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−  [[Category:Large cardinal axioms]]
 +  #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 firstorder formula $\varphi$ such that $L\models\varphi(\aleph_0,\aleph_1...\aleph_n)$ for some $n$. Because of the [[stablestability]] of $\aleph_\omega$, $0^{\#}$ is equivalent to the set of all Gödel numberings of firstorder 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 firstorder definable in $L$, $X\in L_{\omega_1}$. This of course implies that $\aleph_1$ is not firstorder definable in $L$, because $\aleph_1\not\in L_{\omega_1}$. This is already a disproof of $V=L$ (because $\aleph_1$ is firstorder 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 [[ineffabletotally 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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