580 -II. Cardinal arithmetic

February 5, 2009

Homework problem 5. ({\sf ZF}). Show that if \omega\preceq{\mathcal P}(X) then in fact {\mathcal P}(\omega)\preceq{\mathcal P}(X).

Question. If X is Dedekind-finite but {\mathcal P}(X) is Dedekind-infinite, does it follow that there is an infinite Dedekind-finite set Y such that {\mathcal P}(Y)\preceq X?

From now on, until we cover determinacy, we assume the axiom of choice unless stated otherwise.

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Set theory seminar -Stefan Geschke: Cofinalities of algebraic structures

January 6, 2009

This is a short overview of a talk given by Stefan Geschke on November 21, 2008. Stefan’s topic, Cofinalities of algebraic structures and coinitialities of topological spaces, very quickly connects set theory with other areas, and leads to well-known open problems. In what follows, compact always includes Hausdorff. Most of the arguments I show below are really only quick sketches rather than complete proofs. Any mistakes or inaccuracies are of course my doing rather than Stefan’s, and I would be grateful for comments, corrections, etc.

Definition. Let A be a (first order) structure in a countable language. Write {\rm cf}(A) for the smallest \delta such that A=\bigcup_{\alpha<\delta}A_\alpha for a strictly increasing union of proper substructures.

Since the structures A_\alpha need to be proper, {\rm cf}(A) is not defined if A is finite. It may also fail to exist if A is countable, but it is defined if A is uncountable. Moreover, if {\rm cf}(A) exists, then

  1. {\rm cf}(A)\le|A|, and
  2. {\rm cf}(A) is a regular cardinal.

Example 1. Groups can have arbitrarily large cofinality. This is not entirely trivial, as the sets A_\alpha may have size |A|

Question 1. Is every regular cardinal realized this way?

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