Axiom Of Choice?

[ashleyisachild]

I was recently talking about the existence of contradictions with a friend who's studying math. He was citing that an exception to the rule of noncontradiction was the mathematical "axiom of choice". It essentially asserts that something very obviously true is true, but the proof of it necessarily ends in something that contradicts the nature of the real numbers.

I'm no mathematician, so I didn't know how to address this, and I'm pretty sure my friend wasn't mistaken or lying. Does anyone know the solution to this dilemma?

[Lord Poppycock]

. . . I'm pretty sure my friend wasn't mistaken or lying.

It sounds like he was.

[stephen_speicher]

I was recently talking about the existence of contradictions with a friend who's studying math.  He was citing that an exception to the rule of noncontradiction was the mathematical "axiom of choice".  It essentially asserts that something very obviously true is true, but the proof of it necessarily ends in something that contradicts the nature of the real numbers.

I'm no mathematician, so I didn't know how to address this, and I'm pretty sure my friend wasn't mistaken or lying.  Does anyone know the solution to this dilemma?

I do not know whether you friend was lying to you, but he certainly is misusing the Axiom of Choice. This axiom was discovered a century ago in the development of set theory, and it continues to engender much discussion amongst mathematicians today. But it usually takes those involved in "philosophy" to distort the issue in a multitude of ways. Simply put, the Axiom of Choice says that given a collection of non-empty sets, you can create a set containing one element from each of the collection by choosing a member from each set. What this means, _within that set theory_, for the real numbers, does not "contradict their nature," but rather, due to a technical issue known as "well-ordering," the standard ordering of the real numbers is not considered to be well-ordered. So that set theory cannot prescribe a definite function to well-order the reals. So what?

[Randrew]

Wow--what a coincidence! I was JUST about to start a thread on this very topic TODAY! Simply amazing! I think this topic would be more appropriate under Epistemology, but I suppose it's ok if we continue it here.

OK, now I will give my input on this:

The problem with the Axiom of Choice is that, using it, one can prove the infamous Banach-Tarski Paradox which states (actually, implies) that one can take a sphere, cut it into six parts, and rearrange the parts to form a new sphere with double the volume of the original (without creating any new empty space inside the object.)

Equivalently, one can take a pea and rearrange its parts (using only rotations and translations) to form a new object with volume greater than the sun.

So, what do you know of *this*, Mr. Speicher?

There are many equivalent ways of stating AC (set theorist's shorthand for the Axiom of Choice), but the one Stephen gave is probably the most common.

So, the question that Objectivists need to answer is (perhaps): How can AC be re-stated so as not to be misapplied to uncoutable sets in such a way that crazy results like Banach-Tarski paradox?

I recently checked out S. Wagon's book on the paradox, but the technical framework required to understand it is pretty heavy, so it's going to take me some time to finish. The reason I'm doing this is that I want to critically examine the set(s) that are created using AC in order to prove the paradox.

I have already done so with Vitali's construction of a subset of R that is non-Lebesgue measurable, and I found no problem whatsoever with his use of AC. I imagine, though, that Banach-Tarski will be considerably more complicated.

So, are there any mathematicians out there who could explain to me how BT uses AC and what problems there might be with this?

Also, a question for Stephen: what does it mean to well-order the real numbers? Is it anything like the Well-ordering property of the natural numbers?

[stephen_speicher]

Wow--what a coincidence!  I was JUST about to start a thread on this very topic TODAY!  Simply amazing!

Anticipating the future is a hard job, but someone has to do it.

The problem with the Axiom of Choice is that, using it, one can prove the infamous Banach-Tarski Paradox which states (actually, implies) that one can take a sphere, cut it into six parts, and rearrange the parts to form a new sphere with double the volume of the original (without creating any new empty space inside the object.)

Equivalently, one can take a pea and rearrange its parts (using only rotations and translations) to form a new object with volume greater than the sun.

So, what do you know of *this*, Mr. Speicher?

