First note that there exists a natural measure $\mu$ on $P(\omega \times \omega)$, inherited from the Lebesgue measure on the reals (by identifying the reals with $P(\omega)$ and $\omega$ with $\omega \times \omega$ in the natural ways)


Let us consider models of $ZFC$ of the form $(\omega, E),$ where $E \subseteq \omega \times \omega$ is interpreted as the $\in$-relation.


Given a theory $T \supseteq ZFC,$ let us consider the set
$M\_T=\{E \subseteq \omega \times \omega:(\omega, E)$ is a model of $T\}$



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> **Question 1.** Is the set $M\_{ZFC}$ measurable? If so, what is its measure?
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In the case that the answer to the above question is yes, and the measure is positive, then one may ask questions like the following which can measure the truth or falsity of statements like $CH$.



> 
> **Question 2** What is the measure of $M\_{ZFC+CH}?$ What about $M\_{ZFC+\neg CH}?$
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If the answer to question 1 is negative, then one may ask the following:



> 
> **Question 3.** What about questions 1 and 2 if we replace the measure with the outer measure of the required sets?
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Edit: By the answer given by Wojowu, and his suggestion, I would also like to ask the following:


**Question 4.** What happens if we replace $ZFC$ with $ZFC -$foundation? 



On the probability of the truth of the continuum hypothesis
For any $E$ modelling ZFC and for each $n\in\omega$, we must have $(n,n)\not\in E$. Therefore $M\_{ZFC}$ is contained in the cylinder set defined by $(0,0)\not\in E,\dots,(n,n)\not\in E$ which has measure $2^{-n-1}$. Therefore $M\_{ZFC}$ has outer measure zero, thus is measurable and has measure zero.


Had we excluded the diagonal from $\omega\times\omega$, we can't have both $(n,m)\in E,(m,n)\in E$ for $n\neq m$, so we get infinitely many independent conditions each of measure $3/4$, so again we get outer measure zero.


Excluding foundation from ZFC makes this method not work, so it can make for a more interesting question, but perhaps we can use a similar trick.


Edit: Without foundation, let $M\_n$ be the set of those models of ZFC-Foundation for which $n$ represents the empty set. We get infinitely many conditions $(m,n)\not\in E$, so $M\_n$ has outer measure zero. Since $M\_{ZFC-Foundation}$ is a countable union of $M\_n$, it has measure zero too. Since I don't know of a reasonable variant of ZFC which doesn't prove existence of the empty set, I think that closes the case.


