The effort have the potential to pay off handsomely today

That’s just ok. But then:

People try thing, because they just don’t want it enough.

And finally:

Photographic memory! Remember to put in film, or its digital!?

43.614000-116.202000

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This entry was posted on Thursday, January 6th, 2011 at 2:29 pm and is filed under Life. You can follow any responses to this entry through the RSS 2.0 feed.
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Let $C$ be the standard Cantor middle-third set. As a consequence of the Baire category theorem, there are numbers $r$ such that $C+r$ consists solely of irrational numbers, see here. What would be an explicit example of a number $r$ with this property? Short of an explicit example, are there any references addressing this question? A natural approach would […]

Suppose $M$ is an inner model (of $\mathsf{ZF}$) with the same reals as $V$, and let $A\subseteq \mathbb R$ be a set of reals in $M$. Suppose further that $A$ is determined in $M$. Under these assumptions, $A$ is also determined in $V$. The point is that since winning strategies are coded by reals, and any possible run of the game for $A$ is coded by a real, […]

Yes. This is obvious if there are no such cardinals. (I assume that the natural numbers of the universe of sets are the true natural numbers. Otherwise, the answer is no, and there is not much else to do.) Assume now that there are such cardinals, and that "large cardinal axiom" is something reasonable (so, provably in $\mathsf{ZFC}$, the relevant […]

Please send an email to mathrev@ams.org, explaining the issue. (This is our all-purpose email address; any mistakes you discover, not just regarding references, you can let us know there.) Give us some time, I promise we'll get to it. However, if it seems as if the request somehow fell through the cracks, you can always contact one of your friendly edit […]

The characterization mentioned by Mohammad in his answer really dates back to Lev Bukovský in the early 70s, and, as Ralf and Fabiana recognize in their note, has nothing to do with $L$ or with reals (in their note, they indicate that after proving their result, they realized they had essentially rediscovered Bukovský's theorem). See MR0332477 (48 #1080 […]

This is a nice problem, it turned out more interesting than I first thought. Suppose first that $E$ is a measure zero set, and let's show that there is such a sequence of intervals. Since $E$ is measure zero, for each $\epsilon>0$ there is a sequence $\mathcal I_\epsilon$ of open intervals, the sum of whose lengths adds up to less than $\epsilon$. Co […]

The paper MR1029909 (91b:03090). Mekler, Alan H.; Shelah, Saharon. The consistency strength of "every stationary set reflects". Israel J. Math. 67 (1989), no. 3, 353–366, that you mention in the question actually provides the relevant references and explains the key idea of the argument. Note first that $\kappa$ is assumed regular. They refer to MR […]

Start with Conway's base 13 function $c $ (whose range on any interval is all of $\mathbb R $), which is everywhere discontinuous, and note that if $f $ only takes values $0$ and $1$, then $c+f $ is again everywhere discontinuous (since its range on any interval is unbounded). Now note that there are $2^\mathfrak c $ such functions $f $: the characteris […]

Yes, there are such sets. To describe an example, let's start with simpler tasks. If we just want $P\ne\emptyset$ with $P^1=\emptyset$, take $P$ to be a singleton. If we want $P^1\ne\emptyset$ and $P^2=\emptyset$, take $P$ to be a strictly increasing sequence together with its limit $a$. Then $P^1=\{a\}$. If we want $P^2\ne\emptyset$ and $P^3=\emptyset$ […]

The result was proved by Kenneth J. Falconer. The reference is MR0629593 (82m:05031). Falconer, K. J. The realization of distances in measurable subsets covering $R^n$. J. Combin. Theory Ser. A 31 (1981), no. 2, 184–189. The argument is relatively simple, you need a decent understanding of the Lebesgue density theorem, and some basic properties of Lebesgue m […]