only seldom is it so simple as to be a matter of his obviously parroting some timeworn axiom, common to our culture, which he has evidently heard, over and over, from a parent until he experiences it as part of him.

In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that " the product of a collection of non-empty sets is non-empty ".

Informally put, the axiom of choice says that given any collection of bins, each containing at least one object, it is possible to make a selection of exactly one object from each bin.

In many cases such a selection can be made without invoking the axiom of choice ; this is in particular the case if the number of bins is finite, or if a selection rule is available: a distinguishing property that happens to hold for exactly one object in each bin.

Although originally controversial, the axiom of choice is now used without reservation by most mathematicians, and it is included in ZFC, the standard form of axiomatic set theory.

One motivation for this use is that a number of generally accepted mathematical results, such as Tychonoff's theorem, require the axiom of choice for their proofs.

The axiom of choice is avoided in some varieties of constructive mathematics, although there are varieties of constructive mathematics in which the axiom of choice is embraced.

A choice function is a function f, defined on a collection X of nonempty sets, such that for every set s in X, f ( s ) is an element of s. With this concept, the axiom can be stated:

The axiom of choice asserts the existence of such elements ; it is therefore equivalent to:

The statement of the axiom of choice does not specify whether the collection of nonempty sets is finite or infinite, and thus implies that every finite collection of nonempty sets has a choice function.

However, that particular case is a theorem of Zermelo – Fraenkel set theory without the axiom of choice ( ZF ); it is easily proved by mathematical induction.

" In general, it is impossible to prove that F exists without the axiom of choice, but this seems to have gone unnoticed until Zermelo.

") This method cannot, however, be used to show that every countable family of nonempty sets has a choice function, as is asserted by the axiom of countable choice.

If the method is applied to an infinite sequence ( X < sub > i </ sub >: i ∈ ω ) of nonempty sets, a function is obtained at each finite stage, but there is no stage at which a choice function for the entire family is constructed, and no " limiting " choice function can be constructed, in general, in ZF without the axiom of choice.

Until the late 19th century, the axiom of choice was often used implicitly, although it had not yet been formally stated.

This story is an allegory ; the android was primitive scholasticism, which was broken by the Summa of St Thomas, the daring innovator who first substituted the absolute law of reason for arbitrary divinity, by formulating that axiom which we cannot repeat too often, since it comes from such a master: " A thing is not just because God wills it, but God wills it because it is just.

With the omission of the law of the excluded middle as an axiom, the remaining logical system has an existence property which classical logic does not: whenever is proven constructively, then in fact is proven constructively for ( at least ) one particular, often called a witness.

Proofs or constructions using induction and recursion often use the axiom of choice to produce a well-ordered relation that can be treated by transfinite induction.

However, if the relation in question is already well-ordered, one can often use transfinite induction without invoking the axiom of choice.

Though insults are common, and often used in jest, a fundamental axiom of sociology recognizes that derogatory forms of speech make erroneous attributions about the motivation of a person.

The axiom is often used as short-hand for the doctrine, upheld by both the Eastern Orthodox Church and the Roman Catholic Church, that the Church is necessary for salvation (" one true faith ").

Sections s < sub > i </ sub > satisfying the condition of axiom 2 are often called compatible ; thus axioms 1 and 2 together state that compatible sections can be uniquely glued together.

It is due to this intermediate status between ZF and ZF + AC ( ZFC ) that the Boolean prime ideal theorem is often taken as an axiom of set theory.

The axiom systems most often considered in reverse mathematics are defined using axiom schemes called comprehension schemes.

On the other hand, though the above properties guarantee the existence of a categorical equivalence ( given a sufficiently strong version of the axiom of choice in the underlying set theory ), the missing data is not completely specified, and often there are many choices.

Such theologies often involved a more drastic pruning and reinterpretation of traditional belief in order to cohere with the axiom or axioms.

The axiom is often encountered in economics, where it can be used to model a consumer's tendency to choose one brand of product over another.

