Talk:Countability: Difference between revisions
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== "Furthermore, a gap ordinal may have a map to N but this map can not be defined at all using first-order set theory" == |
== "Furthermore, a gap ordinal may have a map to N but this map can not be defined at all using first-order set theory" == |
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Since \(\beta_0\) can be defined in the language of first-order set theory I think there is an injection from it to \(\mathbb N\). Since \(\beta_0\) is in \(L_{\textrm{least stable ordinal}}\), by theorem 7.8 of chapter V of Marek's "[http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and related facts]" (Fundamenta Mathematicae vol. 82, 1974), there is a \(\Sigma_1\) definition of \(\beta_0\) in \(V\) (i.e. there is a \(\Sigma_1\) formula \(\phi(x)\) such that \(\forall x(\phi(x)\leftrightarrow x=\beta_0)\)). Every ordinal \(<\beta_0\) has a \(\Sigma_1\) definition in \(V\) as well. Since satisfaction of \(\Sigma_1\) formulae in \(V\) is definable, the map that takes an ordinal \(\alpha<\beta_0\) to the smallest Gödel- |
Since \(\beta_0\) can be defined in the language of first-order set theory I think there is an injection from it to \(\mathbb N\). Since \(\beta_0\) is in \(L_{\textrm{least stable ordinal}}\), by theorem 7.8 of chapter V of Marek's "[http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of β₂-models of full second order arithmetic and related facts]" (Fundamenta Mathematicae vol. 82, 1974), there is a \(\Sigma_1\) definition of \(\beta_0\) in \(V\) (i.e. there is a \(\Sigma_1\) formula \(\phi(x)\) such that \(\forall x(\phi(x)\leftrightarrow x=\beta_0)\)). Every ordinal \(<\beta_0\) has a \(\Sigma_1\) definition in \(V\) as well. Since satisfaction of \(\Sigma_1\) formulae in \(V\) is definable, the map that takes an ordinal \(\alpha<\beta_0\) to the smallest Gödel-code of a \(\Sigma_1\) formula defining \(\alpha\) is a definable map. [[User:C7X|C7X]] ([[User talk:C7X|talk]]) 01:57, 8 September 2023 (UTC) |
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Latest revision as of 01:57, 8 September 2023
"Furthermore, a gap ordinal may have a map to N but this map can not be defined at all using first-order set theory"[edit source]
Since \(\beta_0\) can be defined in the language of first-order set theory I think there is an injection from it to \(\mathbb N\). Since \(\beta_0\) is in \(L_{\textrm{least stable ordinal}}\), by theorem 7.8 of chapter V of Marek's "Stable sets, a characterization of β₂-models of full second order arithmetic and related facts" (Fundamenta Mathematicae vol. 82, 1974), there is a \(\Sigma_1\) definition of \(\beta_0\) in \(V\) (i.e. there is a \(\Sigma_1\) formula \(\phi(x)\) such that \(\forall x(\phi(x)\leftrightarrow x=\beta_0)\)). Every ordinal \(<\beta_0\) has a \(\Sigma_1\) definition in \(V\) as well. Since satisfaction of \(\Sigma_1\) formulae in \(V\) is definable, the map that takes an ordinal \(\alpha<\beta_0\) to the smallest Gödel-code of a \(\Sigma_1\) formula defining \(\alpha\) is a definable map. C7X (talk) 01:57, 8 September 2023 (UTC)