Weakly compact cardinal: Difference between revisions
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RhubarbJayde (talk | contribs) (Created page with "A weakly compact cardinal is a certain kind of large cardinal. They were originally defined via a certain generalization of the compactness theorem for first-order logic to certain infinitary logics. However, this is a relatively convoluted definition, and there are a variety of equivalent definitions. These include, letting \(\kappa\) be the cardinal in question and assuming \(\kappa^{< \kappa} = \kappa\): * \(\kappa\) is 0-Ramsey. * \(\kappa\) is \(\Pi^1_1\)-indes...") |
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Condition number 4 could be rewritten as \(R(\kappa, \kappa) = \kappa\), where \(R\) is a transfinitary extension of the function used in Ramsey's theorem. |
Condition number 4 could be rewritten as \(R(\kappa, \kappa) = \kappa\), where \(R\) is a transfinitary extension of the function used in Ramsey's theorem. |
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The existence of a weakly compact cardinal is not provable in ZFC - however, if they do, they are very large. In particular, they are inaccessible, Mahlo, \(1\)-Mahlo, hyper-Mahlo and more. However, |
The existence of a weakly compact cardinal is not provable in ZFC - however, if they do, they are very large. In particular, they are inaccessible, Mahlo, \(1\)-Mahlo, hyper-Mahlo and more. However, since "\(\kappa\) is weakly compact" is a \(\Pi^1_2\) property of \(V_\kappa\),<ref>Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. p. 64. doi:[https://doi.org/10.1007%2F978-3-540-88867-3_2 10.1007/978-3-540-88867-3_2]. ISBN 3-540-00384-3</ref> i.e. a \(\Pi_2\) property of \(V_{\kappa+1}\),<ref>J. D. Hamkins, "[https://jdh.hamkins.org/local-properties-in-set-theory/ Local properties in set theory]" (2014), blog post. Accessed 29 August 2023.</ref> a totally reflecting cardinal is larger than the least weakly compact cardinal. |
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