# Zero sharp
**Zero sharp** (0\(^{\sharp}\), pronounced "zero sharp") is, in mathematical set theory, a particular real number that codes the complete first-order theory of the constructible universe (L: the minimal inner model of set theory, built in definable stages) L with respect to a proper class of Silver indiscernibles; its existence is a large-cardinal hypothesis independent of ZFC whose truth means that the universe V is very different from L[1](https://androma.org/page/Set%20Theory%20III%3A%20Large%20Cardinals) • [2](https://neugierde.github.io/cantors-attic/Zero_sharp). The statement "0\(^{\sharp}\) exists" is used as a large-cardinal assumption in its own right, lying below the existence of the [Erdős cardinal](https://www.edgepedia.org/erdos-cardinal), and it is the weakest large-cardinal property known to be incompatible with the axiom of constructibility V = L[3](https://link.springer.com/article/10.1007/s40879-018-0278-1) • [4](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals).
| Key fact | Detail |
|---|---|
| What it is | A real (a set of Gödel numbers of formulas) coding the true sentences of L about the Silver indiscernibles[3](https://link.springer.com/article/10.1007/s40879-018-0278-1) |
| Definability | A Δ\(^{1}_{3}\) subset of ω whose singleton is Π\(^{1}_{2}\)-definable over the reals[4](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals) |
| Origin | Introduced in Jack Silver's 1966 Berkeley Ph.D. dissertation; Solovay independently discovered it and coined the \# notation[4](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals) • [5](https://www.math.ucla.edu/~dam/220c.1.15s/thebook.pdf) |
| Sufficient hypothesis | A measurable cardinal implies 0\(^{\sharp}\) exists; a Ramsey cardinal, or even an ω\(_{1}\)-Erdős cardinal, suffices[5](https://www.math.ucla.edu/~dam/220c.1.15s/thebook.pdf) • [4](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals) |
| Consequence | If 0\(^{\sharp}\) exists, then V ≠ L, there is a nontrivial elementary embedding j : L → L, and L never computes successor cardinals correctly[4](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals) • [6](https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/on-omega-strongly-measurable-cardinals/BC1234011DD1CD1BBFE8B9CA4F3A32E4) |
| Covering dichotomy | If 0\(^{\sharp}\) does not exist, Jensen's covering lemma holds: every uncountable set of ordinals is covered by a constructible set of the same cardinality[7](https://www.cambridge.org/core/books/sets-and-proofs/covering-properties-of-core-models/4F7E429EEE43DB3DC94601A0A811E208) |
| Forcing behavior | No set forcing can add 0\(^{\sharp}\) to a model, and once it exists it cannot be forced out of existence[8](https://math.stackexchange.com/questions/4595269/unforceable-statements-in-zfc) |
## Definition and Silver indiscernibles
A class I of ordinals is a class of indiscernibles for L if for every formula \( \varphi(v_1, \dots, v_n) \) and any two increasing sequences \( \alpha_1 < \dots < \alpha_n \) and \( \beta_1 < \dots < \beta_n \) drawn from I, L satisfies \( \varphi \) on the first sequence exactly when it satisfies \( \varphi \) on the second[1](https://androma.org/page/Set%20Theory%20III%3A%20Large%20Cardinals). Because of these indiscernibles, L acts as though it possessed concealed symmetries; however, a proper class of such ordinals cannot be treated as one real parameter, and thus the coding problem becomes that of packing the indiscernibles' elementary pattern into a single object whose existence can be discussed within ordinary set-theoretic arguments.[1](https://androma.org/page/Set%20Theory%20III%3A%20Large%20Cardinals)
Under the standard definition, 0\(^{\sharp}\) is the set of Gödel codes \( \lceil \varphi \rceil \) of all true sentences \( \varphi \) about L generated by the ω-sequence of indiscernibles[3](https://link.springer.com/article/10.1007/s40879-018-0278-1). In Silver's presentation it is not merely a set of isolated truths about L but a coherent Ehrenfeucht–Mostowski blueprint: for every ordinal θ the blueprint yields a well-founded extensional Ehrenfeucht–Mostowski model generated by a strictly increasing sequence of order indiscernibles, which collapses to constructible levels[9](https://androma.org/theorems/7432).
