Difference between revisions of "Zero sharp"

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[[Category:Large cardinal axioms]]
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#REDIRECT [[Constructible universe#Sharps]]
[[Category:Constructibility]]
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$0^{\#}$ is a [[projective|$\Sigma_3^1$]] real number which cannot be proven to exist in [[ZFC|$\text{ZFC}$]]. It's existence contradicts the [[Axiom of constructibility]], $V=L$. In fact, it's existence is somewhat equivalent to $L$ being completely different from $V$.
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== Definition ==
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$0^{\#}$ is defined as the set of all Gödel numberings of first-order formula $\varphi$ such that $L\models\varphi(\aleph_0,\aleph_1...\aleph_n)$ for some $n$. Because of the [[stable|stability]] of $\aleph_\omega$, $0^{\#}$ is equivalent to the set of all Gödel numberings of first-order formula $\varphi$ such that $L_{\aleph_{\omega}}\models\varphi(\aleph_0,\aleph_1...\aleph_n)$. This definition implies the existence of Silver Indiscernables. Moreover, it implies:
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*Given any set $X\in L$ which is first-order definable in $L$, $X\in L_{\omega_1}$. This of course implies that $\aleph_1$ is not first-order definable in $L$, because $\aleph_1\not\in L_{\omega_1}$. This is already a disproof of $V=L$ (because $\aleph_1$ is first-order definable).
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*For every $\alpha\in\omega_1^L$, every uncountable cardinal is [[Ramsey#iterable|$\alpha$-iterable]], $\geq$ an [[Erdos|$\alpha$-Erdős]], and [[ineffable|totally ineffable]] in $L$.
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*There are $\mathfrak{c}$ many reals which are not constructible (that is, $x\not\in L$).
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The existence of $0^\#$ is implied by:
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* Chang's Conjecture.
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* The existence of an $\omega_1$-iterable cardinal.
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* The negation of the singular cardinal hypothesis ($\text{SCH}$).
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* The [[axiom of determinacy]] ($\text{AD}$).
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== $0^{\#}$ cardinal ==
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$0^{\#}$ exists iff there is a nontrivial [[elementary embedding]] $j:L\rightarrow L$ (by a theorem of Kunen). The critical point of such an embedding is sometimes called a $0^{\#}$ cardinal, and sometimes called a $j:L\rightarrow L$ cardinal. These cardinals do not coincide with measurable cardinals for a long time. While the least measurable cardinal is [[indescribable|$\Sigma_1^2$-describable]], each of these cardinals is totally indescribable. Furthermore, the least measurable cardinal $\kappa$ such that $V_\kappa$ satisfies the existence of a measurable cardinal is not a $j:L\rightarrow L$ cardinal, and the least measurable cardinal $\kappa$ such that $V_\kappa$ satisfies the existence of such a cardinal is not a $j:L\rightarrow L$ cardinal, and so on.
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However, the existence of a measurable suffices to prove the existence and consistency of a $j:L\rightarrow L$ cardinal.
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''More information to be added here.''
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== References ==
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*Jech, Thomas J. Set Theory (The 3rd Millennium Ed.). Springer, 2003.
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Latest revision as of 06:38, 24 November 2017