Update: Regarding to Prof. Hamkins's guidance I added a restrictionrestricted the questions to the "normal" measures to avoid trivial answers.
Update: Regarding to Prof. Hamkins's guidance I added a restriction to the "normal" measures to avoid trivial answers.
Update: Regarding to Prof. Hamkins's guidance I restricted the questions to the "normal" measures to avoid trivial answers.
Are measures of a measurable cardinal measurable? (Edited and Updated Version)
Update: Regarding to Prof. Hamkins's guidance I added a restriction to the "normal" measures to avoid trivial answers.
Definition: Let $\kappa$ be a measurable cardinal. Define:
$\mathbb{M}_{\kappa}:=\lbrace \mu:P(\kappa)\rightarrow \lbrace 0,1\rbrace~|~\mu~\text{is a non-trivial}~\kappa~\text{-additive measure on}~\kappa \rbrace$
$\mathbb{N}_{\kappa}:=\lbrace \mu:P(\kappa)\rightarrow \lbrace 0,1\rbrace~|~\mu~\text{is a non-trivial}~\kappa~\text{-additive normal measure on}~\kappa \rbrace$
A problematic problem on a measurable cardinal $\kappa$ is the number of its two-valued non-trivial $\kappa$-additive normal measures. There are many classic consistency results by Apter, Ben Neria, Friedman, Garti, Hamkins, Kunen, Magidor and others on this topic. Here I want to ask this old question in a new way.
In some sense measures are tools for counting usingby weighing. If we find a measure on a set then we will have some useful information about the weight and size of its subsets and also about its own size which could be a two or real-valued measurable cardinal. Now consider the set of all two-valued non-trivial $\kappa$-additive measures of a measurable cardinal ($\mathbb{M}_{\kappa}$$\mathbb{N}_{\kappa}$) we want to determine the size of this set or its some special subsets (for example the set of normal measures on $\kappa$ ($\mathbb{N}_{\kappa}$))size. A possible way could be trying to find a two-valued non-trivial maximum-additive measure on $\mathbb{M}_{\kappa}$$\mathbb{N}_{\kappa}$. In the other words we are asking about possible measurabilty of the cardinal $|\mathbb{M}_{\kappa}|$$|\mathbb{N}_{\kappa}|$. Now define:
Definition: A measurable cardinal $\kappa$ called "super measurable" if $|\mathbb{M}_{\kappa}|$$|\mathbb{N}_{\kappa}|$ be a measurable cardinal too.
$\forall \lambda~~m(\lambda)\rightarrow |\mathbb{M}_{\lambda}|=\lambda)$$\forall \lambda~~m(\lambda)\rightarrow |\mathbb{N}_{\lambda}|=\lambda)$
$\forall \lambda~~m(\lambda)\rightarrow |\mathbb{M}_{\lambda}|=\lambda^{\oplus})$$\forall \lambda~~m(\lambda)\rightarrow |\mathbb{N}_{\lambda}|=\lambda^{\oplus})$
Remark: Obviously it is impossible to find a direct (within $ZFC$) construction of a non-trivial measure on the measure space $\mathbb{M}_{\kappa}$$\mathbb{N}_{\kappa}$ because it is like proving the existence of a real-valued measurable cardinal so we should think about "relative" measure producing methods. In order to do this we first need a "criterion" to determine that which one of the subsets of $\mathbb{M}_{\kappa}$$\mathbb{N}_{\kappa}$ are "heavy weight" or "large in number". In this direction some approaches could be like these (the arguments are about the space $\mathbb{M}_{\kappa}$, one can redefine a same argument for $\mathbb{N}_{\kappa}$):
Question (4): Is there a known "addition" (or "convolution") operator on $\mathbb{M}_{\kappa}$ (or $\mathbb{N}_{\kappa}$) to "amalgamate" two given two-valued non-trivial $\kappa$-additive (normal) measure (in a non-trivial way) for producing anotherto produce a new measure on $\kappa$?
Are measures of a measurable cardinal measurable?
A problematic problem on a measurable cardinal $\kappa$ is the number of its two-valued non-trivial $\kappa$-additive measures. There are many classic consistency results by Apter, Ben Neria, Friedman, Garti, Hamkins, Kunen, Magidor and others on this topic. Here I want to ask this old question in a new way.
In some sense measures are tools for counting using weighing. If we find a measure on a set then we will have some useful information about the weight and size of its subsets and also about its own size which could be a two or real-valued measurable cardinal. Now consider the set of all two-valued non-trivial $\kappa$-additive measures of a measurable cardinal ($\mathbb{M}_{\kappa}$) we want to determine the size of this set or its some special subsets (for example the set of normal measures on $\kappa$ ($\mathbb{N}_{\kappa}$)). A possible way could be trying to find a two-valued non-trivial maximum-additive measure on $\mathbb{M}_{\kappa}$. In the other words we are asking about possible measurabilty of the cardinal $|\mathbb{M}_{\kappa}|$. Now define:
Definition: A measurable cardinal $\kappa$ called "super measurable" if $|\mathbb{M}_{\kappa}|$ be a measurable cardinal too.
