Weakly compact cardinal: Difference between revisions
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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, 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. |
The existence of a weakly compact cardinal is not provable in ZFC, assuming its existence - 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, or even a \(\Pi^1_2\)-indescribable cardinal, is larger than the least weakly compact cardinal. |
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Note that, unlike the relation between weakly and strongly inaccessible cardinals, and weakly and strongly Mahlo cardinals, strongly compact cardinals are always significantly greater than weakly compact cardinals, both in terms of consistency strength and size. |
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==References== |
==References== |
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Revision as of 20:40, 30 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, assuming its existence - 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, or even a \(\Pi^1_2\)-indescribable cardinal, is larger than the least weakly compact cardinal.
Note that, unlike the relation between weakly and strongly inaccessible cardinals, and weakly and strongly Mahlo cardinals, strongly compact cardinals are always significantly greater than weakly compact cardinals, both in terms of consistency strength and size.
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.