Zero sharp: Difference between revisions
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While "\(0^\sharp\) exists" does not at face value seem to imply the failure of the axiom of constructibility, clauses 2, 3, 5 and 6 more clearly show that this is the case. Also, "\(0^\sharp\) exists" strictly implies the following: |
While "\(0^\sharp\) exists" does not at face value seem to imply the failure of the axiom of constructibility, clauses 2, 3, 5 and 6 more clearly show that this is the case. Also, "\(0^\sharp\) exists" strictly implies the following: |
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* \(\aleph_\omega\) is totally stable - in fact, every uncountable cardinal is totally stable. |
* \(\aleph_\omega\) is totally stable (stable for first-order formulae?) - in fact, every uncountable cardinal is totally stable. |
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* If \(X \in L\) and \(X\) is definable in \(L\) without parameters, then \(X \in L_{\omega_1}\). |
* If \(X \in L\) and \(X\) is definable in \(L\) without parameters, then \(X \in L_{\omega_1}\). |
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* There are only countably many constructible reals - i.e. \(\mathbb{R} \cap L\) is countable. |
* There are only countably many constructible reals - i.e. \(\mathbb{R} \cap L\) is countable. |
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