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Hi Dick,
I think I read the same article you mentioned. It was the cover story in the August 30, 2003 edition of Science News, titled "Infinite Wisdom". I'm not sure it was the same article though because the one I read doesn't mention the Axiom Of Choice by name at all. It only refers to "the standard axioms" of mathematics.
You said that the article you read concerned "a proof that their [sic] exists no system of mathematics which is consistent with the Axiom of Choice." That may be an implication of the new development, but it was not stated that way by Science News. My personal opinion is that Goedel proved a version of that which could be stated, 'There exists no complete system of mathematics which is consistent with the Axiom of Choice.'
As for finding the Science News article, I found it at: http://216.167.111.80/20030830/bob10.asp . However, I had to log in as a subscriber. You'll probably have to have Diantha save one of your SN copies so you can enter your subscriber number which is printed on your mailing label. Without that, you can't access all the complete articles on-line. You can also get to the website through the front door at www.sciencenews.org ,
Thank you for the link to the Axiom of Choice web site. It was very interesting for me to read and it provided some additional interesting links which I explored. Great fun.
After reading your post, I re-read the SN article and made some notes. I'll share them here.
The article begins by describing Cantor's pairing method of determining the cardinal equivalence of two sets. That is, if the elements of the two sets can be set into one to one correspondence with none left over from either set, the two sets are said to be of equal cardinality. That is, they have the same number of elements. If one set has some elements left over, it is "bigger" than the other set.
Ever since I first learned about this, I have had reservations about using it for infinite sets. My objection goes like this: In this method, for two sets to have equal cardinality, it is important that no elements be left over. But if you imagine setting the elements of two infinite sets, say the natural numbers and the even natural numbers, into one-one correspondence, as you reach any particular point you see that one set is getting "used up" faster than the other. For example, in those two sets, after you have formed the pairs (1,2), (2,4), and (3,6) you can see that you are "using up" the evens twice as fast as the naturals. Or, another way of looking at it, at this point we have used up all the evens up to the number 6, but we have used the naturals up only to the number 3. To me that means that so far, the naturals have three elements "left over". Having numbers left over is something we want to avoid so I say we should start worrying about this "error" right away.
If we imagine continuing the process, hoping that this "error" will correct itself somehow, we can make periodic inspections to try to predict the eventual outcome. If we stop when the evens reach 100 we find there are 50 "left over" and if we stop when the evens reach 1000 we find there are 500 naturals left over. Instead of the problem correcting itself, it gets steadily worse as we go. What could possibly convince us that when the pairing is finally done, there are no naturals left over? To me it is completely counter-intuitive, so I have never accepted the notion of defining cardinality of infinite sets in this way. To me, anything that depends on this definition is nonsense. But, since most mathematicians disagree with me, let me continue with my discussion of the SN article with some quotes and comments.
On page 140 the article says, "In 1938, logician Kurt Goedel proved that the continuum hypothesis is consistent with the standard axioms of set theory. Then in 1963, Paul Cohen, now at Stanford University, proved that the opposite of the continuum hypothesis -- the assertion that there is actually an infinite set that is bigger than the set of counting numbers but smaller than the set of real numbers -- is also consistent with the axioms."
The standard set of axioms includes the AC, and the definition of "bigger" is that there are some elements of the "bigger" set left over after pairing them with elements of the "smaller" set as I described above.
I interpret the continuum hypothesis to be the question, 'Does there exist a set bigger than the set of integers and smaller than the continuum?' It is simply an existence question.
Here is where we may brush up against reality and/or metaphysics. What does it mean for a mathematical object to exist?
I think there are three different possible meanings:
1) To a formalist, existence of an object means that it is possible to produce an unambiguous definition of the object which is consistent with everything else that has been developed in your system so far.
2) To a classical Platonist, existence means that such a definition is out there somewhere and we only need to find it.
3) To a Platonist like me, existence means that someone has actually produced such a definition.
Since we don't want to get into metaphysics, I'll stop here. I just want to point out my views on how mathematics brushes up against reality.
SN (page 140) quotes Woodin (the guy who came up with the proof concerning "elegant" axioms): "Cohen's demonstration "caused a foundational crisis...Here we had a question which should have an answer, but it had been proven that there were no means of answering it."
