Compact Cardinal

	Compact Cardinal Compact cardinal may refer to: * Weakly compact cardinal * Subcompact cardinal * Supercompact cardinal * Strongly compact cardinal {{Short pages monitor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Weakly Compact Cardinal In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally called them "not strongly incompact" cardinals.) Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function ''f'': � 2 → there is a set of cardinality κ that is homogeneous for ''f''. In this context, � 2 means the set of 2-element subsets of κ, and a subset ''S'' of κ is homogeneous for ''f'' if and only if either all of 'S''sup>2 maps to 0 or all of it maps to 1. The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below. Every weakly compact cardinal is a reflecting cardinal, and is also a limit of reflecting cardinals. This means also that weakly compact c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Subcompact Cardinal In mathematics, a subcompact cardinal is a certain kind of large cardinal number. A cardinal number ''κ'' is subcompact if and only if for every ''A'' ⊂ ''H''(''κ''+) there is a non-trivial elementary embedding j:(''H''(''μ''+), ''B'') → (''H''(''κ''+), ''A'') (where ''H''(''κ''+) is the set of all sets of cardinality hereditarily less than ''κ''+) with critical point ''μ'' and ''j''(''μ'') = ''κ''. Analogously, ''κ'' is a quasicompact cardinal if and only if for every ''A'' ⊂ ''H''(''κ''+) there is a non-trivial elementary embedding ''j'':(''H''(''κ''+), ''A'') → (''H''(''μ''+), ''B'') with critical point ''κ'' and ''j''(''κ'') = ''μ''. ''H''(''λ'') consists of all sets whose transitive closure has cardinality less than ''λ''. Every quasicompact cardinal is subcompact. Quasicompactness is a strengthening of subcompactness in that it projects large cardinal properties upwards. The relationship is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Supercompact Cardinal In set theory, a supercompact cardinal is a type of large cardinal. They display a variety of reflection properties. Formal definition If ''λ'' is any ordinal, ''κ'' is ''λ''-supercompact means that there exists an elementary embedding ''j'' from the universe ''V'' into a transitive inner model ''M'' with critical point ''κ'', ''j''(''κ'')>''λ'' and :^\lambda M\subseteq M \,. That is, ''M'' contains all of its ''λ''-sequences. Then ''κ'' is supercompact means that it is ''λ''-supercompact for all ordinals ''λ''. Alternatively, an uncountable cardinal ''κ'' is supercompact if for every ''A'' such that , ''A'', ≥ ''κ'' there exists a normal measure over 'A''sup>< ''κ'' with the additional property that every function $\to A$ such that $\ \in U$ is constant on a set in $U$ . Here "const ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]