The class of Silver indiscernibles is a closed unbounded class containing all uncountable cardinals such that, for every uncountable cardinal κ, the intersection I ∩ κ has cardinality κ, is a set of indiscernibles for the structure \( (L_\kappa, \in) \), and every \( a \in L_\kappa \) is definable in \( (L_\kappa, \in) \) from I ∩ κ[10](https://uryaar.com/Digital-garden/Cantors-attic/Indiscernibles). Silver's theorem (1966) gives equivalent formulations: 0\(^{\sharp}\) exists if and only if there is an uncountable regular cardinal γ with an unbounded subset that is a set of indiscernibles for \( L_\gamma \), if and only if there is a closed unbounded proper class C of indiscernibles for L that generates L[5](https://www.math.ucla.edu/~dam/220c.1.15s/thebook.pdf). If the class is chosen so that its ω-th member is as small as possible, it is closed and unbounded and hence uniquely determined[4](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals).
## History
0\(^{\sharp}\) was introduced in Jack Howard Silver's 1966 Ph.D. dissertation at the [University of California](https://www.edgepedia.org/university-of-california), Berkeley, written under [Robert Lawson Vaught](https://www.edgepedia.org/robert-lawson-vaught) and published in abridged form in the Annals of Mathematical Logic in 1971[4](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals) • [11](https://math.berkeley.edu/publications/some-applications-model-theory-set-theory). Robert Solovay independently discovered the set, observed that its singleton is Π\(^{1}_{2}\)-definable over the reals and that 0\(^{\sharp}\) is therefore a Δ\(^{1}_{3}\) subset of ω that is not constructible, and the \# notation originates in his 1967 work[4](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals) • [5](https://www.math.ucla.edu/~dam/220c.1.15s/thebook.pdf). Kenneth Kunen rounded out the analysis by proving the converse direction: the existence of any nontrivial elementary embedding \( i : L \to L \) implies that 0\(^{\sharp}\) exists[4](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals) • [12](https://math.cs.kitami-it.ac.jp/~fuchino/xbooks/The-Higher-Infinite-optimized.pdf).
The background was Scott's 1961 theorem that a measurable cardinal implies V ≠ L, and work of Gaifman and Rowbottom showing that measurable cardinals impose strong structure on L[5](https://www.math.ucla.edu/~dam/220c.1.15s/thebook.pdf) • [4](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals). Silver observed that starting from a [Ramsey cardinal](https://www.edgepedia.org/ramsey-cardinal), or even an ω\(_{1}\)-Erdős cardinal, one obtains an uncountable set of indiscernibles for L, which the condensation property of L extends to a proper class[4](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals).
## Consequences of existence
The existence of 0\(^{\sharp}\) has three important consequences recorded by William Mitchell: the singleton \( \{0^{\sharp}\} \) is definable over the reals by a Π\(^{1}_{2}\) formula; every member of the class I, in particular every uncountable cardinal of V, is in L inaccessible, weakly compact, and indeed has every large cardinal property that can hold in L; and there is a nontrivial elementary embedding \( L \to L \), since every increasing embedding of I into I extends to one, and every elementary embedding of L is determined by its restriction to I[4](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals).
**L and V diverge sharply.** Among the consequences of the Silver indiscernibles is the countability, in V, of the sets of L that are definable over L without parameters, which immediately implies V ≠ L, and more importantly the definability of truth in L[3](https://link.springer.com/article/10.1007/s40879-018-0278-1). Silver showed that if 0^{♯} exists, L then misrepresents even the simplest characteristics of V; for instance, whenever 0^{♯} is present, L misidentifies successor cardinals[6](https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/on-omega-strongly-measurable-cardinals/BC1234011DD1CD1BBFE8B9CA4F3A32E4).
## The covering lemma
Jensen's [covering lemma](https://www.edgepedia.org/covering-lemma), announced in 1974 in handwritten notes titled "Marginalia on a Theorem of Silver", describes the other side of a dichotomy. It asserts that, assuming 0^{
\(\sharp\)} is absent, every uncountable collection A of ordinals can be covered by some set B belonging to L whose cardinality equals that of A.[7](https://www.cambridge.org/core/books/sets-and-proofs/covering-properties-of-core-models/4F7E429EEE43DB3DC94601A0A811E208) • [13](https://doi.org/10.1007/978-1-4020-5764-9_19) Put informally, the universe V either resembles L to a large extent or else is very different from L[14](https://ems.press/journals/owr/articles/1269).
