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 $V$.  
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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 $V$. <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 Indiscernables. Moreover, it implies:
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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 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:
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The existence of $0^\#$ is equivalent to:
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* $(\aleph_\omega)^V$ being regular in $L$.
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* $\Pi^1_1$-[[axiom of projective determinacy|determinacy]] (lightface form).
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* The existence of a nontrivial [[elementary embedding]] $j:L\to L$.
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* The existence of a nontrivial elementary embedding $j:L_\alpha\to L_\beta$ with $\text{crit}(j)<|\alpha|$.
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And is implied by:
 
* Chang's Conjecture.
 
* Chang's Conjecture.
* The existence of an $\omega_1$-iterable cardinal.
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* 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 [[axiom of determinacy]] ($\text{AD}$).
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* The existence of an $\omega_1$-iterable cardinal or of a $\omega_1$-Erdős cardinal.
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* The existence of a weakly compact cardinal $\kappa$ such that $|(\kappa^+)^L|=\kappa$.
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* 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.
  
== $0^{\#}$ cardinal ==
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== Silver indiscernibles ==
  
$0^{\#}$ 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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''More information to be added here.''
  
However, the existence of a measurable suffices to prove the existence and consistency of a $j:L\rightarrow L$ cardinal.
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$0^{\#}$ 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.
  
''More information to be added here.''
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== E.M. blueprints and sharps of arbitrary sets ==
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== Consequence of the nonexistence of $0^\#$ ==
  
== References ==
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{{stub}}
*Jech, Thomas J. Set Theory (The 3rd Millennium Ed.). Springer, 2003.
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{{References}}

Revision as of 05:03, 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]

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 $0^\#$

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References

  1. Jech, Thomas J. Set Theory. Third, Springer-Verlag, Berlin, 2003. (The third millennium edition, revised and expanded) www   bibtex
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