When you transform that pea into a Sun I will give it due consideration. Mathematical methods are not the same thing as physical reality, and when you use concepts of method, especially those that have embedded infinities, you have to be very careful about directly applying that method to the physical world. Unlike some others I do not dismiss set theory out of hand, but I do think that truly foundational mathematics requires something other than set theory at its base. Do not expect any pea to become a Sun within your lifetime.

So, the question that Objectivists need to answer is (perhaps): How can AC be re-stated so as not to be misapplied to uncoutable sets in such a way that crazy results like Banach-Tarski paradox?

Objectivism and Objectivists do not need to answer such a question. Objectivism is a philosophy, not a science. Aside from some basic metaphysical and epistemological principles Objectivism has little to say about abstract mathematics. I for one am not a professional mathematician, though I am interested in and have studied a great deal of this subject. In my opinion time and effort would be more benefically spent in developing a more appropriate foundational base for mathematics.

I recently checked out S. Wagon's book on the paradox, but the technical framework required to understand it is pretty heavy, so it's going to take me some time to finish.  The reason I'm doing this is that I want to critically examine the set(s) that are created using AC in order to prove the paradox.

I have already done so with Vitali's construction of a subset of R that is non-Lebesgue measurable, and I found no problem whatsoever with his use of AC.  I imagine, though, that Banach-Tarski will be considerably more complicated.

What Stan Wagon does not tell you, though, is that Vitali, when he did this almost a century ago, in 1905, based his work on Zermelo's original 1904 presentation, and there were several legitimate concerns voiced against Zermelo's proof at that time. (This is a major part of the reason that Zermelo developed his second proof in 1908.) So Vitali, who developed what is now considered to be a classic paradoxical decomposition, himself used a questionable base at the time.

So, are there any mathematicians out there who could explain to me how BT uses AC and what problems there might be with this?

That is a big task. If you had trouble with Wagon's book you might first try Charles Pinter's Set Theory which is a very nice understandable presentation at the undergraduate level.

Also, a question for Stephen: what does it mean to well-order the real numbers?  Is it anything like the Well-ordering property of the natural numbers?

My comment about the reals had to only with the standard ordering, which is not well-ordered. But, yes, the same would apply. A well-ordered set is just a totally-ordered set where every non-empty subset contains a least member.

[Eric]

The problem with the Axiom of Choice is that, using it, one can prove the infamous Banach-Tarski Paradox ...

Actually, I would say that a more fundamental problem with AC is that it asserts existence without identity. Virtually all of (pure) mathematics divorced existence from identity long ago - and the result is an abstract game with no practical applications.

So, are there any mathematicians out there who could explain to me how BT uses AC and what problems there might be with this?

If you have some training in abstract algebra, a very readable proof of the paradox can be found in:

R. M. Robinson, "On the Decomposition of Spheres.", Fundamenta Mathematicae, V. 34, p. 246-260, 1947.

[ashleyisachild]

What this means, _within that set theory_, for the real numbers, does not "contradict their nature," but rather, due to a technical issue known as "well-ordering," the standard ordering of the real numbers is not considered to be well-ordered. So that set theory cannot prescribe a definite function to well-order the reals. So what?

So, does that mean that most mathematicians who consider the axiom a contradiction are expecting it to "well-order" the reals or something? Also, you seem to be saying that there's nothing wrong with the theory itself, just its metaphysical interpretation. If that were true, how could it lead to a proof of the Banach-Tarski paradox? I'm obviously not expecting a pea to be rearranged to fill volumes larger than the sun, but isn't there some clear-cut error in proof if something implies that it theoretically could?

The problem with the Axiom of Choice is that, using it, one can prove the infamous Banach-Tarski Paradox which states (actually, implies) that one can take a sphere, cut it into six parts, and rearrange the parts to form a new sphere with double the volume of the original (without creating any new empty space inside the object.)

Wow, yeah, my friend mentioned that too, and I decided not to mention it in this thread but it's funny that you did instead.

Aside from some basic metaphysical and epistemological principles Objectivism has little to say about abstract mathematics.

