Meaning of axiom of choice | Babel Free

Definitions

One of the axioms of set theory, equivalent to the statement that an arbitrary direct product of non-empty sets is non-empty; any version of said axiom, for example specifying the cardinality of the number of sets from which choices are made.

Equivalents

Examples

“The axiom of choice is logically equivalent to the assertion that every vector space has a basis.”

“If V = L then the axioms of choice and the continuum hypothesis are both true, and the assertion that a measurable cardinal exists is false.”

“Throughout this work we adopt the Zermelo–Fraenkel (ZF) axioms of set theory with the Axiom of Choice, commonly abbreviated as ZFC. It follows from the work of Gödel and Cohen that if the ZF axioms are consistent, the Axiom of Choice can be neither proved nor disproved from the ZF axioms.”

“To clarify these ideas for the reader, let us show, without the axiom of choice, that a product of finitely many nonempty sets is nonempty: This is done by induction on the number n of sets.[…]The finite axiom of choice is not an axiom, but rather a theorem that can be proved from the other axioms. In contrast, there are weak forms of the axiom of choice that are not provable.”

“Perhaps what this debate about whether to accept the axiom of choice indicates is that the disjunction between regularity and randomness is as fundamental to our conception of the world as that between discreteness and continuity.”

“There is a uniform status quo almost with just a few exceptions of a handful of mathematicians around the world. Most people accept the “infinite choice, infinite decimals” approach to real numbers, and the justification that has been created to substantiate this, and to overcome some of the difficulties that I have shown you is through an elaborate axiomatic framework. Mathematicians are not stupid! They realize that this theory is dubious, and so what they have done is they have created a rather elaborate axiomatic framework. And that axiomatic framework is something that we can study in set theory and logic; and its purported aim is to create a framework for mathematics; but its initial aims were very much directed towards solving the problems, overcoming the difficulties with real numbers as infinite decimals. Prominent amongst these axioms is this Axiom of Choice! The Axiom of Choice manifests itself in mathematics in many ways; but its primary role — OK — its primary role is exactly here at the level of infinite decimals and real numbers. It essentially asserts, as a matter of faith or belief, that it is possible to choose an infinite number of digits arbitrarily and independently, and that a legitimate mathematical object results. This is the key philosophical point. Are we, or are we not, in a position to be able to specify a first digit, a second digit, a third digit, a fourth digit, and so on to infinity? And are we allowed to call that specification a new mathematical object, a real number? The Axiom of Choice says that “Yes we are!”, so it is a statement of belief and its primary objective is to allow the building of this real number system.”

CEFR level