Difference between revisions of "Weakly measurable"

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== Embedding characterizations of weak measurability==
 
== Embedding characterizations of weak measurability==
  
If $(\kappa^+)^{{<}\kappa} = \kappa^+$, then weak measurability can also be characterized in terms of elementary embeddings.  
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If $(\kappa^+)^{{<}\kappa} = \kappa^+$, then weak measurability can also be equivalently characterized in several different ways in terms of elementary embeddings.  
  
(Weak embedding) For every $A \subseteq \kappa^+$, there exists a transitive $M \vDash \text{ZFC}^-$ with $A, \kappa \in M$, a transitive $N$ and an elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$.
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:; Weak embedding characterization : For every $A \subseteq \kappa^+$, there exists a transitive $M \vDash \text{ZFC}^-$ with $A, \kappa \in M$, a transitive $N$ and an elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$.
  
(Embedding) For every transitive set $M$ of size $\kappa^+$ with $\kappa \in M$, there exists a transitive $N$ and an elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$.
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:; Embedding characterization : For every transitive set $M$ of size $\kappa^+$ with $\kappa \in M$, there exists a transitive $N$ and an elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$.
  
(Normal embedding) For every transitive $M \vDash \text{ZFC}^-$ of size $\kappa^+$ closed under ${<}\kappa$ sequences with $\kappa \in M$, there exists a transitive $N$ of size $\kappa^+$ closed under ${<}\kappa$ sequences and a cofinal elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$ such that $N = \{j(f)(\kappa)| f \in M; f: \kappa \longrightarrow M\}$.
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:; Normal embedding characterization : For every transitive $M \vDash \text{ZFC}^-$ of size $\kappa^+$ closed under ${<}\kappa$ sequences with $\kappa \in M$, there exists a transitive $N$ of size $\kappa^+$ closed under ${<}\kappa$ sequences and a cofinal elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$ such that $N = \{j(f)(\kappa)| f \in M; f: \kappa \longrightarrow M\}$.
  
(Normal ZFC embedding) For every $A \subseteq H_{\kappa^+}$ of size $\kappa^+$, there exists a transitive $M \vDash \text{ZFC}$ of size $\kappa^+$ closed under ${<}\kappa$ sequences with $A \subseteq M$ and $\kappa \in M$, a transitive $N$ of size $\kappa^+$ closed under ${<}\kappa$ sequences, and a cofinal elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$ such that $N = \{j(f)(\kappa)| f \in M; f: \kappa \longrightarrow M\}$.
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:; Normal ZFC embedding characterization : For every $A \subseteq H_{\kappa^+}$ of size $\kappa^+$, there exists a transitive $M \vDash \text{ZFC}$ of size $\kappa^+$ closed under ${<}\kappa$ sequences with $A \subseteq M$ and $\kappa \in M$, a transitive $N$ of size $\kappa^+$ closed under ${<}\kappa$ sequences, and a cofinal elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$ such that $N = \{j(f)(\kappa)| f \in M; f: \kappa \longrightarrow M\}$.
  
 
== Weakly measurable cardinals and the core model below $0^{\dagger}$ ==
 
== Weakly measurable cardinals and the core model below $0^{\dagger}$ ==

Revision as of 09:06, 1 January 2012


The weakly measurable cardinals were introduced by Jason Schanker in [Sch11], [Sch11a]. As their name suggests, they provide a weakening of the large cardinal concept of measurability. If the GCH holds at $\kappa$, then the property of the weak measurability of $\kappa$ is equivalent to that of the full measurability of $\kappa$, but when $\kappa^+\lt 2^\kappa$, these concepts can separate. Nevertheless, the existence of a weakly measurable cardinal is equiconsistent with the existence of a measurable cardinal, since if $\kappa$ is weakly measurable, then it is measurable in an inner model.

Formal Definition

A cardinal $\kappa$ is weakly measurable if and only if for every family $\mathcal{A}\subset P(\kappa)$ of size at most $\kappa^+$, there is a nonprincipal $\kappa$-complete filter on $\kappa$ measuring every set in $\mathcal{A}$. (i.e., For every subset $A \in \mathcal{A}$, either $A$ or $\kappa \setminus A$ is in the filter.)

