
Re: Division without the axiom of choice
Posted:
Jan 10, 2013 10:49 PM


On Jan 10, 6:59 pm, Butch Malahide <fred.gal...@gmail.com> wrote: > On Jan 10, 4:15 am, pepste...@gmail.com wrote: > > > Let A and B be sets. Assume ZF without assuming choice. Then, for all [positive] integers n, I believe (correct me if I'm wrong) that, if n x A is equipotent to n x B, then A is equipotent to B. > > Yes, that is correct. More generally, it is proved in ZF that, for any > cardinals a and b, and any nonzero finite cardinal (i.e. natural > number) n, the inequality na <= nb implies a <= b. (This implies the > proposition you asked about, in view of the well known fact that a = b > <> a <= b & b <= a.) > > > There's a famous Conway/Doyle paper which proves this for n = 2 and n = 3. > > However, it doesn't seem rigorous or clear and I have trouble understanding it. > > I don't know the Conway/Doyle paper, and I don't know a proof for n = > 3. A proof for n = 2 has been posted in this newsgroup: > > http://groups.google.com/group/sci.math/msg/1e65b64fee74fe07?hl=en > > > Does anyone know a more axiomatic treatment? (I don't have access to a university, and I'm not in the market for maths purchases, so only free references would be helpful.) > > The following summary and references are cribbed and paraphrased from > p. 174 of Waclaw Sierpinski's book Cardinal and Ordinal Numbers, > second edition revised, Warszawa, 1965. The theorems are theorems of > ZF (standard set theory without the axiom of choice); m and n are > arbitrary cardinals (i.e., if they are infinite, they are not > necessarily alephs); k is a natural number. The theorems you are > interested in are: > > THEOREM 1. If km = kn then m = n. > > THEOREM 2. If km <= kn then m <= n. ***TYPO CORRECTED*** > > F. Bernstein, Untersuchungen aus der Mengenlehre, Math. Annalen 61 > (1905), 117155. [Proves Theorem 1 for k = 2 and outlines a proof for > general k.] > > W. Sierpinski, Sur l'egalite 2m = 2n pour les nombres cardinaux, Fund. > Math. 3 (1922), 116. [Another proof of Theorem 1 for k = 2.] > > W. Sierpinski, Sur l'implication (2m <= 2n) > (m <= n) pour les > nombres cardinaux, Fund. Math. 34 (1947), 148154. [Proof of Theorem 2 > for k = 2.] > > A. Tarski, Cancellation laws in the arithmetic of cardinals, Fund. > Math. 36 (1949), 7792. [Proof of Theorem 2 in general.] > > I guess Tarski's 1949 paper has what you're looking for. I don't know > if it's available as a free etext; I'm inclined to doubt it, but I > haven't looked. On the other hand, I bet your local public library can > get you a copy at nominal cost by interlibrary loan.
Sorry about the typo in the statement of Theorem 2!