My comment about the reals had to only with the standard ordering, which is not well-ordered. But, yes, the same would apply. A well-ordered set is just a totally-ordered set where every non-empty subset contains a least member.

I think that if objectivism sets out basic metaphysical and epistemological principles, it should provide means to support those principles and refute assertions to the contrary. If objectivism (or objectivists) find it important to find the errors of other apparent contradictory assertions, then this one is just as important.

What is the standard ordering? Do mathematicians not take the standard ordering's lack of well-orderedness into account when pointing out the contradictions of AC's lack of ability to well-order the reals? (Sorry if I'm misusing some words and sounding dumb... I'm not too knowledgeable with abstract math terminology.)

Actually, I would say that a more fundamental problem with AC is that it asserts existence without identity.

How does it do that? And but doesn't all of math do that?

[Free Capitalist]

Dr. Speicher, this is closely linked to Godel's Incompleteness Theorem, isn't it? If it wasn't he who authored the Axiom of Choice, do you know the name of the mathematician who did? I was under the impression that Godel published his theory as an answer to Cantor and the 19th thrust of mathematics as logical and causal; by corollary, I assumed that he revolutionized (destroyed?) the field, which was up to that point still dominated by the 19th century approach.

[stephen_speicher]

I think that if objectivism sets out basic metaphysical and epistemological principles, it should provide means to support those principles and refute assertions to the contrary. If objectivism (or objectivists) find it important to find the errors of other apparent contradictory assertions, then this one is just as important.

Objectivism is a philosophy, not a science, and, as I indicated, other than some guiding metaphysical and epistemological principles, it has nothing to say about the details of physics or mathematics. But, if a mathematician tells you that he has proved that contradictions exist, you need not be a mathematician to tell him he is wrong. Or, if a physicist says that he has demonstrated that identity does not hold, without being a physicist you can be sure he has erred. It is not encumbent upon Objectivism or Objectivists to answer in detail to every false claim that is made.

Now, with that said, as I previously indicated, in fact I have studied these issues in some detail. I mentioned before that I do not think that set theory is the proper foundational base for mathematics, but unlike some others I recognize that aspects of the theory do have value. These are very complex technical issues and they require a lot of specialized study if you really want to understand what is right, and what is wrong, with these theories. You simply cannot stomp your foot on the ground and demand that Objectivism, or Objectivists, somehow give you the technical expertise to answer in detail the assertions of a knowledgeable mathematican. Either put forth the effort to study these issues on your own, or be content with answering what is answerable in terms of philosophical principles.

[stephen_speicher]

Dr. Speicher, this is closely linked to Godel's Incompleteness Theorem, isn't it? If it wasn't he who authored the Axiom of Choice, do you know the name of the mathematician who did? I was under the impression that Godel published his theory as an answer to Cantor and the 19th thrust of mathematics as logical and causal; by corollary, I assumed that he revolutionized (destroyed?) the field that was previously dominated by the 19th century approach.

In 1904 Ernst Zermelo was the first to formulate the Axiom of Choice. In his 1904 paper it was not denoted as the Axiom of Choice, but shortly thereafter that is the name Zermelo used. Kurt Goedel was not born until two years after Zermelo's 1904 paper, so he had little to say on this matter for a while. Goedel was to later complete the mathematicization of logic begun earlier by the likes of Georg Cantor, synthesizing what had come before and establishing relative consistency between Zermelo's Axiom of Choice and the Continuum Hypothesis that was initiated by Cantor. Note that set theory had its roots in Cantor, as early as 1873.

p.s. I am not a "Dr."

[Free Capitalist]

Sorry In that case I'll have to use Stephen, against my wishes, as "Mr. Speicher" sounds too formal and distant, and "Prof. Speicher" sounds rather inappropriate for a forum like this. My hand is forced

Back to Goedel, did he really complete Cantor's project? I forgot then, if not Cantor then who was the big late 19th century mathematician who attempted to go the other way and prove the logicity of math? From what I understand, early 20th century mathematicians buckled against his attempt, claiming it would make math "uninteresting" and complained that by that mathematician's plan, everything in math would be unfortunately provable through a finite set of steps. Then, apparently, Godel came along and gave them all a sigh of relief by 'proving' that 100% consistency and logical strictness were impossible in math.

