A weakly compact cardinal (WCC for short) is a certain type of large cardinals with many equivalent definitions, such as this one:
- Let \([x]^2\) be all the 2-element subsets of \(x\). Then an uncountable cardinal \(\alpha\) is weakly compact if and only if, for every function \(f: [\alpha]^2 \rightarrow \{0, 1\}\), there is a set \(S \subseteq \alpha\) such that \(|S| = \alpha\) and \(f\) maps every member of \([S]^2\) to either all 0 or all 1.
- More intuitively, any two-coloring of the edges of the complete graph \(K_\alpha\) contains a monochromatic subgraph isomorphic to \(K_\alpha\).
A WCC is always inaccessible and Mahlo. Thus they cannot be proven to exist in ZFC (assuming it is consistent), and ZFC + "there exists a WCC" is believed to be consistent.
The least WCC (if it exists) is sometimes called "the" weakly compact cardinal \(K\).
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To googologists, \(K\) and other WCCs are mostly useful through ordinal collapsing functions such as Rathjen's Ψ function, and ordinal notations associated to them. In ordinal collapsing functions, weakly inaccessible cardinals are used to diagonalise cardinals, weakly Mahlo cardinals are used to diagonalise weakly inaccessible cardinals, and the weakly compact cardinal is used to diagonalise weakly Mahlo cardinals. Therefore many googologists who do not know the definition of weakly compact cardinals often presume that there were a reasonable sequence \((\Omega,I,M,K,\ldots)\) and its diagonalisation which work in ordinal collapsing functions in a desired way, although there is not such a known sequence.
See also[]
Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function) · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function · Arai's psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · Berkeley cardinal · (Club Berkeley cardinal · limit club Berkeley cardinal)
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)