$0^{\sharp}$ (zero sharp) is a $\Sigma_3^1$ real number (see [[Projective|projective hierarchy]]) which, under the existence of many [[Indiscernibles]] (a statement independent of $\text{ZFC}$), has a certain number of properties that contradicts the <a href="L" class="mw-redirect" title="L">axiom of constructibility</a> and implies that, in short, $L$ and $V$ are "*very different*". Technically, under the standard definition of $0^\sharp$ as a (real number encoding a) set of formulas, $0^\sharp$ provably exists in $\text{ZFC}$, but lacks all its important properties. Thus the expression "$0^\sharp$ exists" is to be understood as "$0^\sharp$ exists *and* there are uncountably many Silver indiscernibles". ### Definition of $0^{\sharp}$ We say that *$0^{\sharp}$ exists* if for some ordinal $\delta$, there is an uncountable set of [[indiscernibles]] for $L_{\delta}$. This implies that there is a class of [[Indiscernibles#Silver indiscernibles|Silver indiscernibles]], and that for every uncountable cardinals $\kappa<\lambda$, $L_{\kappa}\prec L_{\lambda} \prec L$. Then $0^{\sharp}$ is defined as the set of all Gödel numberings of first-order formula $\varphi$ such that $L_{\aleph_{\omega}}\models\varphi(\aleph_1,...,\aleph_n)$ for some $n$. ### Equivalences of $0^♯s existence The following statements are equivalent: - There is an uncountable set of Silver indiscernibles (i.e. "$0^♯$ exists") - There is a proper class of Silver indiscernibles (unboundedly many of them). - There is a unique well-founded remarkable E.M. set (see below). - There is a nontrivial [[Elementary embedding]] $j:L\to L$. - There is a proper class of nontrivial elementary embeddings $j:L\to L$. - There is a nontrivial elementary embedding $j:L_\alpha\to L_\beta$ with $\text{crit}(j)<\|\alpha\|$. - Jensen's Covering Theorem fails (see below). - $L$ is thin, i.e. $\|L\cap V_\alpha\|=\|\alpha\|$ for all $\alpha\geq\omega$. - $\Sigma^1_1$-<a href="Axiom_of_projective_determinacy" class="mw-redirect" title="Axiom of projective determinacy">determinacy</a> (lightface form). - $\aleph_\omega$ is regular (hence weakly inaccessible) in $L$. ### Consequences of $0^♯s existence If $0^♯$ exists then: - $L_{\aleph_\omega}\prec L$ and so $0^♯$ also corresponds to the set of the Gödel numberings of first-order formulas $\varphi$ such that $L\models\varphi(\aleph_1...\aleph_n)$ - In fact, $L_\kappa\prec L$ for every Silver indiscernible, and thus for every uncountable cardinal. - 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). - For every $\alpha\in\omega_1^L$, every Silver indiscernible (and in particular every uncountable cardinal) in $L$ is a Silver cardinal, $\alpha$-[[Ramsey#$ alpha$-iterable cardinal|iterable]], $\geq$ an $\alpha$-[[Erdos|Erdős]], [[Ineffable]] and [[Remarkable#Completely remarkable $n$-remarkable cardinals|completely remarkable]] and has most other virtual large cardinal properties and other large cardinal properties consistent with $V=L$.{% cite Gitmana Bagaria2017a %} - There are only countably many reals in $L$, i.e. $\|\mathbb{R}\cap L\|=\aleph_0$ in $V$. - By [[Elementary embedding#Absoluteness]] (The hypothesis can be weakened, because one can chop at off the universe at any Silver indiscernible and use reflection.):{% cite Gitman2018 %} - $L$, equipped with only its definable classes, is a model of the <a href="Generic_Vop%C4%9Bnka%27s_Principle" class="mw-redirect" title="Generic Vopěnka&#39;s Principle">generic Vopěnka principle</a>. - In $L$ there are numerous [[Rank_into_rank.md#Virtually_rank-into-rank|virtually rank-into-rank]] embeddings $j : V_θ^L → V_θ^L$, where $θ$ is far above the supremum of the critical sequence. - Therefore every Silver indiscernible - is [[Extendible.md#Virtually_extendible_cardinals|virtually extendible]] in $L$ for every definable class $A$ - and is the critical point of virtual rank-into-rank embeddings with targets as high as desired and fixed points as high above the critical sequence as desired. - There is a class-forcing notion $\mathbb{P}$ definable in $L$, such that in any $L$-generic extension $L[C]$ by this forcing, $\text{GBC}$ and the generic Vopěnka principle hold, yet [[ORD_is_Mahlo]].