How did axiom of choice prove the principle of good order?

"axiom of choice" may not be heard by most people; Even students who study math and science may not have been exposed to it, but listen more and use less. However, this "axiom of choice" is an axiom that has puzzled the whole field of mathematics for many years, and its rationality is still inconclusive. Some people think this is obvious and simple. However, when we savor its content and use carefully, we will not only find its infinite use, but also begin to question our understanding of this axiom and even begin to doubt its authenticity. Axiom of choice is such a puzzling axiom. Now let's see what it is.

There are many equivalent forms of "axiom of choice". The following is a simple description:

Axiom of choice let c be a set composed of nonempty sets. Then, we can choose an element from each set in C to form a new set.

In order to make readers understand better, here are some examples:

1. If C is the set of all non-empty subsets of {1, 2, 3, …}, then we can define a new set so that its elements are the smallest elements of each set in C.

2a。 If C are all real number intervals with finite length and non-zero, then we can define a new set so that its elements are the midpoint of each interval in C. ..

It seems reasonable, but the above example may be more mathematical and difficult to understand. Now let's use a more realistic example.

3a。 If you put a few piles of apples in front. Then, we can choose one apple from each pile and put them in a new pile.

After reading this example, you may understand it better, but pay attention to the so-called "several piles", which may be infinite piles, and each pile may have infinite apples, so it can be changed to.

3b。 If you put an infinite number of apples in front of you, there are an infinite number of apples in each pile. Then, we can choose one apple from each pile and put them in a new pile.

This is the axiom of choice. Seems to make sense. Since there are apples in each pile, you can of course choose an apple from each pile. No matter how many apples there are in each pile, you must be able to do it.

But in this pile of apples, which one should I choose? Maybe someone will say, "Pick any one!" But what is "casual"? Can you state it more specifically? Does this "casual" method necessarily exist? If we look at this problem from a mathematical point of view, according to axiom of choice,

2b。 If c is all real number intervals with non-zero length, then we can define a new set so that its elements are points on each interval in c. ..

What if we look closely at 2b, "the point of every interval in C"? The biggest point? The smallest point? The middle point? None of them exist, because "real interval with non-zero length" contains infinite interval, so there may not be such concepts as "maximum", "minimum" and "middle". So, how to state the method in detail? Will this method not exist?

Some people may think that even if we can't state the method, we can't deny or give up this axiom, because there are many "existence theorems" in mathematics, which only point out the existence of an event and can't describe the sought method in detail. For example, the mean value theorem and Rolle theorem have all been proved to be true existence theorems.

In addition, the failure to specify the method may also be due to human language barriers, that is, they just can't express themselves in human language. Just as the greatest writers only express it in sentences that they think are most appropriate, they may be limited by language and cannot fully reflect their inner thoughts. This is called "unspeakable".

But of course, "axiom of choice" is 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 with the most basic tools, but often some non-basic theories are used in these proofs, such as "well-ordering principle" and "Zorn Lemma".

All sets in the well-ordered principle are also well-ordered sets. In other words, for each set, there is a sorting method, so that all its subsets also have the smallest elements.

Zon Lemma If a poset is an inductive poset, then it must have the largest element. In other words, if every chain of a poset has an upper bound, then 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 "Zorn's Lemma" cannot be proved with 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 disprove "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 axiom system that we commonly use now is compatible with axiom of choice, that is to say, axiom of choice's logical contradiction cannot be obtained by ZF axiom system. 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 non-acceptance. 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 theory can be said to violate the laws of physics, because it can divide a unit sphere (radius 1) into finite parts, at least five parts, and then recombine it through 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. This theory means that if we choose to accept "axiom of choice", then "Barna-"

This actually involves another mathematical concept ── measurable set. The theory of "Barnach-Taskey" is the result of the non-existence of measurable set. 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.

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 there are some results that "axiom of choice" must use.

Bertrand Russell, a famous philosopher and mathematician, once said, "If a pair of socks comes from an infinite pair, we need a axiom of choice, but if it is a shoe, it is unnecessary. Because the shoes are divided into two parts, there is no difference between socks, and I don't know how to choose. In addition, if you only have a limited number of pairs of socks, you can logically dispense with "axiom of choice".

Jerry Boehner once said: "axiom of choice is obviously right; The principle of well sequence is obviously incorrect; Who can decide "Zuo Si Lemma"? Although this is a joke, it can be seen that the intuition of road overpass does not necessarily follow mathematical thinking. Mathematically, these three propositions are equivalent, but for axiom of choice, many mathematicians intuitively think that they are correct. For "well-ordering principle", many mathematicians also think there is a problem; The "Zoos Lemma" is so complicated that many mathematicians can't judge by intuition alone.

"axiom of choice" is indeed a mysterious axiom. Although it looks shallow, it has a wonderful effect, even extraordinary effect. 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; In the end, you have no idea what this means. 」