The Axioms required to make the theoretical machinery operate are set out tersely and powerfully, so that all permissible operations within the theory can be traced rigorously back to these axioms, rules, and primitive notions.

The strongest appeal of the Copernican formulation consisted in just this: ideally, the justification for dealing with special problems in particular ways is completely set out in the basic ' rules ' of the theory.

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 ".

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.

* ZF – Zermelo – Fraenkel set theory omitting the Axiom of Choice.

* ZFC – Zermelo – Fraenkel set theory, extended to include the Axiom of Choice.

These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it.

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.

For finite sets X, the axiom of choice follows from the other axioms of set theory.

A proof requiring the axiom of choice may establish the existence of an object without explicitly defining the object in the language of set theory.

For example, while the axiom of choice implies that there is a well-ordering of the real numbers, there are models of set theory with the axiom of choice in which no well-ordering of the reals is definable.

It is possible to prove many theorems using neither the axiom of choice nor its negation ; such statements will be true in any model of Zermelo – Fraenkel set theory ( ZF ), regardless of the truth or falsity of the axiom of choice in that particular model.

In constructive set theory, however, Diaconescu's theorem shows that the axiom of choice implies the law of the excluded middle ( unlike in Martin-Löf type theory, where it does not ).

Thus the axiom of choice is not generally available in constructive set theory.

A cause for this difference is that the axiom of choice in type theory does not have the extensionality properties that the axiom of choice in constructive set theory does.

Some results in constructive set theory use the axiom of countable choice or the axiom of dependent choice, which do not imply the law of the excluded middle in constructive set theory.

Because of independence, the decision whether to use of the axiom of choice ( or its negation ) in a proof cannot be made by appeal to other axioms of set theory.

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 set of all automorphisms of an object forms a group, called the automorphism group.

* In elementary arithmetic, the set of integers, Z, considered as a group under addition, has a unique nontrivial automorphism: negation.

If in the third identity we take H = G, we get that the set of commutators is stable under any endomorphism of G. This is in fact a generalization of the second identity, since we can take f to be the conjugation automorphism.

The set of all automorphisms is a subset of End ( X ) with a group structure, called the automorphism group of X and denoted Aut ( X ).

The composition of two automorphisms is again an automorphism, and with this operation the set of all automorphisms of a group < var > G </ var >, denoted by Aut (< var > G </ var >), forms itself a group, the automorphism group of < var > G </ var >.

In other words, an automorphism of E / F is an isomorphism α from E to E such that α ( x ) = x for each x in F. The set of all automorphisms of E / F forms a group with the operation of function composition.

This is a consequence of the first isomorphism theorem, because Z ( G ) is precisely the set of those elements of G that give the identity mapping as corresponding inner automorphism ( conjugation changes nothing ).

A hypergraph automorphism is an isomorphism from a vertex set into itself, that is a relabeling of vertices.

The set of automorphisms of a hypergraph H (= ( X, E )) is a group under composition, called the automorphism group of the hypergraph and written Aut ( H ).

A deck transformation or automorphism of a cover p: C → X is a homeomorphism f: C → C such that p o f = p. The set of all deck transformations of p forms a group under composition, the deck transformation group Aut ( p ).

Over an algebraically closed field, this and its triple cover are the only forms ; however, over other fields, there are often many other forms, or “ twists ” of E < sub > 6 </ sub >, which are classified in the general framework of Galois cohomology ( over a perfect field k ) by the set H < sup > 1 </ sup >( k, Aut ( E < sub > 6 </ sub >)) which, because the Dynkin diagram of E < sub > 6 </ sub > ( see below ) has automorphism group Z / 2Z, maps to H < sup > 1 </ sup >( k, Z / 2Z ) = Hom ( Gal ( k ), Z / 2Z ) with kernel H < sup > 1 </ sup >( k, E < sub > 6, ad </ sub >).

There are 23 orbits of them under the automorphism group of the Leech lattice, and these orbits correspond to the 23 Niemeier lattices other than the Leech lattice: the set of vertices of deep hole is isometric to the affine Dynkin diagram of the corresponding Niemeier lattice.

One can define a projective space axiomatically in terms of an incidence structure ( a set of points P, lines L, and an incidence relation I specifying which points lie on which lines ) satisfying certain axioms – an automorphism of a projective space thus defined then being an automorphism f of the set of points and an automorphism g of the set of lines, preserving the incidence relation ,< ref group =" note ">" Preserving the incidence relation " means that if point p is on line l then is in ; formally, if then .</ ref > which is exactly a collineation of a space to itself.

Given a fundamental domain for the group action ( for the full, orientation-reversing symmetry group, a ( 2, 3, 7 ) triangle ), the reflection domains ( images of this domain under the group ) give a tiling of the quartic such that the automorphism group of the tiling equals the automorphism group of the surface – reflections in the lines of the tiling correspond to the reflections in the group ( reflections in the lines of a given fundamental triangle give a set of 3 generating reflections ).

When each number of successes X is paired with its probability of occurrence Af, the set of pairs Af, is a probability function called a binomial distribution.

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:

: For any set X of nonempty sets, there exists a choice function f defined on X.

: Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.

This guarantees for any partition of a set X the existence of a subset C of X containing exactly one element from each part of the partition.

For example, after having established that the set X contains only non-empty sets, a mathematician might have said " let F ( s ) be one of the members of s for all s in X.

For example, suppose that X is the set of all non-empty subsets of the real numbers.

If we try to choose an element from each set, then, because X is infinite, our choice procedure will never come to an end, and consequently, we will never be able to produce a choice function for all of 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.

If the automorphisms of an object X form a set ( instead of a proper class ), then they form a group under composition of morphisms.

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X