$\forall \lambda~~m(\lambda)\rightarrow |\mathbb{M}_{\lambda}|=\lambda)$
$\forall \lambda~~m(\lambda)\rightarrow |\mathbb{M}_{\lambda}|=\lambda^{\oplus})$
Remark: Obviously it is impossible to find a direct (within $ZFC$) construction of a non-trivial measure on the measure space $\mathbb{M}_{\kappa}$ because it is like proving the existence of a real-valued measurable cardinal so we should think about "relative" measure producing methods. In order to do this we first need a "criterion" to determine that which one of the subsets of $\mathbb{M}_{\kappa}$ are "heavy weight" or "large in number". In this direction some approaches could be like these:
Question (4): Is there a known "addition" (or "convolution") operator on $\mathbb{M}_{\kappa}$ to "amalgamate" two given two-valued non-trivial $\kappa$-additive measure (in a non-trivial way) for producing another measure on $\kappa$?
Are measures of a measurable cardinal measurable? (Edited and Updated Version)
Update: Regarding to Prof. Hamkins's guidance I added a restriction to the "normal" measures to avoid trivial answers.
Definition: Let $\kappa$ be a measurable cardinal. Define:
$\mathbb{M}_{\kappa}:=\lbrace \mu:P(\kappa)\rightarrow \lbrace 0,1\rbrace~|~\mu~\text{is a non-trivial}~\kappa~\text{-additive measure on}~\kappa \rbrace$
$\mathbb{N}_{\kappa}:=\lbrace \mu:P(\kappa)\rightarrow \lbrace 0,1\rbrace~|~\mu~\text{is a non-trivial}~\kappa~\text{-additive normal measure on}~\kappa \rbrace$
A problematic problem on a measurable cardinal $\kappa$ is the number of its two-valued non-trivial $\kappa$-additive normal measures. There are many classic consistency results by Apter, Ben Neria, Friedman, Garti, Hamkins, Kunen, Magidor and others on this topic. Here I want to ask this old question in a new way.
In some sense measures are tools for counting by weighing. If we find a measure on a set then we will have some useful information about the weight and size of its subsets and also about its own size which could be a two or real-valued measurable cardinal. Now consider the set $\mathbb{N}_{\kappa}$ we want to determine its size. A possible way could be trying to find a two-valued non-trivial maximum-additive measure on $\mathbb{N}_{\kappa}$. In the other words we are asking about possible measurabilty of the cardinal $|\mathbb{N}_{\kappa}|$. Now define:
Definition: A measurable cardinal $\kappa$ called "super measurable" if $|\mathbb{N}_{\kappa}|$ be a measurable cardinal too.
$\forall \lambda~~m(\lambda)\rightarrow |\mathbb{N}_{\lambda}|=\lambda)$
$\forall \lambda~~m(\lambda)\rightarrow |\mathbb{N}_{\lambda}|=\lambda^{\oplus})$
Remark: Obviously it is impossible to find a direct (within $ZFC$) construction of a non-trivial measure on the measure space $\mathbb{N}_{\kappa}$ because it is like proving the existence of a real-valued measurable cardinal so we should think about "relative" measure producing methods. In order to do this we first need a "criterion" ���to determine that which one of the subsets of $\mathbb{N}_{\kappa}$ are "heavy weight" or "large in number". In this direction some approaches could be like these (the arguments are about the space $\mathbb{M}_{\kappa}$, one can redefine a same argument for $\mathbb{N}_{\kappa}$):
Question (4): Is there a known "addition" (or "convolution") operator on $\mathbb{M}_{\kappa}$ (or $\mathbb{N}_{\kappa}$) to "amalgamate" two given two-valued non-trivial $\kappa$-additive (normal) measure (in a non-trivial way) to produce a new measure on $\kappa$?
Are measures of a measurable cardinal measurable?
A problematic problem on a measurable cardinal $\kappa$ is the number of its two-valued non-trivial $\kappa$-additive measures. There are many classic consistency results by Apter, Ben Neria, Friedman, Garti, Hamkins, Kunen, Magidor and others on this topic. Here I want to ask this old question in a new way.
In some sense measures are tools for counting using weighing. If we find a measure on a set then we will have some useful information about the weight and size of its subsets and also about its own size which could be a two or real-valued measurable cardinal. Now consider the set of all two-valued non-trivial $\kappa$-additive measures of a measurable cardinal ($\mathbb{M}_{\kappa}$) we want to determine the size of this set or its some special subsets (for example the set of normal measures on $\kappa$ ($\mathbb{N}_{\kappa}$)). A possible way could be trying to find a two-valued non-trivial maximum-additive measure on $\mathbb{M}_{\kappa}$. In the other words we are asking about possible measurabilty of the cardinal $|\mathbb{M}_{\kappa}|$. Now define:
Definition: A measurable cardinal $\kappa$ called "super measurable" if $|\mathbb{M}_{\kappa}|$ be a measurable cardinal too.
