What are some motivating consequences of the axiom of choice (or its omission)? I know that weak forms of choice are sometimes required for interesting results like Banach-Tarski; what are some important consequences of a strong formulation of the axiom of choice?

# [Math] Motivating implications of the axiom of choice

axiom-of-choiceaxiomseducationset-theory

#### Related Solutions

An interesting question, that would take many pages to begin to answer! We make a small disjointed series of comments.

In the last few years, there has been a systematic program, initiated by Friedman and usually called Reverse Mathematics, to discover precisely how much we need to prove various theorems. The rough answer is that for many important things, we need very much less than ZFC. For many things, full ZF is vast overkill. Small fragments of second-order arithmetic, together with very limited versions of AC, are often enough.

About the Axiom of Choice, for a fair bit of basic analysis, it is pleasant to have Countable Choice, or Dependent Choice, at least for some kinds of sets. We really want, for example, sequential continuity in the reals to be equivalent to continuity. One could do this without full DC, but DC sounds not unreasonable to many people who have some discomfort with the full AC. This was amusingly illustrated in the early $20$-th century. A number of mathematicians who had publicly objected to AC turned out to have unwittingly used some form of AC in their published work.

Next, bases. For finite dimensional vector spaces, there is no problem, we do not need any form of AC (though amusingly we do to prove that the Dedekind definition of finiteness is equivalent to the usual definition.)

For *some* infinite dimensional vector spaces, we cannot prove the existence of a basis in ZF (I guess I have to add the usual caveat "if ZF is consistent"). However, an algebraic basis is not usually what we need in analysis. For example, we often express nice functions as $\sum_0^\infty a_nx^n$. This is an infinite "sum." The same remark can be made about Fourier series. True, we would use an algebraic basis for $\mathbb{R}$ over $\mathbb{Q}$ to show that there are strange solutions to the functional equation $f(x+y)=f(x)+f(y)$. But are these strange solutions of any conceivable use in Physics?

Finally, why should the Banach-Tarski result be unacceptable to a physicist *as* physicist? It is easy to show that the sets in the decomposition cannot be all measurable. In mathematical models of physical situations, do non-measurable sets of points in $\mathbb{R}^3$ ever appear?

This is a historical issue really.

Originally set theory was developed by Cantor and the well ordering principle was somewhat assumed in the background (e.g. Cantor's proof of the Cantor-Bernstein theorem was a corollary from the fact that every two cardinalities are comparable).

In 1904 Zermelo formulated the axiom of choice and proved its equivalence to the well ordering principle. He later formulated more axioms which described our intuition about sets, therefore removing the "naivity" from the Cantorian set theory. He did not add the axiom of foundations, nor the schema of replacement. Those were the result of Skolem and Fraenkel which were popularized by von Neumann.

The axiom of choice remained controversial, the thought that the continuum can be well-ordered was mind boggling and caused many people feel uneasy about this axiom. Further results like the Banach-Tarski paradox did not help to accept this axiom either.

Prior to set theory most mathematics was somewhat constructive in the sense that things were finitely generated or approximated by finitary means (e.g. limits of sequences). It requires quite the leap of faith to go from things you can pretty much write down to things which you cannot describe but only prove their existence. In this sense the axiom of choice augments the way we do mathematics by allowing us to discuss objects which we cannot describe in full.

It was questionable, therefore, whether this axiom is even consistent with the rest of the axioms of set theory. GĂ¶del proved this consistency in the late 1940's while Cohen proved the consistency of its negation in the 1960's (it is important to remark that if we allow non-set elements to exist then Fraenkel already proved these things in the 1930's).

Nowadays it is considered normal to assume the axiom of choice, but there are natural situations in which one would like to remove it or find himself in universes where the axiom of choice does not hold. This makes questions like "How much choice is needed here?" important for these contexts.

**Some things to read:**

## Best Answer

Each of the following is equivalent to the Axiom of Choice:

Every vector space (over any field) has a basis.

Every surjection has a right inverse.

Zorn's Lemma.

The first is

extremelyimportant and useful. The third is used all the time, especially in algebra, also very important and useful.You could write an entire book on important consequences (and equivalents) of the Axiom of Choice.

Unfortunately, any publisher worth his salt would reject it, since both have already been written:

Rubin, Herman; Rubin, Jean E.

Equivalents of the axiom of choice.North-Holland Publishing Co., Amsterdam 1963 xxiii+134 pp.Rubin, Herman; Rubin, Jean E.

Equivalents of the axiom of choice. II.Studies in Logic and the Foundations of Mathematics, 116. North-Holland Publishing Co., Amsterdam, 1985. xxviii+322 pp. ISBN: 0-444-87708-8Howard, Paul; Rubin, Jean E.

Consequences of the axiom of choice.Mathematical Surveys and Monographs, 59. American Mathematical Society, Providence, RI, 1998. viii+432 pp. ISBN: 0-8218-0977-6These books are probably not the best place to start, though; the first book is okay, listing some of the most important equivalents as they were known before Cohen's work, but at least the last is pretty difficult to slough through.

If you want a good introduction to the Axiom of Choice and some idea of its uses, Horst Herrlich's

Axiom of Choice, Lecture Notes in Mathematics v. 1876, Springer-Verlag (2006) ISBN: 3-540-30989-6 is pretty good, discussing some of the bad things that happen if you don't have AC, some of the bad things that happen if youdohave AC, and some alternative axioms that contradict AC but lead to very nice theorems.