Since then people have used his Theorem to try and undercut the validity of logic itself, which is very similar to mathematics in its method.

[Randrew]

What Stan Wagon does not tell you, though, is that Vitali, when he did this almost a century ago, in 1905, based his work on Zermelo's original 1904 presentation, and there were several legitimate concerns voiced against Zermelo's proof at that time. (This is a major part of the reason that Zermelo developed his second proof in 1908.) So Vitali, who developed what is now considered to be a classic paradoxical decomposition, himself used a questionable base at the time.

Actually, I read about the construction not in Wagon but in Measure and Integral by Zygmund and Wheeden.

Alright, then, I won't give the full proof here since it's too technical, but let me say what the set was and then tell me if you see anything wrong with it:

For any real number x, let E(x) (actually, E subscripted with x) be the set of all reals of the form x+r, where r is any rational number. This is an equivalence relation that partitions the entire real line, so for any two real numbers x and y, either E(x) and E(y) are identical or they are disjoint.

Now we construct a set E by selecting exactly one member from each equivalence class (i.e., let E be a set that consists of exactly one member from each of the aforementioned equivalence classes.) Using some technical machinery and a cool lemma about measurable sets, we can show that E is nonmeasurable (which, of course, I will not do here.)

So, my question is: why can we not say that E exists? Obviously it is an abstraction, so perhaps I am misusing the word "exist," but what I mean is that it "exists" as a linguistic tool that we can use to describe (approximate) reality, just like adjectives, just like other mathematical and logical abstractions.

I know that there are some apparent problems with set theory (like Russell's paradox(es)), but it is such a simple and *useful* tool, so there *has* to be some way to salvage it for the sake of an objective foundation for mathematics.

You simply cannot stomp your foot on the ground and demand that Objectivism, or Objectivists, somehow give you the technical expertise to answer in detail the assertions of a knowledgeable mathematican

I hope you weren't referring to me here

If so, then I apologize, for I didn't mean to come across this way. That is, I don't see the Banach-Tarski paradox as a threat to Objectivism, I just think that a proper foundation of mathematics (whatever it is and whenever it is formulated) should clearly address this apparent contradiction, and it seems to me that the clearest way to get to the heart of the matter here would be to examine the (possibly loose) application of AC to uncountable sets.

this is closely linked to Godel's Incompleteness Theorem, isn't it?

Um, I don't think so. That's a separate issue, even more fascinating than Banach-Tarski (in my opinion.)

Just as ashleyisachild's friend tried to assert that the Axiom of Choice disproves the law of noncontradiction, there are some who would assert that Godel's Essential Incompleteness Theorem disproves the law of excluded middle. What Godel's theorem states, roughly, is that there exist *true* statements that *cannot be proven* and false statements that cannot be disproven (all of this within a given set of axioms, of course.) However, Excluded Middle still holds: hence, if statement S is "unreachable" (that is, not able to be proven or disproven by accepted methods of axiom and inference), then it is either 1) True, but unreachable, or 2) False, but unreachable, but it is still either T or F.

[stephen_speicher]

Back to Goedel, did he really complete Cantor's project? I forgot then, if not Cantor then who was the big late 19th century mathematician who attempted to go the other way and prove the logicity of math?

There was the logisitic school, mainly Bertrand Russell and Alfred North Whitehead, who initially thought that mathematics was derivable from logic. And the great Henri Poincare, who was famously in opposition to set theory. And then, of course, there was David Hilbert, who early-on sought to provide a foundational basis for the number system independent of set theory. But I'm not sure just what you have in mind.

[stephen_speicher]

Actually, I read about the construction not in Wagon but in Measure and Integral by Zygmund and Wheeden.

