Recall that the Cantor set is defined as the intersection where

and is obtained by removing from each closed interval that makes up its open middle third, so

,

,

etc. Each is the union of closed intervals, each of length .

Let’s prove that is the interval . (Cf. Abbott, Understanding analysis, Exercise 3.3.6.)

1.

The usual proof consists in showing inductively that for all . This is easy: Note first that

,

where

and

.

This equality is verified by induction. Using this, we can use induction again to verify that, indeed, for all .

We clearly have that . To prove the converse, for each and each , pick such that . The sequence of is bounded, so it has a convergent subsequence . The corresponding subsequence has itself a convergent subsequence . One argues that their limit values belong to , because they belong to each , since these sets are nested and closed. Finally, it follows immediately that as well.

2.

A very elegant different argument is obtained by using an alternative characterization of : Note that each can be written in base three as

where each is , , or . By induction, one easily verifies that iff it admits such an expansion with . It follows that iff it admits an expansion where no is .

Given , we have , so we can write where the ternary expansion of has only s and s (so ), and the expansion of has only s and s: If

,

we can set where unless , in which case as well, and similarly where unless , in which case as well.

We then have that , and both and are in .

This construction has the further advantage of making clear that the typical admits continuum many () representations as sum of two members of : If we can split (where the expansions of only have s and s), we can set

.

This gives us as many representations as subsets of .

3.

The related problem of describing appears to be much more complicated. See here and here.

The description below comes from József Beck. Combinatorial games. Tic-tac-toe theory, Encyclopedia of Mathematics and its Applications, 114. Cambridge University Press, Cambridge, 2008, MR2402857 (2009g:91038). Given a finite set $S$ of points in the plane $\mathbb R^2$, consider the following game between two players Maker and Breaker. The players alternat […]

Yes. This is a consequence of the Davis-Matiyasevich-Putnam-Robinson work on Hilbert's 10th problem, and some standard number theory. A number of papers have details of the $\Pi^0_1$ sentence. To begin with, take a look at the relevant paper in Mathematical developments arising from Hilbert's problems (Proc. Sympos. Pure Math., Northern Illinois Un […]

I am looking for references discussing two inequalities that come up in the study of the dynamics of Newton's method on real-valued polynomials (in one variable). The inequalities are fairly different, but it seems to make sense to ask about both of them in the same post. Most of the details below are fairly elementary, they are mostly included for comp […]

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 […]

Not necessarily. That $\mathfrak m$ is consistently singular is proved in MR0947850 (89m:03045) Kunen, Kenneth. Where $\mathsf{MA}$ first fails. J. Symbolic Logic 53(2), (1988), 429–433. There, Ken shows that $\mathfrak{m}$ can be singular of cofinality $\omega_1$. (Both links above are behind paywalls.)

No, the rank of a set $x$ is the least $\alpha$ such that $x\in V_{\alpha+1}$. Note that if $\alpha$ is limit, any $x\in V_\alpha$ belongs to some $V_\beta$ with $\beta

The real numbers are the usual thing. Surreal numbers are not real numbers, so no, they are not an example of non-constructible reals. Any real $r$ can be written as an infinite sequence $(n;d_1,d_2,\dots)$ where $n$ in an integer and the $d_i$ are digits. Whether the real is rational, constructible or not, is irrelevant. Any rational number, in fact, any al […]

Following Tomas's suggestion, I am posting this as an answer: I encountered this problem while directing a Master's thesis two years ago, and again (in a different setting) with another thesis last year. I seem to recall that I somehow got to this while reading slides of a talk by Paul Pollack. Anyway, I like to deduce the results asked in the prob […]

This is a beautiful and truly fundamental result, and so there are several good quality presentations. Try MR1321144. Kanamori, Akihiro. The higher infinite. Large cardinals in set theory from their beginnings. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, 1994. xxiv+536 pp. ISBN: 3-540-57071-3, or any of the newer editions (the 2003 second ed […]

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