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Topic: Division without the axiom of choice
Replies: 10   Last Post: Jan 12, 2013 4:51 PM

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Butch Malahide

Posts: 894
Registered: 6/29/05
Re: Division without the axiom of choice
Posted: Jan 10, 2013 7:59 PM
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On Jan 10, 4:15 am, 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:

> 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 <= n then m <= n.

F. Bernstein, Untersuchungen aus der Mengenlehre, Math. Annalen 61
(1905), 117-155. [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), 1-16. [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), 148-154. [Proof of Theorem 2
for k = 2.]

A. Tarski, Cancellation laws in the arithmetic of cardinals, Fund.
Math. 36 (1949), 77-92. [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.

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