Zero sharp: Difference between revisions
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* \(0^\sharp\) exists.
* There is an uncountable set \(X\) of ordinals with no \(Y \in L\) so that \(X \subseteq Y\) and \(|X| = |Y|\) (\(L\) does not have the [[covering property]]).<ref>Any text about Jensen's covering theorem
* For all \(\alpha\), \(|V_\alpha \cap L| = |L_\alpha|\).
* Every game whose payoff set is a \(\Sigma^1_1\) subset of Baire space is [[Axiom of determinacy|determined]].
* \(\aleph_\omega^V\) is regular in \(L\).
* There is a nontrivial elementary embedding \(j: L \to L\).<ref>Many papers about 0 sharp</ref>
* There is a proper class of nontrivial elementary
* For some \(\alpha, \beta\), there is a nontrivial elementary embedding \(j: L_\alpha \to L_\beta\) with critical point below \(|\alpha|\).
* Every uncountable cardinal is inaccessible in \(L\).<ref name=":0">W. H. Woodin, [https://arxiv.org/abs/1605.00613 The HOD dichotomy], p.1</ref>
* There is a singular cardinal \(\gamma\) so that \((\gamma^+)^L < \gamma\).<ref name=":0" />
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:
* \(\aleph_\omega\) is totally stable (stable for first-order formulae?) - in fact, every uncountable cardinal is totally stable.
* If \(X \in L\) and \(X\) is definable in \(L\) without parameters, then \(X \in L_{\omega_1}\).
* There are only countably many constructible reals - i.e. \(\mathbb{R} \cap L\) is countable.
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If \(0^\sharp\) exists, then it is defined as the (real corresponding to) the set of \(\ulcorner \varphi \urcorner\) where \(\varphi\) is a first-order formula with \(n\) free variables and \(L_{\aleph_\omega} \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Since \(0^\sharp\) exists, this is equivalent to \(L \models \varphi(\aleph_0, \aleph_1, \cdots, \aleph_n)\). Note that, since \(L_{\aleph_\omega}\) is a set, \(\mathrm{ZFC}\) can define a truth predicate for it, and so the existence of \(0^\sharp\) as a mere set of formulas is trivial. It is interesting only when there are is a proper class of Silver indiscernibles.
Alternatively, \(0^\sharp\) may be defined as a sound mouse (iterable premouse), or as an Ehrenfeucht-Mostowski blueprint.
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