580 -Some choiceless results (3)

[Updated December 3. The previous proof that there is a canonical bijection for all infinite ordinals was seriously flawed. Thanks to Lorenzo Traldi for pointing out the problem.]

5. Specker’s lemma.

This result comes from Ernst Specker, Verallgemeinerte Kontinuumshypothese und Auswahlaxiom, Archiv der Mathematik 5 (1954), 332-337. I follow Akihiro Kanamori, David Pincus, Does GCH imply AC locally?, in Paul Erdős and his mathematics, II (Budapest, 1999), Bolyai Soc. Math. Stud., 11, János Bolyai Math. Soc., Budapest, (2002), 413-426 in the presentation of this and the following result. The Kanamori-Pincus paper, to which we will return next lecture, has several interesting problems, results, and historical remarks, and I recommend it. It can be found here.

580 -Some choiceless results (2)

There are a few additional remarks on the Schröder-Bernstein theorem worth mentioning. I will expand on some of them later, in the context of descriptive set theory.

The dual Schröder-Bernstein theorem (dual S-B) is the statement “Whenever are sets and there are surjections from onto and from onto then there is a bijection between and .”

* This follows from the axiom of choice. In fact, is equivalent to: Any surjective function admits a right inverse. So the dual S-B follows from choice and the S-B theorem. 

* The proofs of S-B actually show that if one has injections and , then one has a bijection contained in . So the argument above gives the same strengthened version of the dual S-B. Actually, over , this strengthened version implies choice. This is in Bernhard Banaschewski, Gregory H. Moore, The dual Cantor-Bernstein theorem and the partition principle, Notre Dame J. Formal Logic 31 (3), (1990), 375-381. 

* If is onto, then there is 1-1, so if there are surjections in both directions between and , then and have the same size. Of course, this is possible even if and do not.

Open question. () Does the dual Schröder-Bernstein theorem imply the axiom of choice?

* The dual S-B is not a theorem of .