Axiom of determinacy: Difference between revisions
Small cardinals under AD
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Lightface and boldface analytic determinacy are actually significantly weaker than the existence of a Woodin cardinal, which is significantly weaker than \(\mathbf{\Pi}^1_n\)-determinacy for \(n > 1\). In particular, lightface analytic determinacy is equiconsistent with the existence of [[Zero sharp|\(0^\sharp\)]], and boldface analytic determinacy is equiconsistent with the existence of [[Sharp|\(r^\sharp\)]] for all real numbers \(r\).
Assuming AD, \(\aleph_1\) and \(\aleph_2\) are [[Measurable cardinal|measurable]], \(\aleph_n\) is singular for all \(2<n<\omega\), and \(\aleph_{\omega+1}\) is measurable.<ref>T. Jech, "About the Axiom of Choice". In ''Handbook of Mathematical Logic'', Studies in Logic and the Foundations of mathematical vol. 90, ed. J. Barwise (1977)</ref><sup>p. 369</sup>
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