Controversy in axiom of choice

But "axiom of choice" is certainly not that simple. Its incredible and wonderful usage and the results it leads to are just the beginning.

Proving axiom of choice is not easy. One of the reasons is that axiom of choice is not only a simple mathematical proposition, but also involves a more basic mathematics-set theory. Set theory is the basic theory of mathematics, so there are few tools to prove it.

Many mathematicians try to prove axiom of choice, hoping to prove it with the most basic tools, but often in these proofs, some non-basic theories are used, such as "Good Order Theorem" and "Zorn Lemma".

well ordering theorem

All the equipment can be ordered well. In other words, for each set, there is a sorting method, so that all its subsets have the smallest elements.

Zorn's Lemma

If a poset is an inductive poset, it must have the largest element. In other words, if each chain in a poset has an upper bound in the original poset, the poset must have the largest element. Even if these theories are interpreted literally, it is not easy to judge their authenticity. In fact, "well-ordering principle" and "Zuo En Lemma" cannot be proved by basic tools. So far, no one can prove "axiom of choice" with basic tools.

The more interesting result is that "axiom of choice", "well-ordering principle" and "Zuo En Lemma" are all equivalent propositions, that is, they describe the same event. Over the years, many equivalent propositions of "axiom of choice" have been found, but the webmaster has not counted them. Some books can write about 30 equivalent propositions, and webmasters have collected some equivalent propositions (English versions) for netizens' reference, but human beings just wander between these propositions. Therefore, it is not easy to prove or deny "axiom of choice" in mathematics, so mathematicians shift their goals and look at its compatibility from the logical system. In fact, it has been proved that the ZF axiomatic system that we commonly use now is compatible with axiom of choice, that is to say, the logical contradiction with ZF axiomatic system cannot obtain "axiom of choice". If we choose to accept "axiom of choice", there will be an axiomatic system containing "axiom of choice", which is generally called "ZFC axiomatic system"; Otherwise, it will not be accepted in the axiomatic system and will not be accepted as a "theorem" until it can be proved.

However, this debate is still not over, because this axiom is not just a question of acceptance or rejection. If this axiom is abandoned, many beautiful and almost "common sense" results will be abandoned at the same time; But in fact, it is quite different from many "common sense".

One of the well-known unreasonable results is the "Barnah-Talsky Paradox", or "the problem of dividing the ball". This paradox can be said to violate the laws of physics, because it says that a unit sphere (with a radius of 1) can be divided into a finite number of points (at least five points) and then reassembled by some rigid body movements, that is, rotation and translation. But after the combination, it actually becomes two unit spheres, that is, the volume has doubled. The proof of this paradox must be used in axiom of choice. That is to say, if we choose to accept axiom of choice, Barna-Taskey Paradox is a theorem, but is it possible in reality?

This actually involves another mathematical concept ── measurable set. The "Barnach-Taskey Paradox" is the result of the existence of unpredictable sets. If we accept "axiom of choice", we must accept the unpredictable set. If axiom of choice is not accepted, it may be more reasonable to assume that all sets are Lebesgue measurable.

However, if we give up axiom of choice, there will be some unreasonable situations. These conditions depend on the selected model that does not conform to axiom of choice. For example, in Cohen's model, there is a discontinuous function at point x0, but for any series {an} with limit x0, the limit of {bn=f(an)} is f(x0). In other words, the function value can be approximated to f(x0) by any sequence that approximates x0, which is precisely the embodiment of "continuity". Some models even deny "binary countable axiom of choice" (axiom of choice is established on countable binary sets), which is equivalent to "the union of countable disjoint binary sets is countable"! In a word, "axiom of choice" is a controversial proposition, and most mathematicians accept this axiom, because many useful results can be drawn from it. There is no logical contradiction in using this axiom anyway. However, for logicians or set theorists, this is a problem that must be solved. Some people will suggest using a weak "countable axiom of choice" instead. There are indeed many results that can be proved by countable axiom of choice, but this only sidesteps the problem temporarily, and some results must be proved by "axiom of choice".

Bertrand Russell, a famous philosopher and mathematician, once said, "If you choose one pair of infinite socks, we need' axiom of choice', but if you change it into shoes, it is unnecessary." Because shoes can be divided into two parts, socks are no different, I don't know how to choose. In addition, if there are only a limited number of pairs of socks, there is no need to use "axiom of choice" logically.

Jerry Boehner once said: "axiom of choice is obviously right; The "good order principle" is obviously incorrect; Who can decide "Zon Lemma"? "Although this is a joke, it can be seen that Taoist intuition does not necessarily follow mathematical thinking. Mathematically, these three propositions are equivalent, but for axiom of choice, many mathematicians intuitively know that it is correct; For "well-ordering principle", many mathematicians think there is a problem; Zon lemma is so complicated that many mathematicians can't judge by intuition.

"axiom of choice" is indeed a mysterious axiom. Although it looks simple, the function is wonderful and even the effect is extraordinary. Some people voted for it, while others were skeptical. I believe that the discussion and research on this axiom will continue, so let's see how mathematicians solve it. Finally, the webmaster ended with a sentence from Russell, who once said about "axiom of choice":

"At first it seemed to understand; But the more you think about it, the more strange the inference from this axiom seems to become; Finally, you have no idea what this means. "