Difference between revisions of "Zero sharp"

From Cantor's Attic
Jump to: navigation, search
m
Line 1: Line 1:
 
[[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]], $\text{V=L}$. In fact, it's existence is somewhat equivalent to $\text{L}$ being completely different from $\text{V}$.  
+
$0^{\#}$ 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 $V$.  
  
 
== Definition ==
 
== Definition ==
  
$0^{\#}$ is defined as the set of all Gödel numberings of first-order formula $\varphi$ such that $\text{L}\models\varphi(\aleph_0,\aleph_1...\aleph_n)$ 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 $\text{L}_{\aleph_{\omega}}\models\varphi(\aleph_0,\aleph_1...\aleph_n)$. This definition implies the existence of Silver Indiscernables. Moreover, it implies:
+
$0^{\#}$ 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:
  
*Given any set $X\in\text{L}$ which is first-order definable in $\text{L}$, $X\in\text{L}_{\omega_1}$. This of course implies that $\aleph_1$ is not first-order definable in $\text{L}$, because $\aleph_1\not\in\text{L}_{\omega_1}$. This is already a disproof of $\text{V=L}$ (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).
*For every $\alpha\in\omega_1^\text{L}$, every uncountable cardinal is [[Ramsey#iterable|$\alpha$-iterable]], $\geq$ an [[Erdos|$\alpha$-Erdős]], and [[ineffable|totally ineffable]] in $\text{L}$.
+
*For every $\alpha\in\omega_1^L$, every uncountable cardinal is [[Ramsey#iterable|$\alpha$-iterable]], $\geq$ an [[Erdos|$\alpha$-Erdős]], and [[ineffable|totally ineffable]] in $L$.
*There are $\mathfrak{c}$ many reals which are not constructible (that is, $x\not\in\text{L}$).
+
*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 implied by:
Line 19: Line 19:
 
== $0^{\#}$ cardinal ==
 
== $0^{\#}$ cardinal ==
  
$0^{\#}$ exists iff there is a nontrivial [[elementary embedding]] $j:\text{L}\rightarrow\text{L}$ (by a theorem of Kunen). The critical point of such an embedding is sometimes called a $0^{\#}$ cardinal, and sometimes called a $j:\text{L}\rightarrow\text{L}$ 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 $j:\text{L}\rightarrow\text{L}$ cardinal, and the least measurable cardinal $\kappa$ such that $V_\kappa$ satisfies the existence of such a cardinal is not a $j:\text{L}\rightarrow\text{L}$ cardinal, and so on.
+
$0^{\#}$ 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.
  
However, the existence of a measurable suffices to prove the existence and consistency of a $j:\text{L}\rightarrow\text{L}$ cardinal.
+
However, the existence of a measurable suffices to prove the existence and consistency of a $j:L\rightarrow L$ cardinal.
  
 
''More information to be added here.''
 
''More information to be added here.''

Revision as of 13:16, 11 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 $V$.

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 Indiscernables. 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 implied by:

  • Chang's Conjecture.
  • The existence of an $\omega_1$-iterable cardinal.
  • The negation of the singular cardinal hypothesis ($\text{SCH}$).
  • The axiom of determinacy ($\text{AD}$).

$0^{\#}$ cardinal

$0^{\#}$ 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 $\Sigma_1^2$-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.

However, the existence of a measurable suffices to prove the existence and consistency of a $j:L\rightarrow L$ cardinal.

More information to be added here.

References

  • Jech, Thomas J. Set Theory (The 3rd Millennium Ed.). Springer, 2003.