Update. I present here a quick sketch of the solution of Exercise 3.(b). See Lecture 18, where it is shown that the result actually holds in , although the proof uses choice.

Let and be two well-orderings of a set . We want to find a subset of of the same size as where the two well-orderings coincide. Let . By combining with an isomorphism between and its order type, we may assume that is an ordinal and . By restricting attention to the subset of , we may assume is a well-ordering of . By further restricting to the subset of of order type under , we may assume that as well.

Assume first that is regular. The result follows easily. The desired set can be built by a straightforward recursion: Given and a sequence of elements of increasing under both well-orderings, regularity ensures that the sequence is bounded under both well-orderings, and we can find which is larger than all the previous under both orderings.

The argument for singular is slightly more delicate. Namely, we may not be able to carry out the construction above since the sequence could be unbounded in one of the orderings when . We circunvent the problem by only considering ordinals whose cofinality is larger than the cofinality of . Notice that if an increasing sequence of order type is unbounded in an ordinal of cofinality , then .

To implement this idea, let be an increasing sequence of regular cardinals cofinal in , with . Consider the subset . It must contain a subset of size where coincides with . By the remark above, this subset is bounded in . Let denote the shortest initial segment of containing . By removing from the set , we are left with a set of size , and any ordinal there is larger than the elements of under both orderings. The induction continues this way, by considering at stage a set of size .

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Tuesday, May 20th, 2008 at 12:40 am and is filed under 116c: Set theory. You can follow any responses to this entry through the RSS 2.0 feed.
You can leave a response, or trackback from your own site.

(1) Patrick Dehornoy gave a nice talk at the Séminaire Bourbaki explaining Hugh Woodin's approach. It omits many technical details, so you may want to look at it before looking again at the Notices papers. I think looking at those slides and then at the Notices articles gives a reasonable picture of what the approach is and what kind of problems remain […]

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 […]

First of all, $f(z)+e^z\ne 0$ by the first inequality. It follows that $e^z/(f(z)+e^z)$ is entire, and bounded above. You should be able to conclude from that.

Yes. The standard way of defining these sequences goes by assigning in an explicit fashion to each limit ordinal $\alpha$, for as long as possible, an increasing sequence $\alpha_n$ that converges to $\alpha$. Once this is done, we can define $f_\alpha$ by diagonalizing, so $f_\alpha(n)=f_{\alpha_n}(n)$ for all $n$. Of course there are many possible choices […]

I disagree with the advice of sending a paper to a journal before searching the relevant literature. It is almost guaranteed that a paper on the fundamental theorem of algebra (a very classical and well-studied topic) will be rejected if you do not include mention on previous proofs, and comparisons, explaining how your proof differs from them, etc. It is no […]

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