In mathematics, the axiom of regularity ( also known as the axiom of foundation ) is one of the axioms of Zermelo – Fraenkel set theory and was introduced by.

With the axiom of dependent choice ( which is a weakened form of the axiom of choice ), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true.

Hence, the axiom of regularity is equivalent, given the axiom of dependent choice, to the alternative axiom that there are no downward infinite membership chains.

In addition to omitting the axiom of regularity, non-standard set theories have indeed postulated the existence of sets that are elements of themselves.

Given the other ZF axioms, the axiom of regularity is equivalent to the axiom of induction.

Let A be a set, and apply the axiom of regularity to

While this approach can serve to preserve the axiom of extensionality, the axiom of regularity will need an adjustment instead.

There are many situations where another condition of topological spaces ( such as normality, paracompactness, or local compactness ) will imply regularity if some weaker separation axiom, such as preregularity, is satisfied.

Of course, one can easily find regular spaces that are not T < sub > 0 </ sub >, and thus not Hausdorff, such as an indiscrete space, but these examples provide more insight on the T < sub > 0 </ sub > axiom than on regularity.

The axiom of regularity, which is one of the axioms of Zermelo – Fraenkel set theory, asserts that all sets are well-founded.

In ZFC, there is no infinite descending ∈- sequence by the axiom of regularity.

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.

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.

This definition of cofinality relies on the axiom of choice, as it uses the fact that every non-empty set of cardinal numbers has a least member.

For example, Euclid assumed implicitly that any line contains at least two points, but this assumption cannot be proved from the other axioms, and therefore must be an axiom itself.

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.

The first axiom asserts the existence of at least one member of the set " number ".

The axiom AX is now justified based on the principle that what will predictably happen every time this experiment is performed, should at the very least be possible.

: Means that if A and B are in all respect identical the consumer will consider a to be at least as good as ( is weakly preferred ) to B. Alternatively, the axiom can be modified to read that the consumer is indifferent with regard to A and B.

( Although it was perhaps implied, he did not include an axiom requiring at least one subset to be independent.

If sheaf values are in the category of sets, applying the local identity axiom to the empty family shows that over the empty set, there is at most one section, and applying the gluing axiom to the empty family shows that there is at least one section.

This is a relatively weak large cardinal axiom since it amounts to saying that ∞ is 1-inaccessible in the language of the next section, where ∞ denotes the least ordinal not in V, i. e. the class of all ordinals in your model.

The only real number axiom that does not follow easily from the definitions is the completeness of ≤, i. e. the least upper bound property.

Formally, assuming the axiom of choice, cardinality of a set X is the least ordinal α such that there is a bijection between X and α.

* an axiom scheme asserting that all polynomials of odd degree have at least one root,

A decision method is an axiomatic system that contains at least one action axiom.

During the conversion, it may be useful to put all the applications of modus ponens to axiom 1 at the beginning of the deduction ( right after the H → H step ).

This fact is not very useful in practice since there are many such extensions, inconsistent with each other, and also since proving such operators exist requires invoking the axiom of choice or its equivalents, such as Zorn's lemma.

( Separating the definitions in this way is useful in the absence of the axiom of choice ; see the next section.

This last diagram also illustrates a useful intuitive interpretation of the octahedral axiom.