Ah, another coincidence. I was just re-reading Lebesgue's original Measure and the Integral

So, my question is: why can we not say that E exists?  Obviously it is an abstraction, so perhaps I am misusing the word "exist," but what I mean is that it "exists" as a linguistic tool that we can use to describe (approximate) reality, just like adjectives, just like other mathematical and logical abstractions.

Sure it exists, but it is a nonmeasurable set with a nonmeasurable function. Not all of mathematics automatically applies to reality.

I know that there are some apparent problems with set theory (like Russell's paradox(es)), but it is such a simple and *useful* tool, so there *has* to be some way to salvage it for the sake of an objective foundation for mathematics.

Yes, some of it is of value, but that, in and of itself, does not mean that it forms a proper foundational basis. I truly believe that such a foundation is to be found elsewhere, but that does not mean we need to trash all of set theory.

Also, in regard to Goedel, you should realize that Goedel lost contact with reality in more ways than one. He was withdrawn, depressed, and suffered from episodes of hypochondria and paranoia. Goedel had the radiator and ice box in his apartment removed because he thought they were giving off a poison gas. Most of Goedel's work has more to do with some aspects of mathematical formalism, as opposed to saying something about the real world. Goedel was a Platonist, an ontological Platonist, and he believed, for instance, that objects such as sets existed in the same manner as physical objects. Goedel once said, of his own work, that it "... had not established any boundaries for the powers of human reason, but rather for the possibilities of pure formalism in mathematics." Here is a short story about Goedel I once posted.

*************************

If you are familiar with Cantor's continuum problem, in the late 40's Goedel wrote a paper on the subject. People like Poincare' and Weyl were not accepting of Cantor's set theory, and Goedel stated that whatever one's philosophical view, the issue was whether the power of the continuum was formably deducible from a set of axioms. Goedel thought that the issue would be determined to be undecidable, based on axioms considered so far. However, he thought that "even if one should succeed in proving undemonstrability", that would "by no means settle the question", at least for someone like Goedel who thought that "a more profound understanding of the concepts underlying logic and mathematics", meaning entities which inhabited a Platonic universe of their own, might "enable us to recognize hitherto unknown" axioms. This paper of Goedel's became a manifesto for mathematical Platonism. Russell once said "Goedel turned out to be an unadulterated Platonist."

Now, almost twenty years later, Paul Cohen claims to have proved the Axiom of Choice was independent of ZF set theory, and the Continuum Hypothesis independent of ZFC. This is what Goedel had conjectured, but had been unable to prove. Cohen was a mathematician, not a logician, and to fend off criticism he sought Goedel's blessing. Goedel made suggestions, Cohen was awarded the Fields Medal a few years later, and Cohen's approach was quickly extended and used to demonstrate the independence from ZFC of many related statements. At the time, Goedel told Cohen "You have achieved the most important progress in set theory since its axiomatization." He said that Cohen's proofs were "in all essential respects ... _the_ best possible".

Yet, coincidentally, Cohen's discovery came at a time when Goedel was revising the paper I mentioned, on Cantor's continum problem, from twenty years earlier. He mentions Cohen's work (as a postscript) in the paper revision, but he did not retract what he said earlier about Cantor's continuum hypothesis, and insisted from new axioms it would be disproved. Goedel held on to his Platonism, and it affected his analysis. He used a geometric analogy, stating that the truth or falsity of Euclid's fifth postulate "retains its meaning if the primitive terms are taken ... as referring to the behavior of rigid bodies, rays of light, etc." He thought, likewise, though sets where remote from sense experience, that they would be perceptible by mathematical intuition. According to Goedel, these "new mathematical intuitions" were "perfectly possible." So much for logic and reality.

These are not my 'interpretations' of facts, I am just relaying facts as presented in many sources, though they may not identify the principles as clearly as one would like. Anyone can verify this stuff for themselves. Here are three references that you can use, if you have the stomach to do so.

"Goedel Remembered", the Goedel-Symposium in Salzburg-July 10-12, 1983, edited by P. Weingartner and L. Schmetterer, Bibliopolis, 1987.

"Logical Dilemmas--the Life and Work of Kurt Goedel", J. Dawson Jr., A K Peters, Ltd., 1997.

