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.

As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy.

For each step S in the deduction which is a premise in Γ ( a reiteration step ) or an axiom, we can apply modus ponens to the axiom 1, S →( H → S ), to get H → S.

Rather than starting with experience, Aristotle begins a priori with the law of noncontradiction as the fundamental axiom of an analytic philosophical system.

The Greeks distinguished between nomos ( νόμος, " law "), and arché ( αρχή, " starting rule, axiom, principle ").

That's not a serious issue for a modern axiomatic mathematician, since the implication of axiom is now a starting point for theory rather than a self-evident plank in a platform based on intuition.

Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X.

Now it is easy to convince oneself that the set X could not possibly be measurable for any rotation-invariant countably additive finite measure on S. Hence one couldn't expect to find an algorithm to find a point in each orbit, without using the axiom of choice.

The second point cannot in any reasonable way be combined with the first point, as long as the axiom system is genuinely finitary.

This point of view regards as an incredibly rich set given to us by one bold new axiom, which can never be approached by any piecemeal process of construction.

* Euclidean geometry, under Hilbert's axiom system the primitive notions are point, line, plane, congruence, betweeness and incidence.

* Euclidean geometry, under Peano's axiom system the primitive notions are point, segment and motion.

Formalism is thus silent on the question of which axiom systems ought to be studied, as none is more meaningful than another from a formalistic point of view.

Indeed, it is not hard to see that it is equivalent to the Boolean prime ideal theorem ( BPIT ), a well-known intermediate point between the axioms of Zermelo-Fraenkel set theory ( ZF ) and the ZF theory augmented by the axiom of choice ( ZFC ).

The seventh axiom does not allow construction of further point.

However, since every point ( a < sub > 1 </ sub >, ..., a < sub > n </ sub >) is the zero set of the polynomials x < sub > 1 </ sub >-a < sub > 1 </ sub >, ..., x < sub > n </ sub >-a < sub > n </ sub >, points are closed and so every variety satisfies the T < sub > 1 </ sub > axiom.

The full axiom system proposed has point, line, and line containing point as primitive notions:

#( Affine axiom of parallelism ) Given a point A and a line r, not through A, there is at most one line through A which does not meet r.

# Playfair's axiom: Given a line and a point not on, there exists exactly one line containing such that

axiomatizes affine geometry ( over the reals ) as ordered geometry together with an affine form of Desargues's theorem and an axiom stating that in a plane there is at most one line through a given point not meeting a given line.

Pasch's axiom and the exterior angle theorem still hold for an omega triangle, defined by two points in hyperbolic space and an omega point.

One of the axioms is the so-called dimension axiom: if is a single point, then for all, and.

Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset with the property that all of its translates by G are disjoint from X and from each other.

Every segment is congruent to itself ; that is, we always have AB ≅ AB. We can state the above axiom briefly by saying that every segment can be laid off upon a given side of a given point of a given straight line in one and only one way.

This is not an axiom of infinity in the usual sense ; if Infinity does not hold, the closure of exists and has itself as its sole additional member ( it is certainly infinite ); the point of this axiom is that contains no additional elements at all, which boosts the theory from the strength of second order arithmetic to the strength of Morse – Kelley set theory with the proper class ordinal a weakly compact cardinal.

c ) Everything which exists is better than anything non-existent ( by the first point: since it is more rational, it also has more reality ), and, consequently, it is the best possible being in the best of all possible worlds ( by the axiom: " That which contains more reality is better than that which contains less reality ").

Rather, he regarded the notion of " effective calculability " as merely a " working hypothesis " that might lead by inductive reasoning to a " natural law " rather than by " a definition or an axiom ".

This theorem showed that axiom systems were limited when reasoning about the computation which deduces their theorems.