineffable cardinals remarkable cardinals α  Erdos cardinals for countable α 0 not a cardinal γ  iterable, γ  Erdos cardinals for uncountable
 of 1  iterable cardinals which in turn is below remarkable cardinals which in turn is below ω  Erdos cardinals A list of large cardinal axioms by consistency
 is consistent, then ZFC no such cardinal exists is consistent. A remarkable observation about large cardinal axioms is that they appear to occur
 Louisville finished a remarkable season by defeating third  ranked Florida 33 23 in the Allstate Sugar Bowl, giving the Cardinals their 11th win for just
 Archbishop of New York from 1984 until his death in 2000, and was created a cardinal in 1985. He previously served as a U.S. Navy chaplain 1952 1979, including
 was an Italian Catholic priest and diplomat for the Holy See, who became Cardinal Secretary of State. He was the most important figure behind the Vatican s
 The 1948 Chicago Cardinals season was the 29th season in franchise history. The Cardinals won the Western division on the final weekend at Wrigley Field
 Measurable cardinal N  huge cardinal Ramsey cardinal Rank  into  rank Remarkable cardinal Shelah cardinal Strong cardinal Strongly inaccessible cardinal Subtle
 Cordeiro 19 January 1918   11 February 1994 was the first Pakistani cardinal Cordeiro was educated at St Patrick s High School, Karachi, the University

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Axiom of determinacy 

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