"Reflections on Kurt Goedel", H. Wang, MIT Press, 1987.

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[ewv]

The axiom of choice is one of those principles that, depending on how it is used, is either obvious or incomprehensible. The problems show up in its use with bizarre infinite sets and can usually be attributed to the arbitrary manipulation of floating abstractions involving the infinite, not the axiom of choice itself. This is a problem of epistemology involving the proper formulation of concepts, and cannot be dealt with plunging in at an advanced level of mathematics without a lot of prior work in the technical mathematics, its historical evolution showing why and how these ideas arose, and validating a proper conceptual approach to mathematics starting from the beginning at an elementary level and working up through the advanced abstractions.

[Eric]

There was the logisitic school, mainly Bertrand Russell and Alfred North Whitehead, who initially thought that mathematics was derivable from logic. And the great Henri Poincare, who was famously in opposition to set theory. And then, of course, there was David Hilbert, who early-on sought to provide a foundational basis for the number system independent of set theory.

Let's not forget Brouwer. At the beginning of last century, he put up a noble fight against both formalism and logicism by stressing meaning in mathematics. Unfortunately, his theory (the poorly-named "Intuitionism") included some rather bizarre philosphical speculation which ultimately undermined his efforts, but his insight into what's wrong with classical mathematics and his attempt to fix it should be applauded.

[Eric]

Sure it exists, but it is a nonmeasurable set with a nonmeasurable function.

The "construction" of the set E above cannot be carried out by you or me or any finite creature. So, how can E be said to "exist"?

In the words of Errett Bishop: "If God has mathematics of his own that needs to be done, let him do it himself."

Not all of mathematics automatically applies to reality.

So, what then does it apply to? And, what good is it?

[Randrew]

Also, in regard to Goedel, you should realize that Goedel lost contact with reality in more ways than one. He was withdrawn, depressed, and suffered from episodes of hypochondria and paranoia. Goedel had the radiator and ice box in his apartment removed because he thought they were giving off a poison gas.

And let's not forget how he died: he starved to death because he was convinced his wife was trying to poison him (so I heard.)

Just because I accept the Incompleteness Theorem does not mean that I agree with his personal philosophy or way of life. Or were you issuing a warning that too much dwelling on set-theoretical problems can lead to insanity?

The "construction" of the set E above cannot be carried out by you or me or any finite creature. So, how can E be said to "exist"?

The set of even numbers is infinite, so there is no way that a "finite creature" could ever list them all. But that is a countable set, so...

...what about the real numbers? Do you not believe that this set exists? But I may still not be getting to the point, so...

...let's talk about E: E is a subset of the real numbers. The points in E exist along the real line. Since the real numbers exist, E must also exist.

Ok, I'm getting uncomfortable using the word "exist" so liberally. I own Introduction to Objectivist Epistemology but have not read it yet, so I will search there for the Objectivist meaning of existence. (To save time, if possible: can anyone point me to a specific section of it that deals with this issue? Should I be using a word other than "exist" in reference to mathematical abstractions?)

Exercise: Prove that real analysis is Really Anal. (Hint: use the Law of Identity.)

[ewv]

Randrew: The set of even numbers is infinite, so there is no way that a "finite creature" could ever list them all. But that is a countable set, so...what about the real numbers? Do you not believe that this set exists?

You don't have to be able to list all the referents of a concept to form the concept; the process by which concepts are formed in terms of essential characteristics which permit the concepts to be open-ended, covering all referrents that exist, have ever existed or may ever exist in the future, is dicussed in IOE.

Randrew: Ok, I'm getting uncomfortable using the word "exist" so liberally. I own Introduction to Objectivist Epistemology but have not read it yet, so I will search there for the Objectivist meaning of existence. (To save time, if possible: can anyone point me to a specific section of it that deals with this issue? Should I be using a word other than "exist" in reference to mathematical abstractions?)

Without understanding Objectivist epistemology you should feel "uncomfortable" with using the word "exist" in that way -- the relation of concepts to existence is the central problem of epistemology and is poorly understood in mathematics especially.

