Editing Remarkable cardinal

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

A [[cardinal number|cardinal]] ''κ'' is called remarkable if for all [[regular cardinal]]s ''θ'' > ''κ'', there exist ''π'', ''M'', ''λ'', ''σ'', ''N'' and ''ρ'' such that

# ''π'' : ''M'' → ''H''<sub>''θ''</sub> is an [[elementary embedding]]
# ''M'' is [[countable]] and [[transitive set|transitive]]
# ''π''(''λ'') = ''κ''
# ''σ'' : ''M'' → ''N'' is an elementary embedding with [[critical point (set theory)|critical point]] ''λ''
# ''N'' is countable and transitive
# ''ρ'' = ''M'' ∩ '''[[Ordinal number|Ord]]''' is a [[regular cardinal]] in ''N''
# ''σ''(''λ'') &gt; ''ρ''
# ''M'' = ''H''<sub>''ρ''</sub><sup>''N''</sup>, i.e., ''M'' ∈ ''N'' and ''N'' ⊨ "''M is the set of all sets that are hereditarily smaller than &rho;''"

Equivalently, <math>\kappa</math> is remarkable if and only if for every <math>\lambda>\kappa</math> there is <math>\bar\lambda<\kappa</math> such that in some [[Forcing (set theory)|forcing]] extension <math>V[G]</math>, there is an elementary embedding <math>j:V_{\bar\lambda}^V\rightarrow V_\lambda^V</math> satisfying <math>j(\operatorname{crit}(j))=\kappa</math>. Although the definition is similar to one of the definitions of [[supercompact cardinal]]s, the elementary embedding here only has to exist in <math>V[G]</math>, not in <math>V</math>.

==See also==
*[[Hereditarily countable set]]

==References==
{{refbegin}}
*{{Citation | last1=Schindler | first1=Ralf | title=Proper forcing and remarkable cardinals | url=https://www.math.ucla.edu/~asl/bsl/0602/0602-003.ps | doi=10.2307/421205 | mr=1765054  | year=2000 | journal=The Bulletin of Symbolic Logic | issn=1079-8986 | volume=6 | issue=2 | pages=176–184| jstor=421205 | citeseerx=10.1.1.297.9314 | s2cid=1733698 }}
*{{Citation | last1=Gitman | first1=Victoria | title=Virtual large cardinals | url=http://nylogic.org/wp-content/uploads/virtualLargeCardinals.pdf  | year=2016 }}
{{refend}}

[[Category:Large cardinals]]


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