{% cite Gitman2018 %} - Proof includes a lemma stating: For any ordinal $δ$ and any natural number (of the meta-theory — this lemma is a scheme) $n$, if $D_{δ,n} ⊂ \mathbb{P}$ is the collection of conditions $c$ for which there is an ordinal $θ$ such that - $L_θ ≺_{Σ_n} L$, - $c ∩ θ$ is $L_θ$-generic for $\mathbb{P}^{L_θ}$ and - in some forcing extension of $L$, there is an elementary embedding $j : ⟨ L_θ , ∈, c ∩ θ ⟩ → ⟨ L_θ , ∈, c ∩ θ ⟩$ with critical point above $δ$, then $D_{δ,n}$ is a definable dense subclass of $\mathbb{P}$ in $L$. - There is a definable class-forcing notion in $L$, such that in the corresponding $L$-generic extension, $\text{GBC}$ holds, the generic Vopěnka scheme holds, but $\text{Ord}$ is not definably Mahlo, because there is a $∆_2$-definable club class avoiding the regular cardinals. - There is a class-forcing extension $L[G]$ of the constructible universe in which the generic Vopěnka principle holds (so $gVP(κ, \mathbf{Σ_{n+1}})$ and $gVP(Π_n)$ hold for any $κ$ and $n$), but there are no $Σ_2$-reflecting cardinals and hence no remarkable cardinals (or $n$-remarkable cardinals).{% cite Gitman2018 %} ### Statements implying $0^♯s existence The existence of $0^♯$ is implied by: - [[Chang's_conjecture]] - Both $\omega_1$ and $\omega_2$ being singular (requires $\neg\text{AC}$). - The negation of the singular cardinal hypothesis ($\text{SCH}$). - The existence of an $\omega_1$-iterable cardinal or of a $\omega_1$-Erdős cardinal. - The existence of a weakly compact cardinal $\kappa$ such that $|(\kappa^+)^L|=\kappa$. - The existence of some uncountable regular cardinal $\kappa$ such that every constructible $X\subseteq\kappa$ either contains or is disjoint from a closed unbounded set. Note that if $0^♯$ exists then for every Silver indiscernible (in particular for every uncountable cardinal) there is a nontrivial [[Elementary embedding]] $j:L\rightarrow L$ with that indiscernible as its critical point. Thus if any such embedding exists, then a proper class of those embeddings exists. ### Nonexistence of $0^\sharp$, Jensen's Covering Theorem ### EM blueprints and alternative characterizations of $0^\sharp$ An **EM blueprint** (Ehrenfeucht-Mostowski blueprint) $T$ is any theory of the form $\{\varphi:(L_\delta;\in,\alpha_0,\alpha_1...)\models\varphi\}$ for some ordinal $\delta>\omega$ and $\alpha_0<\alpha_1<\alpha_2...$ which are indiscernible in the structure $L_\delta$. Roughly speaking, it's the set of all true statements about $\alpha_0,\alpha_1,\alpha_2...$ in $L_\delta$. For an EM blueprint $T=\{\varphi:(L_\delta;\in,\alpha_0,\alpha_1...)\models\varphi\}$ **the theory $T^{-}$** is defined as $\{\varphi:L_\delta\models\varphi\}$ (the set of truths about any definable elements of $L_\delta$). Then, **the structure $\mathcal{M}(T,\alpha)=(M(T,\alpha);E)\models T^{-}$** has a very technical definition, but it is indeed uniquely (up to isomorphism) the only structure which satisfies the existence of a set $X$ of $\mathcal{M}(T,\alpha)$-ordinals such that: 1. $X$ is a set of indiscernibles for $\mathcal{M}(T,\alpha)$ and $(X;E)\cong\alpha$ ($X$ has order-type $\alpha$ with respect to $\mathcal{M}(T,\alpha)$) 2. For any formula $\varphi$ and any $x<y<z...$ with $x,y,z...\in X$, $\mathcal{M}(T,\alpha)\models\varphi(x,y,z...)