Notation: For any cardinal $\kappa$ consider $\kappa^{\oplus}$ to be the least measurable cardinal greater than $\kappa$. Also let $m(x)$ and $sm(x)$ to be the abbriviations for the phrases "$x$ is a measurable cardinal" and "$x$ is a super measurable cardinal" respectively.
Question (1): Can "all" measurable cardinals be super measurable? In the other words is the following statement true?
$Con(ZFC+\exists~\text{some measurable cardinal})\Longrightarrow $ $Con(ZFC+\exists~\text{some measurable cardinal}~+$
$\forall \lambda~~m(\lambda)\rightarrow sm(\lambda))$
Question (2): Can "all" measurable cardinals be super measurable and moreover there are "many" such cardinals? In the other words is the following statement true?
$Con(ZFC+\exists~\text{class many measurable cardinals})\Longrightarrow $ $Con(ZFC+\exists~\text{class many measurable cardinals} +$
$\forall \lambda~~m(\lambda)\rightarrow sm(\lambda))$
Question (3): Can "all" measurable cardinals be super measurable and moreover there are "many" such cardinals with some "control on size" of measure space? In the other words are the following statements true?
**(a)** $Con(ZFC+\exists~\text{class many measurable cardinals} +$
$\forall \lambda~~m(\lambda)\rightarrow sm(\lambda)) \Longrightarrow Con(ZFC+\exists~\text{class many measurable cardinals} +$
$\forall \lambda~~m(\lambda)\rightarrow |\mathbb{M}_{\lambda}|=\lambda)$
**(b)** $Con(ZFC+\exists~\text{class many measurable cardinals} +$
$\forall \lambda~~m(\lambda)\rightarrow sm(\lambda)) \Longrightarrow Con(ZFC+\exists~\text{class many measurable cardinals} +$
$\forall \lambda~~m(\lambda)\rightarrow |\mathbb{M}_{\lambda}|=\lambda^{\oplus})$
Remark: Obviously it is impossible to find a direct (within $ZFC$) construction of a non-trivial measure on the measure space $\mathbb{M}_{\kappa}$ because it is like proving the existence of a real-valued measurable cardinal so we should think about "relative" measure producing methods. In order to do this we first need a "criterion" to determine that which one of the subsets of $\mathbb{M}_{\kappa}$ are "heavy weight" or "large in number". In this direction some approaches could be like these:
(a) We know that every measurable cardinal $\kappa$ has some non-invariant subsets by measures in $\mathbb{M}_{\kappa}$ (for example the set of all successor ordinals in $\kappa$). So we can say a subset $S\subseteq \mathbb{M}_{\kappa}$ to be "large" (measure $1$) iff for all non $\mathbb{M}_{\kappa}$-invariant set $X\subseteq \kappa$ there are at least two measures $\mu_{1},\mu_{2}\in S$ such that the pair $(\mu_{1},\mu_{2})$ "unfolds" the hidden non-invariance of $X$ (i.e. $\mu_{1}(X)=0$ and $\mu_{2}(X)=1$).
(b) Another criterion for the "largness" of a subset $S\subseteq \mathbb{M}_{\kappa}$ could be defining a notion of "addition" on $\mathbb{M}_{\kappa}$ and considering the "convergence" of the series $\sum_{i\in S}\mu_{i}$ in the space $\mathbb{M}_{\kappa}$ and in the last step defining a natural two-valued measure on $\mathbb{M}_{\kappa}$ like $\Omega :P(\mathbb{M}_{\kappa})\longrightarrow \lbrace 0,1\rbrace$ such that $\forall S\subseteq \mathbb{M}_{\kappa}~~~~~\Omega(S)=1\Longleftrightarrow \sum_{i\in S}\mu_{i} \notin \mathbb{M}_{\kappa}$
(c) Unfortunately both of above approaches are "direct" measure constructions and don't work perfectly. Perhaps one can refine them by considering some "special" two-valued $\kappa$-additive measure $\mu_{0}$ on $\kappa$ as a "shepherd measure" which controls the non-triviality and maximum-additivity of measures on $\mathbb{M}_{\kappa}$ like $\Omega$ when this measures are "defined" from $\mu_{0}$ in some way.
Question (4): Is there a known "addition" (or "convolution") operator on $\mathbb{M}_{\kappa}$ to "amalgamate" two given two-valued non-trivial $\kappa$-additive measure (in a non-trivial way) for producing another measure on $\kappa$?