The lemma has striking consequences for cardinal arithmetic: if 0\(^{\sharp}\) does not exist, then L computes successors of singular cardinals correctly, and combinatorial principles Jensen proved in L, such as the square principle at singular cardinals, hold in V[7](https://www.cambridge.org/core/books/sets-and-proofs/covering-properties-of-core-models/4F7E429EEE43DB3DC94601A0A811E208). The covering lemma is also used to obtain lower bounds for the consistency strength of set-theoretic statements that do not mention large cardinals[6](https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/on-omega-strongly-measurable-cardinals/BC1234011DD1CD1BBFE8B9CA4F3A32E4). Extensions of the covering lemma to larger inner models, including the weak covering lemma basic to inner models beyond measurable cardinals, form a central part of inner model theory[13](https://doi.org/10.1007/978-1-4020-5764-9_19).
## Forcing and absoluteness
The existence of 0\(^{\sharp}\) is upwards absolute: there is no set forcing that adds 0\(^{\sharp}\) to a model of ZFC[8](https://math.stackexchange.com/questions/4595269/unforceable-statements-in-zfc). One argument uses Shoenfield absoluteness, which makes Σ\(^{1}_{2}\) and Π\(^{1}_{2}\) sentences invariant under set forcing, together with the definability of 0\(^{\sharp}\) from Silver indiscernibles that survive the extension[8](https://math.stackexchange.com/questions/4595269/unforceable-statements-in-zfc) • [15](https://math.stackexchange.com/questions/5127899/can-we-force-0-sharp-to-exist). Conversely, "smallness statements" about canonical inner models, such as 0\(^{\sharp}\) and other sharps, cannot be forced into inexistence once they exist, because the canonical inner models are robust under generic extensions[8](https://math.stackexchange.com/questions/4595269/unforceable-statements-in-zfc).
Class forcing is more subtle. M. C. Stanley, a set theorist at [San Jose State University](https://www.edgepedia.org/san-jose-state-university), proved in the [Journal of Symbolic Logic](https://www.edgepedia.org/journal-of-symbolic-logic) (1998) that 0\(^{\sharp}\) can be invisibly class generic, treating the class-forcing case in detail[16](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/70FED51C572CB3391037221FA5EA45E0/S0022481200014274a.pdf/invisible-genericity-and-0-.pdf) • [15](https://math.stackexchange.com/questions/5127899/can-we-force-0-sharp-to-exist). Sy D. Friedman of the Kurt Gödel Research Center characterized 0\(^{\sharp}\) in forcing terms as well: the universe being L-saturated for ω\(_{1}\)-forcings is equivalent to the existence of 0\(^{\sharp}\)[17](https://www.logic.univie.ac.at/~dsyfriedman/papers/char.0-sharp.pdf).
## Determinacy and generalizations
The Martin–Harrington theorem connects 0\(^{\sharp}\) with determinacy: in ZF, the determinacy of analytic (Σ\(^{1}_{1}\)) games holds if and only if 0\(^{\sharp}\) exists[18](https://arxiv.org/abs/1503.04000). Donald Martin had shown that the existence of sharps implies analytic determinacy, and [Leo Harrington](https://www.edgepedia.org/leo-harrington) of the University of California, Berkeley proved the converse in 1978[19](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/analytic-determinacy-and-0/BBF9EC7CF0907F5DC79F9B1C0A4CB2F2). Harrington's theorem is proved in two steps, through Harrington's Principle, and the equivalence with 0\(^{\sharp}\) requires fourth-order arithmetic; third-order arithmetic plus Harrington's Principle is equiconsistent with ZFC plus a remarkable cardinal and does not imply that 0\(^{\sharp}\) exists[18](https://arxiv.org/abs/1503.04000).
The sharp construction relativizes and iterates. For any set \( a \subseteq \omega \), \( a^{\sharp} \) is defined as the theory of a closed unbounded class of indiscernibles for the structure \( (L[a], \in, a) \), and the construction can be iterated by setting \( 0^{\alpha\sharp} = \langle 0^{\nu\sharp} : \nu < \alpha \rangle \) for countable ordinals α, giving the beginning of a hierarchy of inner models[4](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals). Derrick DuBose characterized the iterated sharps \( 0^{(k)\sharp} \) in terms of determinacy of pointclasses[20](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/equivalence-of-determinacy-and-iterated-sharps/648AB34A67DDE377E21658B70CA75E1B), and Yizheng Zhu established the descriptive set-theoretic representation of the higher mice \( M_n^{\sharp} \), called \( 0^{(n+1)\sharp} \)[21](https://arxiv.org/abs/1706.00661). Beyond L, 0\(^{\dagger}\) (zero pistol) is the sharp for a strong cardinal, the minimal sound active mouse with a strong-cardinal extender[2](https://neugierde.github.io/cantors-attic/Zero_sharp).
