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 Feb 4 comment Rational valued functions on the Cantor set with $\int_{C} f^{3}d\mu=1$ The natural question follows: can we do a function which is locally constant around no point? Feb 3 comment Dimensions of dual vector spaces Also: I suspect one can — and I suspect you know how to — manufacture an example where $V$ and $V^*$ have the same dimension on one side. If so, you should maybe state this, and give an example. (And if not, you should maybe add this as an easier or preliminary question.) Feb 3 comment Dimensions of dual vector spaces Is this the sort of example of the "perhaps less well known" fact? If $F$ is the field of rational functions over $\mathbb{Q}$ in $2^{\aleph_0}$ indeterminates, which has dimension $2^{\aleph_0}$ over $\mathbb{Q}$ (I think), and if $V$ has dimension $\aleph_0$ over $F$ then $V^*$ has dimension $2^{\aleph_0}$ over $F$, so they both have dimension $2^{\aleph_0}$ over $\mathbb{Q}$ (so they have the same cardinal and abelian group structure). Right? Feb 3 accepted Do the analogies between metamathematics of set theory and arithmetic have some deeper meaning? Feb 3 comment Pointwise convergence for continuous functions The keyword you want to search for is "Baire class 1" functions. See for example Kechris's Classical Descriptive Set Theory (Springer GTM 156), §24.B. One characterization is that $f\colon\mathbb{R}\to\mathbb{R}$ is pointwise limit of continuous functions iff $f^{-1}(U)$ is a countable union of closed sets for every open set $U$ (op. cit., 24.10). Its set of continuous points is then a comeager (hence dense) $G_\delta$ (op. cit., 24.14). Feb 3 comment Pointwise convergence for continuous functions Amusingly, the function $\mathbf{1}_{\mathbb{Q}}$ taking the value $1$ on the rationals and $0$ on the irrationals, which is discontinuous everywhere, is not (as per smyrlis's answer) pointwise limit of continuous functions (=Baire class 1), but it is (Baire class 2) pointwise limit of functions like this one which are themselves pointwise limits of continuous functions. So pointwise limits of pointwise limits are not necessarily pointwise limits. Feb 1 comment Rank of a fat random matrix To put Robert Israel's answer differently, non-full-rank matrices are a (singular) algebraic subvariety of $\mathbb{C}^{n\times k}$ that is not the full space, so it has codimension at least $1$ and Lebesgue measure zero: with probability $1$ a random matrix has full rank, you don't need to take a limit. (Over finite fields, of course, things would be different.) Jan 25 answered Wanted, dead or alive: Have you seen this curve? (circular variant of cardioid) Jan 25 comment $T$-nilpotent ideals See Lam, A First Course in Noncommutative Rings (2d edition Springer 2001, GTM 131), §23 ("Perfect and Semiperfect Rings"), esp. around definition 23.18 (of left-perfect rings) and theorem 23.20 (Bass criterion). Jan 25 comment $T$-nilpotent ideals I suppose you are well aware of this, but just in case other readers might be, like me, a little forgetful, a ring is left-perfect iff $J(R)$ is left T-nilpotent and $R/J(R)$ is semisimple, i.e., left-Artinian (where $J(R)$ is the Jacobson radical). Jan 24 comment Wanted, dead or alive: Have you seen this curve? (circular variant of cardioid) If you say the straight-line version draws a cardioid, then the curve you describe, or at least the "inside" half of it, should be the transform of the cardioid under the transformation that takes the Beltrami-Klein model of projective space to the Poincaré disk model (because the points on the unit circle are preserved by this transformation, which then maps the line segment to the circle arc, and tangency will be preserved). Maybe there is no better description. Jan 24 comment Wanted, dead or alive: Have you seen this curve? (circular variant of cardioid) You should try to compute a polar equation for the curve, this will make it easier to look up. Try to find it in this site, which seems to list an unreasonable number of historical curves and constructions thereof, and, if you can't, try contacting its author. Jan 18 comment “Partial-computably isomorphic” sets Great! Knowing the True Name of a concept unlocks a lot of power to search about it. Thank you also for the final comment, I had wondered about that among other things. Jan 18 accepted “Partial-computably isomorphic” sets Jan 18 revised “Partial-computably isomorphic” sets point out some easy facts and a connection with isomorphism in the effective topos Jan 18 comment Reducing Consistency of $PA$ $\mathsf{PA}$ proves the consistency of $I\Sigma_n$ for all $n$, so it does not prove the equivalence between its own consistency and that of $I\Sigma_n$ for any $n$. Also, I don't think $\mathsf{HA}$ should really be considered a subtheory of $\mathsf{PA}$, as the logic is different. Jan 16 answered Between mu- and primitive recursion Jan 16 comment “Partial-computably isomorphic” sets The $\sim$-class of $\mathbb{N}$ is exactly the set of infinite computably enumerable sets (given an infinite c.e. set we can find a 1-to-1 enumeration of it, proving the equivalence; conversely, the function $f$ in the condition $\mathbb{N}\sim E$ shows that $E$ is c.e.). I should probably have mentioned this. So all infinite c.e. sets are identified but not, say, with the complements of c.e. sets or anything of degree $0''$. Jan 16 asked “Partial-computably isomorphic” sets Jan 3 revised Is the absolute of a compact space the projective limit of the Stone-Čech compactifications of its open dense subsets? fix reference in Fine-Gillman-Lambek (I was trying to re-read this and noticed this mistake)