I composed this after reading your original post; however when I came to post it I read your answer to Mike so I posted it here. I have no arguments with anything you said.
I only comment here because I just recently read a news article concerning a proof that their exists no system of mathematics which is consistent with the Axiom of Choice. The proof is via a means completely new to mathematics: concerns some kind of set of elements which the guy calls "elegant" axioms which the article did not define (except to say that the method was unusual to mathematics). From what I read, I got the impression that an elegant axiom was one which is consistent with the Axiom of Choice and which sufficed to provide a paradox free derivation of a mathematics. Somehow the guy goes through a proof that the existence of an "elegant" axiom produces a paradox (only my impression you understand). His paper has been in review by a number of mathematical societies for about two years now. As of the moment, the proof has not been accepted but it is apparently being taken very seriously.
I thought the article was in my latest issue (or maybe last weeks issue) of the "Science News" book review section. Diantha has already thrown it out so I can't check that. I looked at the "Science News" web site and couldn't find any reference to it (perhaps I read it somewhere else). I wish I could find a reference to it for you.
In looking for it, I ran across this reference which I thought people here might find interesting.
Particularly the concise statement of the Axiom of Choice.
Axiom of Choice: Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S.
The whole issue seems to me (and of course, my mathematics though extensive is pretty rote and unthoughtout) one of applying the consequences to reality or dealing with the implied consequences in reality. Mathematically, I think I have to go with Harv and Eric Schechter that mathematics is just a game played with arbitrary rules and, so long as the rules are themselves not inconsistent with one another, anything is allowed. Applying the consequences to reality is another thing entirely. I think I made that clear in my treatise which I stated that the number of observations and the knowable data making up those observations had to be finite. I wasn't aware that you and I ever had a serious disagreement on this issue at all.
Fundamentally the solution in real applications is to use "infinite" or "infinity" to describe the limiting characteristics of an algorithm and not to think of the term as being a number: i.e., there is no such usable number as infinity; however, there is always one a bit bigger than the last one you thought of.
I would like to slowly and carefully go into detail about why I think denying the AC is a sensible thing to do, and I would like to have you slowly and carefully tell me why you think it is nonsense.
We have discussed this before and I think you know my answer: it is nonsense if we are just playing the game of describing procedures but it is far from nonsense if we are talking about obtaining a result. I look forward to your discussion and will comment when I think something should be said.
Another quote from Eric's web site!
Jerry Bona once said,
The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?
This is a joke. In the setting of ordinary set theory, all three of those principles are mathematically equivalent -- i.e., if we assume any one of those principles, we can use it to prove the other two. However, human intuition does not always follow what is mathematically correct. The Axiom of Choice agrees with the intuition of most mathematicians; the Well Ordering Principle is contrary to the intuition of most mathematicians; and Zorn's Lemma is so complicated that most mathematicians are not able to form any intuitive opinion about it.
I think there is a lot being said in that joke!
Have fun guys -- Dick