$ iff $\mathcal{M}(T,\alpha)\models\varphi(\alpha_0,\alpha_1,\alpha_2...)$ where $\alpha_0,\alpha_1...$ are the indiscernibles used in the EM blueprint. 3. If lt;$ is an $\mathcal{M}(T,\alpha)$-definable $\mathcal{M}(T,\alpha)$-well-ordering of $\mathcal{M}(T,\alpha)$, then: $\mathcal{M}(T,\alpha)=\{\min{}_<^{\mathcal{M}(T,\alpha)}\{x:\mathcal{M}(T,\alpha)\models\varphi[x,a,b,c...]\}:\varphi\in\mathcal{L}_\in\text{ and } a,b,c...\in X\}$ $0^\sharp$ is then defined as the **unique** EM blueprint $T$ such that: 1. $\mathcal{M}(T,\alpha)$ is isomorphic to a transitive model $M(T,\alpha)$ of ZFC for every $\alpha$ 2. For any infinite $\alpha$, the set of indiscernibles $X$ associated with $M(T,\alpha)$ can be made cofinal in $\text{Ord}^{M(T,\alpha)}$. 3. The $L_\delta$-indiscernibles $\beta_0<\beta_1...$ can be made so that if lt;$ is an $M(T,\alpha)$-definable well-ordering of $M(T,\alpha)$, then for any $(m+n+2)$-ary formula $\varphi$ such that $\min_<^{M(T,\alpha)}\{x:\varphi[x,\beta_0,\beta_1...\beta_{m+n}]\}<\beta_m$, then: $\min{}_<^{M(T,\alpha)}\{x:\varphi[x,\beta_0,\beta_1...\beta_{m+n}]\}=\min{}_<^{M(T,\alpha)}\{x:\varphi[x,\beta_0,\beta_1...\beta_{m-1},\beta_{m+n+1}...\beta_{m+2n+1}]\}$ If the EM blueprint meets 1. then it is called *well-founded*. If it meets 2. and 3. then it is called *remarkable.* **Lemma.** The property "$\Sigma$ is a well-founded remarkable E.M. set" is absolute for every inner model of ZF. Hence $M \vDashquot;$0^{\sharp}$ exists" if and only if $0^{\sharp} \in M$ in which case $\left(0^{\sharp}\right)^M=0^{\sharp}$. If $0^\sharp$ exists (i.e. there is a well-founded remarkable EM blueprint) then it happens to be equivalent to the set of all $\varphi$ such that $L\models\varphi[\kappa_0,\kappa_1...]$ for some uncountable cardinals $\kappa_0,\kappa_1...<\aleph_\omega$. This is because the associated $M(T,\alpha)$ will always have $M(T,\alpha)\prec L$ and furthermore $\kappa_0,\kappa_1...$ would be indiscernibles for $L$. $0^\sharp$ exists interestingly iff some $L_\delta$ has an uncountable set of indiscernables. If $0^\sharp$ exists, then there is some uncountable $\delta$ such that $M(0^\sharp,\omega_1)=L_\delta$ and $L_\delta$ therefore has an uncountable set of indiscernables. On the other hand, if some $L_\delta$ has an uncountable set of indiscernables, then the EM blueprint of $L_\delta$ is $0^\sharp$. ### Sharps of arbitrary sets ### Generalisations $0^\dagger$ (zero dagger) is a set of integers analogous to $0^\sharp$ and connected with inner models of [measurability](Measurable.md "Measurable").{% cite Kanamori1990 %} $0^{sword}$ is connected with nontrivial [[Mitchell rank and Mitchell order]]. $¬ 0 ^{sword}$ (*not zero sword*) means that there is no <a href="Mouse" class="mw-redirect" title="Mouse">mouse</a> with a measure of Mitchell order gt; 0$.{% cite Sharpe2011 %} $0^\P$ (zero pistol) is connected with [strong](Strong.md "Strong") cardinals. $¬ 0^\P$ (*not zero pistol*) means that a [[Core_model]] may be built with a strong cardinal, but that there is no class of indiscernibles for it that is closed and unbounded in $\mathrm{Ord}$).{% cite Sharpe2011 %} $0^¶$ is “the sharp for a strong cardinal”, meaning the minimal sound active mouse $\mathcal{M}$ with $M \| \mathrm{crit}(\dot F^{\mathcal{M}}) \models \text{“There exists a strong cardinaly”}$, with $\dot F^{\mathcal{M}}$ being the top extender of $\mathcal{M}$.{% cite Nielsen2018 %} ## Additional References - {% cite Jech2003 %} - user46667, *Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)*, URL (version: 2014-03-17): <a href="https://mathoverflow.net/q/156940" class="external free">https://mathoverflow.net/q/156940</a> - {% cite Chang1971 %}