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 Sep 13 revised Intersection of Borel subset with line deleted 2 characters in body Sep 13 comment $\overline{E}\subset \bigcup_{x\in E}\overline{B_x(r_x)}$ if $E\subset \bigcup_{x\in E}\overline{B_x(r_x)}$ In that case $\overline E = [0,1]$ is covered by the union if we choose the $r_x$ for $x\in (0,1)$ in the same way as above. But we could also take an enumeration of $E\cap (0,1) = \{ x_1,x_2,x_3,.. \}$ and let $r_{x_i}' = \frac{r_{x_i}}{2^{i+1}}$ as well as $r_{0} = r_{1} = \frac{1}{8}$ Then the union of the $\overline {B_{x}(r_x')}$ has measure at most $\frac{3}{4}$ and thus cannot cover $[0,1]$. Sep 13 answered $\overline{E}\subset \bigcup_{x\in E}\overline{B_x(r_x)}$ if $E\subset \bigcup_{x\in E}\overline{B_x(r_x)}$ Sep 13 comment Intersection of Borel subset with line Absolutely. Thanks for telling me, I corrected it. Sep 13 revised Intersection of Borel subset with line edited body Sep 13 comment Intersection of Borel subset with line Right, it is a bit easier to start with closed sets rather than with open for the proof, so I changed that. Sep 13 revised Intersection of Borel subset with line added 2 characters in body Sep 13 comment Intersection of Borel subset with line @Carl Mummert: Please have a look at the question again. It isn't asking about the projection of the Borel set, but about the sets $A(y)$ for various $y\in\mathbb{R}$, there is a difference. Sep 13 answered Intersection of Borel subset with line Sep 12 revised 1. Measurable sets : uncountable union, 2. null set is disjoint from translated rationals added 349 characters in body Sep 12 answered 1. Measurable sets : uncountable union, 2. null set is disjoint from translated rationals Sep 12 awarded Custodian Sep 12 reviewed Reviewed Proving a Square Root of a Symmetric Matrix Sep 12 awarded Informed Sep 11 comment Partial derivative and dependent variables The function $f$ might be given by a formula that technically (mathematically) still makes sense for such $a,b$ that do not satisfy the constraint. Then $\partial f /\partial a$ would be the local rate of change of the value of $f$ if we just increase $a$ independently. That doesn't give us any result for the context of the constraint, but we could compute the directional derivative, which is the gradient of f multiplied with a unit vector indicating a direction. This gives us the rate of change of $f$ if we change $a$ and $b$ along the line with that direction-this makes sense in context. Sep 8 comment What is the Sequence? $1+\frac{n(n+1)}{2}$ Sep 4 comment The sequence $x_{n+1}=ax_{n}+b$ converges to where? A nitpick: Your rule only works for $x_0=1$, otherwise it's $x_0 a^n$ instead of $a^n$. Sep 2 comment Is the critical point of an embedding of a model of set theory inaccessible in it? Yeah, I meant the second version with the constant symbol, sorry I wasn't being clear enough on that. Ah but I don't think "that model itself is ill-founded" is an issue, because that's just talking about being externally illfounded, in contrast the axiom of foundation holds inside that model, so just for sake of consistency of the theory, that should be irrelevant. (We might as well take that overarching model to be V. ) Sep 2 comment Is the critical point of an embedding of a model of set theory inaccessible in it? Sure there are such models that contain no transitive models. But saying that there exists a model of ZFC with a transitive element that is a model is different to saying that every model of ZFC contains a model like that. Showing that every finite subset of a theory is consistent is by the compactness theorem enough to show that the theory is consistent. In this case, the theory would be "ZFC+(there exists a transitive model of ZFC)". Sep 2 comment Is the critical point of an embedding of a model of set theory inaccessible in it? But we can show from the consitency of ZFC the consistency of ZFC+there is a transitive model: Given a finite subset of ZFC, by reflection in any model of ZFC there is a $\theta$ so that $V_{\theta}$ satisfies this subset, so ZFC+(there is transitive M with $M\models$(this subset)) is consistent.