Definition (1): A unique measurable cardinal is a measurable cardinal with a unique normal measure.
Remark (1): Unique measurable cardinals have a very special property. Note We know that if $\kappa$ is a measurable cardinal and $\mu$ be a two-valued non-trivial trivial$\kappa$-additive measure on it then the corresponding inner model produced by Mostowski collapse of Scott's ultraproduct ($M_{\kappa,\mu}$) is dependent on both $\kappa$ and $\mu$. BySo by changing measures on a certain measurable cardinal we can produce many different inner models. But on a unique measurable cardinal $\kappa$, changing measures don't give us a new inner modelsEven by transitivity if two such inner models be isomorphic then they are equal. Because if $\mu_{1}$ and $\mu_{2}$ be two Rudin-Kiesler equivalent measures onNow a question is:
"Is it possible to have a measurable cardinal $\kappa$ then we have $M_{\kappa,\mu_{1}}\cong M_{\kappa, \mu_{2}}$ and by transitivity of these models it is clear that $M_{\kappa,\mu_{1}}= M_{\kappa, \mu_{2}}$. So if $\kappa$ bewhich its inner models be independent from its measures?"
Precisely:
Definition (1): A "unique measurable cardinal" is a unique measurable cardinal with a unique normal measurewhich for all two-valued non-trivial $\mu_{0}$, then any other measure on $\kappa$ is Rudin-Kiesler equivalent with $\mu_{0}$ and so $M_{\kappa , \mu_{0}}$ will be the unique inner model which one can produce by Scott's constructionadditive measures like $\mu$ and $\mu'$ on it we have $M_{\kappa , \mu}=M_{\kappa , \mu'}$. WeWe show it by $M_{\kappa}$this unique inner model by $M_{\kappa}$.
Question (1): Is the following statement true?
$Con(ZFC+ \text{there exists a measurable cardinal})\Longrightarrow$ $Con(ZFC+ \text{there exists a unique measurable cardianl})$
Now the main question is about the behavior of the "well defined" function $\kappa \mapsto M_{\kappa}$:
Definition (2): Define:
The collection of all unique measurable cardinals:
$UM:=\lbrace \kappa~|~\kappa~\text{is a unique measurable cardinal}\rbrace$
The (informal) collection of all inner models of unique measurable cardinals:
$IUM:=\lbrace M_{\kappa}~|~\kappa \in UM \rbrace$
Question (12): Is the function $\kappa \mapsto M_{\kappa}$ from $UM$ to $IUM$ (strictly) increasing? In the other words which one of the following statements are true?
$(1)~\forall \kappa,\lambda \in UM~~~~~(\kappa < \lambda \longrightarrow M_{\kappa}\subseteq M_{\lambda})$
$(2)~\forall \kappa,\lambda \in UM~~~~~(\kappa < \lambda \longrightarrow M_{\kappa}\subsetneq M_{\lambda})$