Embedding characterizations of weak measurability

If $(\kappa^+)^{{<}\kappa} = \kappa^+$, then weak measurability can also be equivalently characterized in several different ways in terms of elementary embeddings.

Weak embedding characterization 
For every $A \subseteq \kappa^+$, there exists a transitive $M \vDash \text{ZFC}^-$ with $A, \kappa \in M$, a transitive $N$ and an elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$.
Embedding characterization 
For every transitive set $M$ of size $\kappa^+$ with $\kappa \in M$, there exists a transitive $N$ and an elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$.
Normal embedding characterization 
For every transitive $M \vDash \text{ZFC}^-$ of size $\kappa^+$ closed under ${<}\kappa$ sequences with $\kappa \in M$, there exists a transitive $N$ of size $\kappa^+$ closed under ${<}\kappa$ sequences and a cofinal elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$ such that $N = \{j(f)(\kappa)| f \in M; f: \kappa \longrightarrow M\}$.
Normal ZFC embedding characterization 
For every $A \subseteq H_{\kappa^+}$ of size $\kappa^+$, there exists a transitive $M \vDash \text{ZFC}$ of size $\kappa^+$ closed under ${<}\kappa$ sequences with $A \subseteq M$ and $\kappa \in M$, a transitive $N$ of size $\kappa^+$ closed under ${<}\kappa$ sequences, and a cofinal elementary embedding $j: M \longrightarrow N$ with critical point $\kappa$ such that $N = \{j(f)(\kappa)| f \in M; f: \kappa \longrightarrow M\}$.

Weakly measurable cardinals and the core model below $0^{\dagger}$

If $\kappa$ is weakly measurable, then $\kappa$ is measurable in Mitchell's core model $K^m$ (core model below $0^{\dagger}$).

Weakly measurable cardinals and forcing

Weakly measurable cardinals $\kappa$ are invariant under forcing of size less than $\kappa$ and forcing that adds no new subsets of $\kappa^+$. Many other preservation results for these large cardinals are unknown. For example, it is an open question as to whether we can always force to an extension where a weakly measurable cardinal $\kappa$ from the ground model remains weakly measurable and becomes indestructible by the further forcing to add a Cohen subset of $\kappa$. However, if $\kappa$ is measurable in the ground model, we inherit all of the indestructibility results we can get for its weak measurability from its full measurability and more. In particular, we will be able to force to an extension where $\kappa$ is measurable, the GCH holds, and the weak measurability of $\kappa$ is preserved by the further forcing to add any number of Cohen subsets of $\kappa$. Starting with a measurable cardinal $\kappa$, this result allows us to force to an extension where we preserve the weak measurability of $\kappa$ and yet make the GCH fail first at $\kappa$. Since the GCH cannot fail first at a measurable cardinal, this will also be a forcing extension where $\kappa$ is no longer measurable.

Place in the large cardinal hierarchy

In terms of consistency strength, weakly measurable cardinals occupy the same place as measurable cardinals in the large cardinal hierarchy. In terms of size, the possibilities for these large cardinals are still being investigated. Because measurable cardinals must be weakly measurable, and weakly measurable cardinals must be weakly compact, we are provided with strict upper and lower bounds on their sizes with respect to these large cardinal notions. In the presence of the GCH, weakly measurable cardinals and measurable cardinals coincide so their sizes are the same in this case. At the opposite extreme, it was left as an open question in [Sch11] and [Sch11a] as to whether the least weakly measurable cardinal could also be the least weakly compact cardinal. Despite being left open, there are promising developments that are being undertaken jointly by Gitik, Hamkins, and Schanker, which are aimed at this possibility.

References

  • [Sch11] Schanker, J. A. (2011), Weakly measurable cardinals. Mathematical Logic Quarterly, 57: 266–280. doi: 10.1002/malq.201010006
  • [Sch11a] Schanker, J. A. (2011), Weakly Measurable Cardinals and Partial Near Supercompactness. Dissertation submitted to CUNY Graduate Center.