Difference between revisions of "Zero sharp"
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[[Category:Large cardinal axioms]] | [[Category:Large cardinal axioms]] | ||
[[Category:Constructibility]] | [[Category:Constructibility]] | ||
− | $0^{\#}$ 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 $V$. | + | $0^{\#}$ 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 $V$. <cite>Jech2003:SetTheory</cite> |
== Definition == | == Definition == | ||
− | $0^{\#}$ 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 | + | $0^{\#}$ 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 indiscernibles. Moreover, it implies: |
*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). | *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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*There are $\mathfrak{c}$ many reals which are not constructible (that is, $x\not\in L$). | *There are $\mathfrak{c}$ many reals which are not constructible (that is, $x\not\in L$). | ||
− | The existence of $0^\#$ is implied by: | + | The existence of $0^\#$ is equivalent to: |
+ | |||
+ | * $(\aleph_\omega)^V$ being regular in $L$. | ||
+ | * $\Pi^1_1$-[[axiom of projective determinacy|determinacy]] (lightface form). | ||
+ | * The existence of a nontrivial [[elementary embedding]] $j:L\to L$. | ||
+ | * The existence of a nontrivial elementary embedding $j:L_\alpha\to L_\beta$ with $\text{crit}(j)<|\alpha|$. | ||
+ | And is implied by: | ||
* Chang's Conjecture. | * 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 negation of the singular cardinal hypothesis ($\text{SCH}$). | ||
− | * The | + | * 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. | ||
− | == | + | == Silver indiscernibles == |
− | + | ''More information to be added here.'' | |
− | + | $0^{\#}$ exists iff there is a nontrivial [[elementary embedding]] $j:L\rightarrow L$ (by a theorem of Kunen). If any such embedding exists, then a proper class of those embeddings exists and every Silver indiscernible (in particular every uncountable cardinal) is the critical point of a such embedding. In $L$, every Silver indiscernible is [[indescribable|totally indescribable]], [[ineffable|totally ineffable]], and much more. | |
− | + | == E.M. blueprints and sharps of arbitrary sets == | |
+ | |||
+ | == Consequence of the nonexistence of $0^\#$ == | ||
− | + | {{stub}} | |
− | + | {{References}} |
Revision as of 06:03, 17 November 2017
$0^{\#}$ is a $\Sigma_3^1$ real number which cannot be proven to exist in $\text{ZFC}$. It's existence contradicts the Axiom of constructibility, $V=L$. In fact, it's existence is somewhat equivalent to $L$ being completely different from $V$. [1]
Contents
Definition
$0^{\#}$ 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 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 indiscernibles. Moreover, it implies:
- 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 uncountable cardinal is $\alpha$-iterable, $\geq$ an $\alpha$-Erdős, and totally ineffable in $L$.
- There are $\mathfrak{c}$ many reals which are not constructible (that is, $x\not\in L$).
The existence of $0^\#$ is equivalent to:
- $(\aleph_\omega)^V$ being regular in $L$.
- $\Pi^1_1$-determinacy (lightface form).
- The existence of a nontrivial elementary embedding $j:L\to L$.
- The existence of a nontrivial elementary embedding $j:L_\alpha\to L_\beta$ with $\text{crit}(j)<|\alpha|$.
And 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.
Silver indiscernibles
More information to be added here.
$0^{\#}$ exists iff there is a nontrivial elementary embedding $j:L\rightarrow L$ (by a theorem of Kunen). If any such embedding exists, then a proper class of those embeddings exists and every Silver indiscernible (in particular every uncountable cardinal) is the critical point of a such embedding. In $L$, every Silver indiscernible is totally indescribable, totally ineffable, and much more.
E.M. blueprints and sharps of arbitrary sets
Consequence of the nonexistence of $0^\#$
This article is a stub. Please help us to improve Cantor's Attic by adding information.
References
- Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www bibtex