You can use the term "exist" in its technical mathematical meaning, provided you do not also interpret that with any kind of Platonist meaning as if numbers and other abstract mathematical "entities" refer directly to some kind of physical object. In IOE you will not find a discussion of "mathematical existence" specifically, but you will find that mathematics is classified as a science of method and how abstract concepts of method in general are (indirectly) related to existents in reality. But don't just jump to the discussions of existence or existents, or "concepts of method"; you will have to understand the whole explanation from the beginning.

[stephen_speicher]

Let's not forget Brouwer.  At the beginning of last century, he put up a noble fight against both formalism and logicism by stressing meaning in mathematics.  Unfortunately, his theory (the poorly-named "Intuitionism") included some rather bizarre philosphical speculation which ultimately undermined his efforts, but his insight into what's wrong with classical mathematics and his attempt to fix it should be applauded.

The first Intuitionist was really Kronecker, who, unfortunately, countered what he saw as the mysticism of Cantor's transfinite numbers with holding the whole numbers as "the work of God." I did not mean to slight Brouwer (who can legitimately be called the founder of modern Intuitionism), but unfortunately much of Brouwer's views, like Kronecker, severely undercut his positive position. Casting out the law of excluded middle, not to mention his view of how logic stands in relation to mathematics, is not too copacetic. Brouwer's own work in topology contradicted his Intuitionist views.

[stephen_speicher]

The "construction" of the set E above cannot be carried out by you or me or any finite creature. So, how can E be said to "exist"?

It exists as an abstraction, specifically a concept of method.

So, what then does it apply to? And, what good is it?

There are many weaker propositions which lead to a nonmeasurable sets of the reals, and since the Axiom of Choice is independent of other axioms of set theory, then, if nothing else, this solidifies those weaker propositions.

But, look, I really do not want to be put in the position of defending what I have already stated is indefensible. I have made clear that a different foundation than set theory is required for mathematics, and unfortunately I do not have one available. But, I just refuse to dismiss the entirety of set theory out of hand. If others find no use for it at all in their studies of mathematics, so be it. But, I find that sad, since then I believe they are missing out on a great deal of value.

[Eric]

Ok, I'm getting uncomfortable using the word "exist" so liberally.  Should I be using a word other than "exist" in reference to mathematical abstractions?

Mathematical existence is a tricky subject. I believe that most mathematicians don't really think about what they mean when they say "there exists". To them, it is just part of, and therefore gets its meaning from, the mathematical formalism. This is obviously backwards. Meaning must come first.

Mathematics is full of so-called "pure existence" or "ideal existence" theorems, i.e., theorems that assert the existence of some object without giving the slightest clue as to how to find it. I believe that such theorems should be distinguished from ones which actually construct the object in question, since there is an obvious difference in pragmatic content. For example, if I need to find a root of an equation, a theorem that asserts the "pure" existence of a solution is of very little value compared to a theorem that constructively finds the root.

Yet, today, these two different concepts are both lumped into the mathematicians' "there exists". If any distinction is maintained, it is that pure existence proofs are generally considered more "elegant" than constructive proofs - "elegant" apparently meaning less useful.

I personally don't think that "pure existence" is existence. Existence is identity, and absolutely nothing has been identified in a pure existence proof. This is why I deny that the set E above exists: the Axiom of Choice asserts only pure existence.

This is not to say that "pure existence" theorems are useless. They tell us, for example, to not bother looking for a counterexample and to focus our efforts on constructing a real solution.

[Eric]

Casting out the law of excluded middle ... is not too copacetic.

Brouwer showed that there are limits to the applicability of the law of excluded middle (LEM). To me, this is not surprising. Aristotle formulated LEM for finite situations. It's far from obvious that it should still apply when dealing with the infinite.

LEM also leads one to trouble when dealing with existence. Would you say "Either God exists or he doesn't"? "God exists" is arbitrary - neither true nor false - so LEM doesn't apply. Can't mathematicians also make arbitrary assertions? To me, there's very little difference between the statements "The Reals can be well ordered" and "God exists".