Me: Why the surprise? Why the crisis? In my opinion, Goedel proved that there inevitably were such questions in systems containing the AC and Cohen simply scared one up.
SN: "Does it even make sense to say the continuum hypothesis is true or false?"
Me: Does it even make sense to say the Pythagorean Theorem is true or false? If, by 'true', you simply mean that you can present a formal proof within the axioms, then it makes sense to say that each of these propositions are either true or false. But if, by 'true', you mean it is consistent with reality, then in both cases you have a difficult, if not impossible, job to do before you can even entertain the question. That job is to unambiguously define each and every term in the proposition in the context of reality. For example, the Pythagorean Theorem deals with the concept of lines. What, exactly, is a "real" line? Is there any such thing as a line in reality? And if so, does it have all the same properties as the lines of geometry? I don't think those questions are at all trivial.
SN: "To formalists, it makes no sense...The hypothesis must be inherently vague."
Me: I think Wittgenstein showed that ALL language statements are "inherently vague".
SN: "To Platonists...the axioms are insufficient."
Me: To me, a Platonist of sorts, the axioms are deficient. The bad apple in the barrel is the AC. I doubt that any axiom can allow for the definition of infinite sets without introducing inconsistencies into the system.
SN (p. 141): "Mathematicians have long known that there is no all-powerful axiom that can answer every question about Cantor's hierarchy [of sizes of infinite sets]."
Me: Im not surprised. When you talk about "every question" you are talking about a LARGE playground. To me, mathematical objects are like objects of our knowledge about reality. And both of these are like the pairing of set elements I talked about earlier. They all have the feature that as you incrementally increase knowledge, you open up more questions than you answered. It is like the familiar image of the inside of a circle representing what we know, the outside of the circle representing what we don't know, and all the unanswered questions which we can sensibly articulate are around the perimeter. As knowledge increases by including previously unanswered questions into the circle, the perimeter increases bringing on more unanswered questions than we had before.
"Elegant" axioms -- defined as those sufficient to settle the continuum hypothesis -- according to Woodin, all make the hypothesis false. To me, using the above imagery, this simply means that the Elegant axioms are on the perimeter of that circle and there are a ton of yet-to-be comprehended questions way beyond the circle that are generated by the AC and the notion of infinity.
On page 141, SN says that one Joel Hamkins says "Woodin's novel approach of sidestepping the search for the right axiom doesn't conform to the way mathematicians thought the continuum hypothesis would be settled."
I'm not surprised at this either. I think most breakthroughs in science, mathematics, and probably every other field as well, are made by people who don't conform to the way everyone else expects things to be done. It reminds me of the current search for a TOE. There are several different approaches, e.g. superstring theory, supergravity, etc. But the best bet at the moment is M-Theory which is sort of an aglommeration of all of the others. That's not the customary way of approaching the problem. Incidentally, as other readers of this post may not know, your "The Foundations of Reality" approach to understanding reality also breaks the conventional mold.
So much for the Science News article. If you find that you read some other article than this one, please let me know. It would be interesting to read that one, especially if the Axiom of Choice were specifically dealt with in the article.
Turning to what you wrote in your post, you wrote:
"The whole issue seems to me...one of applying the consequences to reality or dealing with the implied consequences in reality."
I think that's exactly right. It's fine to build esoteric theoretical structures and see where they lead. But we will learn nothing about reality from that effort if the underlying axioms don't make sense in isomorphism to something real. I don't think the AC makes sense in any real sense.
This is the same as saying that observations of the world are, and must be, finite. You are right: you and I have never "had a serious disagreement on this issue at all."
You said, "[T]here is no such usable number as infinity; however, there is always one a bit bigger than the last one you thought of."
I would amend this to say that there always CAN BE one a bit bigger. All it takes is for someone to define it. But, and here is where my brand of Platonism kicks in, there is no such bigger number unless and until someone actually does expressly define it.
I liked the joke you quoted from Eric's web site. There was a lot of good fun in both that site and the ones he linked to. Thanks again for letting me in on it.
Warm regards,
Paul
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