In his proof that the axiom of infinity holds in Quine’s New Foundations, Specker associated to each set a tree, now called the Specker tree of . (The name is due to Randall Holmes.)

In order to make sense of the definition without needing to appeal to the axiom of choice, let us define an equivalence relation on subsets of by saying that iff . The nodes of the tree are equivalence classes of subsets of under . Specifically, at the root we have the class of , and the immediate successors of a class are the classes such that .

Of course, under choice we can simply use cardinals as nodes, rather than the classes .

If is a set, let , the Hartogs function of , be the set of all ordinals that inject into , so is the first ordinal that cannot inject. Sierpiński proved that and therefore . This observation gives us immediately that the Specker tree of any set is well-founded.

Under choice, it follows that all trees have finite rank. This is because any branch through the tree is finite (its nodes being a strictly decreasing sequence of cardinals). Moreover, the smallest cardinal in a branch is larger than the smallest cardinal in any larger branch. But an infinite-rank tree must have arbitrarily large branches, so there is no smallest cardinal in the tree, which is absurd.

Remarkably, it is still open whether in the absence of choice it is consistent that there is a Specker tree of infinite rank.

Typeset using LaTeX2WP. Here is a printable version of this post.

43.614000-116.202000

Advertisements

Like this:

LikeLoading...

Related

This entry was posted on Friday, February 11th, 2011 at 2:53 pm and is filed under 580: Topics in 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.

The only reference I know for precisely these matters is the handbook chapter MR2768702. Koellner, Peter; Woodin, W. Hugh. Large cardinals from determinacy. In Handbook of set theory. Vols. 1, 2, 3, 1951–2119, Springer, Dordrecht, 2010. (Particularly, section 7.) For closely related topics, see also the work of Yong Cheng (and of Cheng and Schindler) on Harr […]

As other answers point out, yes, one needs choice. The popular/natural examples of models of ZF+DC where all sets of reals are measurable are models of determinacy, and Solovay's model. They are related in deep ways, actually, through large cardinals. (Under enough large cardinals, $L({\mathbb R})$ of $V$ is a model of determinacy and (something stronge […]

Throughout the question, we only consider primes of the form $3k+1$. A reference for cubic reciprocity is Ireland & Rosen's A Classical Introduction to Modern Number Theory. How can I count the relative density of those $p$ (of the form $3k+1$) such that the equation $2=3x^3$ has no solutions modulo $p$? Really, even pointers on how to say anything […]

(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 […]

It is not possible to provide an explicit expression for a non-linear solution. The reason is that (it is a folklore result that) an additive $f:{\mathbb R}\to{\mathbb R}$ is linear iff it is measurable. (This result can be found in a variety of places, it is a standard exercise in measure theory books. As of this writing, there is a short proof here (Intern […]

Very briefly: Yes, there are several programs being developed that can be understood as pursuing new axioms for set theory. For the question itself of whether pursuing new axioms is a reasonably line of inquiry, see the following (in particular, the paper by John Steel): MR1814122 (2002a:03007). Feferman, Solomon; Friedman, Harvey M.; Maddy, Penelope; Steel, […]

This is a very interesting question and the subject of current research in set theory. There are, however, some caveats. Say that a set of reals is $\aleph_1$-dense if and only if it meets each interval in exactly $\aleph_1$-many points. It is easy to see that such sets exist, have size $\aleph_1$, and in fact, if $A$ is $\aleph_1$-dense, then between any tw […]

Say that the triangle is $ABC$. The vector giving the median from $A$ to $BC$ is $(AC+AB)/2$. Similarly, the one from $B$ to $AC$ is $(BA+BC)/2$, and the one from $C$ to $BA$ is $(CB+CA)/2$. Adding these, we get zero since $CB=-BC$, etc.

The usual definition of a series of nonnegative terms is as the supremum of the sums over finite subsets of the index set, $$\sum_{i\in I} x_i=\sup\biggl\{\sum_{j\in J}x_j:J\subseteq I\mbox{ is finite}\biggr\}.$$ (Note this definition does not quite work in general for series of positive and negative terms.) The point then is that is $a< x