ⓘ Iterable cardinal
In mathematics, an iterable cardinal is a type of large cardinal introduced by Gitman, and Sharpe and Welch, and further studied by Gitman and Welch. Sharpe and Welch defined a cardinal κ to be iterable if every subset of κ is contained in a weak κ model M for which there exists an M ultrafilter on κ which allows for wellfounded iterations by ultrapowers of arbitrary length. Gitman gave a finer notion, where a cardinal κ is defined to be α iterable if ultrapower iterations only of length α are required to wellfounded.
 that 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
 field of set theory, a large cardinal property is a certain kind of property of transfinite cardinal numbers. Cardinals with such properties are, as the
 totally ineffable cardinals remarkable cardinals α  Erdos cardinals for countable α 0 not a cardinal γ  iterable γ  Erdos cardinals for uncountable
 In mathematics, a Mahlo cardinal is a certain kind of large cardinal number. Mahlo cardinals were first described by Paul Mahlo 1911, 1912, 1913 As
 whose elements are in M. Huge cardinals were introduced by Kenneth Kunen 1978 In what follows, jn refers to the n  th iterate of the elementary embedding
 mathematics, limit cardinals are certain cardinal numbers. A cardinal number λ is a weak limit cardinal if λ is neither a successor cardinal nor zero. This
 Woodin cardinals K is maximal, universal, and fully iterable This implies that for every iterable extender model M called a mouse there is an elementary
 the mathematical discipline of set theory, a cardinal characteristic of the continuum is an infinite cardinal number that may consistently lie strictly between
 showed that iterated forcing can construct models where Martin s axiom holds and the continuum is any given regular cardinal In iterated forcing, one
 Cardinal is the second studio album by American rock band Pinegrove, released February 12, 2016 on Run for Cover. Pinegrove formed in Montclair, New Jersey
 an added condition of iterability referring to the existence of wellfounded iterated ultrapowers a mouse is then an iterable premouse. The notion of
 alternative to large cardinal axioms. The Fundamental Theorem of Proper Forcing, due to Shelah, states that any countable support iteration of proper forcings

Large cardinal 
Axiom of determinacy 

Berkeley cardinal 
Core model 
Critical point (set theory) 
Extender (set theory) 

Extendible cardinal 
Grothendieck universe 
Huge cardinal 
Indescribable cardinal 
Ineffable cardinal 
Kunens inconsistency theorem 
Mahlo cardinal 
Measurable cardinal 

Rankintorank 
Reinhardt cardinal 
Remarkable cardinal 
Shelah cardinal 
Shrewd cardinal 
Strong cardinal 
Strongly compact cardinal 
Quasicompact cardinal 
Unfoldable cardinal 
Weakly compact cardinal 
Wholeness axiom 
Woodin cardinal 
Zero sharp 

Film 

Television show 

Game 

Sport 

Science 

Hobby 

Travel 

Technology 

Brand 

Outer space 

Cinematography 

Photography 

Music 

Literature 

Theatre 

History 

Transport 

Visual arts 

Recreation 

Politics 

Religion 

Nature 

Fashion 

Subculture 

Animation 

Award 

Interest 