## References
1. [Set Theory III: Large Cardinals, Androma](https://androma.org/page/Set%20Theory%20III%3A%20Large%20Cardinals)
2. [Zero sharp, Cantor's Attic](https://neugierde.github.io/cantors-attic/Zero_sharp)
3. [Set theory and the analyst, European Journal of Mathematics (Springer)](https://link.springer.com/article/10.1007/s40879-018-0278-1)
4. [William J. Mitchell, Inner Models and Large Cardinals (ResearchGate)](https://www.researchgate.net/publication/2356814_Inner_Models_And_Large_Cardinals)
5. [Set theory lecture notes (sharps chapter), UCLA](https://www.math.ucla.edu/~dam/220c.1.15s/thebook.pdf)
6. [Ben-Neria and Hayut, On ω-strongly measurable cardinals, Forum of Mathematics, Sigma (2023)](https://www.cambridge.org/core/journals/forum-of-mathematics-sigma/article/on-omega-strongly-measurable-cardinals/BC1234011DD1CD1BBFE8B9CA4F3A32E4)
7. [Covering Properties of Core Models, in Sets and Proofs (Cambridge University Press)](https://www.cambridge.org/core/books/sets-and-proofs/covering-properties-of-core-models/4F7E429EEE43DB3DC94601A0A811E208)
8. [Unforceable statements in ZFC, Mathematics Stack Exchange](https://math.stackexchange.com/questions/4595269/unforceable-statements-in-zfc)
9. [Silver Indiscernibles from 0♯ — Statement & Proof, Androma](https://androma.org/theorems/7432)
10. [Indiscernibles, Ur Ya'ar (Cantor's Attic mirror)](https://uryaar.com/Digital-garden/Cantors-attic/Indiscernibles)
11. [Some Applications of Model Theory in Set Theory, UC Berkeley Mathematics Department](https://math.berkeley.edu/publications/some-applications-model-theory-set-theory)
12. [A. Kanamori, The Higher Infinite (Springer Monographs in Mathematics)](https://math.cs.kitami-it.ac.jp/~fuchino/xbooks/The-Higher-Infinite-optimized.pdf)
13. [The Covering Lemma, Springer Nature Link](https://doi.org/10.1007/978-1-4020-5764-9_19)
14. [Feinstrukturtheorie und Innere Modelle, EMS Press](https://ems.press/journals/owr/articles/1269)
15. [Can we force 0♯ to exist?, Mathematics Stack Exchange](https://math.stackexchange.com/questions/5127899/can-we-force-0-sharp-to-exist)
16. [M. C. Stanley, Invisible genericity and 0#, Journal of Symbolic Logic 63 (1998)](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/70FED51C572CB3391037221FA5EA45E0/S0022481200014274a.pdf/invisible-genericity-and-0-.pdf)
17. [Sy D. Friedman, A characterisation of 0# in terms of forcing](https://www.logic.univie.ac.at/~dsyfriedman/papers/char.0-sharp.pdf)
18. [Cheng and Schindler, Harrington's principle over higher order arithmetic, arXiv](https://arxiv.org/abs/1503.04000)
19. [Leo Harrington, Analytic determinacy and 0#, Journal of Symbolic Logic 43 (1978)](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/analytic-determinacy-and-0/BBF9EC7CF0907F5DC79F9B1C0A4CB2F2)
20. [Derrick Albert DuBose, The equivalence of determinacy and iterated sharps, Journal of Symbolic Logic](https://www.cambridge.org/core/journals/journal-of-symbolic-logic/article/abs/equivalence-of-determinacy-and-iterated-sharps/648AB34A67DDE377E21658B70CA75E1B)
21. [Yizheng Zhu, The higher sharp III: An EM blueprint of 0^{3♯}, arXiv](https://arxiv.org/abs/1706.00661)
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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Number systems › Ordinal and cardinal numbers › Large cardinals › Sharps and mice*
*Initially written Sep 17, 2026 · Reviewed: Sep 19, 2026; Sep 30, 2026; Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*
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