[stephen_speicher]

...

This reminds of one fellow who also wanted to cast out the law of excluded middle. He argued that probability says there is a certain liklihood that X will occur, but the law of excluded middle says that X will either occur, or not. He preferred probability, so the law of excluded middle had to go.

Brouwer's intuitionist logic rejects the law of excluded middle, and in this case I agree with Hilbert who was quoted as saying "Forbidding a mathematician to make use of of the principle of excluded middle is like forbidding an astronomer his telescope or a boxer the use of his fists." [*] Aristotle got it right in the first place, and I really have no interest in any sort of discussion where the law of excluded middle is cast aside.

[*] H. Weyl, American Mathematical Society Bulletin, 50, p. 637, 1944.

[ewv]

Eric: Brouwer showed that there are limits to the applicability of the law of excluded middle (LEM). To me, this is not surprising. Aristotle formulated LEM for finite situations. It's far from obvious that it should still apply when dealing with the infinite.

The law of the Excluded Middle as formulated and advocated by Aristotle was not and is not limited to the finite -- it applies, for example, to the integers. The Law of the Excluded Middle is a corollary of the Law of Identity and therefore applies to any meaningful statement. Properly conceived, the concept of the mathematical infinite is valid and the Law of the Excluded Middle therefore applies to it.

Brouwer didn't "show" there are limits to the Law of the Excluded Middle, nor in particular did he restrict it to the finite. He advocated that it is inapplicable to infinite sets for which a constructive existence proof is lacking. His aim was to get epistemological control over mathematical concepts of the infinite, which had long been a cause of epistemological problems in mathematics. There would be nothing wrong with arguing that the Law of the Excluded Middle (or other logical principles for that matter) is inapplicable to meaningless concepts, from which nothing can be derived, but his restriction of infinite sets to only those with constructive existence proofs missed the essence of the problem and resulted in over-kill.

Nevertheless, his critique generated a lot of philosophical sympathy from mathematicians. Even Hilbert took it seriously. He rejected Brouwer's proscription, but Brouwer's critique was the motive for Hilbert's formulation of what became known as Formalism in the foundations of mathematics -- his goal was to validate the infinite in mathematics using only finite means in his "metamathematics". Hilbert's approach didn't work either and was also epistemologically flawed, partly because he lacked a valid philosophical concept of abstraction and mathematical abstraction in particular (but his the motive and meaning of his "Formalist" approach to the problem was a lot more serious and plausible than the "meaningless game of symbol manipulation" commonly associated with Formalism today.)

LEM also leads one to trouble when dealing with existence. Would you say "Either God exists or he doesn't"? "God exists" is arbitrary - neither true nor false - so LEM doesn't apply. Can't mathematicians also make arbitrary assertions? To me, there's very little difference between the statements "The Reals can be well ordered" and "God exists".

The problem with applying the Law of the Excluded Middle to the question of "God exists" is not that it is an arbitrary assertion, but that it isn't an assertion at all because the subject of the sentence is undefined (and described in contradictory terms), and meaningless, making the alleged statement itself literally meaningless. It raises the immediate question: "What exists?" -- "What is it you claim exists"?

Mathematicians can make arbitrary assertions or otherwise concoct alleged statements out of meaningless concepts, too, e.g., Bertrand Russell's famous example of the set of all sets that don't include themselves. The underlying problem there was the self-referential invocation of the "Law of the Unexcluded Anything" that is essentially Cantor's original principle that a set can be validly defined by any arbitrary condition on its alleged elements. But that does not mean that well-defined mathematical concepts like the well-ordering of the reals are meaningless or equivalent to mystical assertions of undefined beings (like Cantor's own philosophical reification of the infinite). "Well-ordering" is a technical concept with a clear meaning to a mathematician. The cause of paradoxical or confused meaning in attempts at advanced concepts of mathematical method can be very subtle and should not all be thrown out as some kind of arbitrary mysticism.