This is often stated as a Mayer-Vietoris axiom: for any CW complex W covered by two subcomplexes U and V, and any elements u ∈ F ( U ), v ∈ F ( V ) such that u and v restrict to the same element of F ( U ∩ V ), there is an element w ∈ F ( W ) restricting to u and v, respectively.

In fact, the axiom of regularity is often called the foundation axiom since it can be proved within ZFC < sup >−</ sup > ( that is, ZFC without the axiom of regularity ) that well-foundedness implies regularity.

In 1971, before Martin obtained his proof, Harvey Friedman showed that any proof of Borel determinacy must use the axiom of replacement in an essential way, in order to iterate the powerset axiom transfinitely often.

By constructing a valid model for a subsystem without a specific axiom, we show that the omitted axiom is independent if its correctness does not necessarily follow from the subsystem.

This last axiom is omitted from the definition of a ring: it follows automatically from the other ring axioms.

The assumption that ZF is consistent is harmless because adding another axiom to an already inconsistent system cannot make the situation worse.

One argument given in favor of using the axiom of choice is that it is convenient to use it because it allows one to prove some simplifying propositions that otherwise could not be proved.

In class theories such as Von Neumann – Bernays – Gödel set theory and Morse – Kelley set theory, there is a possible axiom called the axiom of global choice which is stronger than the axiom of choice for sets because it also applies to proper classes.

The most common way to formalize this is by defining a field as a set together with two operations, usually called addition and multiplication, and denoted by + and ·, respectively, such that the following axioms hold ; subtraction and division are defined implicitly in terms of the inverse operations of addition and multiplication :< ref group =" note "> That is, the axiom for addition only assumes a binary operation The axiom of inverse allows one to define a unary operation that sends an element to its negative ( its additive inverse ); this is not taken as given, but is implicitly defined in terms of addition as " is the unique b such that ", " implicitly " because it is defined in terms of solving an equation — and one then defines the binary operation of subtraction, also denoted by "−", as in terms of addition and additive inverse.

( Subset is not used in the formal definition above because the axiom of power set is an axiom that may need to be stated without reference to the concept of subset.

This issue arises in first order theories of arithmetic, such as Peano arithmetic, because the principle of mathematical induction is expressed as an infinite set of axioms ( an axiom schema ).

In this case, G is indeed a theorem in T ’, because it is an axiom.

Consider the theory obtained by adding a new constant symbol ε to the language and adjoining to Σ the axiom ε > 0 and the axioms ε < 1 / n for all positive integers n. Clearly, the standard real numbers R are a model for every finite subset of these axioms, because the real numbers satisfy everything in Σ and, by suitable choice of ε, can be made to satisfy any finite subset of the axioms about ε.

However, if we introduce a formal system where modus ponens is simply an axiom, then we are to abide by it simply, because it is so.

In advanced mathematics, some functions exist because of an axiom, such as the Axiom of Choice.

* " The axiom of choice must be wrong because it implies the Banach-Tarski paradox, meaning that geometry contradicts common sense.

It was Otto Stolz who gave the axiom of Archimedes its name because it appears as Axiom V of Archimedes ’ On the Sphere and Cylinder.

Note, that the third axiom is actually redundant, because the second and fourth axioms imply is also an identity, and identities are unique.

According to the Church-Turing thesis, any effectively calculable function is calculable by a Turing machine, and thus a set S is recursively enumerable if and only if there is some algorithm which yields an enumeration of S. This cannot be taken as a formal definition, however, because the Church-Turing thesis is an informal conjecture rather than a formal axiom.

Nearly every proof which explicitly relies on the axiom of choice is non-constructive in nature because this axiom is fundamentally non-constructive.

For example, when the neolithic reality of The Living Land invaded North America, soldiers found that their guns and radios no longer worked because the tech axiom of the cosm only allowed for a neolithic level of technology.

Intuition suggests to many people that any subset S of the unit disk ( or unit line ) should have a measure, because one can throw darts at the disk ( see Freiling's axiom of symmetry ), and the probability of landing in S is the measure of the set.