Weakly compact cardinal: Difference between revisions
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.
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,
==References==
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Revision as of 21:24, 29 August 2023
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 \(\Pi^1_1\)-indescribable.
- \(\kappa\) is \(\kappa\)-unfoldable.
- The partition property \(\kappa \to (\kappa)^2_2\) holds.
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.
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 \(V_\kappa\),[1] i.e. a \(\Pi_2\) property of \(V_{\kappa+1}\),[2] a totally reflecting cardinal is larger than the least weakly compact cardinal.
References
- ↑ Kanamori, Akihiro (2003). The Higher Infinite: Large Cardinals in Set Theory from Their Beginnings (2nd ed.). Springer. p. 64. doi:10.1007/978-3-540-88867-3_2. ISBN 3-540-00384-3
- ↑ J. D. Hamkins, "Local properties in set theory" (2014), blog post. Accessed 29 August 2023.