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.

Basic theories, such as arithmetic, real analysis and complex analysis are often introduced non-axiomatically, but implicitly or explicitly there is generally an assumption that the axioms being used are the axioms of Zermelo – Fraenkel set theory with choice, abbreviated ZFC, or some very similar system of axiomatic set theory like Von Neumann – Bernays – Gödel set theory, a conservative extension of ZFC.

They are a set of axioms strong enough to prove many important facts about number theory and they allowed Gödel to establish his famous second incompleteness theorem.

The contributions of Kurt Gödel in 1940 and Paul Cohen in 1963 showed that the hypothesis can neither be disproved nor be proved using the axioms of Zermelo – Fraenkel set theory, the standard foundation of modern mathematics, provided ZF set theory is consistent.

One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when many, such as Bertrand Russell, A. N. Whitehead and David Hilbert, were pioneering the use of logic and set theory to understand the foundations of mathematics.

In work on Zermelo – Fraenkel set theory, the notion of class is informal, whereas other set theories, such as Von Neumann – Bernays – Gödel set theory, axiomatize the notion of " class ", e. g., as entities that are not members of another entity.

Another approach is taken by the von Neumann – Bernays – Gödel axioms ( NBG ); classes are the basic objects in this theory, and a set is then defined to be a class that is an element of some other class.

Gödel and Paul Cohen showed that this hypothesis cannot be proved or disproved using the standard axioms of set theory.

) If formulated in von Neumann – Bernays – Gödel set theory, the surreal numbers are the largest possible ordered field ; all other ordered fields, such as the rationals, the reals, the rational functions, the Levi-Civita field, the superreal numbers, and the hyperreal numbers, are subfields of the surreals ; it has also been shown that the maximal class hyperreal field is isomorphic to the maximal class surreal field.

These set theories cannot prove their own Gödel sentences – provided that they are consistent, which is generally believed.

Then it is not possible to define, in ZFC, the set of all ( Gödel numbers of ) formulas that define real numbers.

If is indexed by a set consisting of all the natural numbers or a finite subset of them, then it is easy to set up a simple one to one coding ( or Gödel numbering ) from the free group on to the natural numbers, such that we can find algorithms that, given, calculate, and vice versa.

A Gödel numbering is a precomplete numbering on the set of computable functions so the generalized theorem yields the Kleene recursion theorem as a special case.

Many researchers in axiomatic set theory have subscribed to what is known as set-theoretical Platonism, exemplified by mathematician Kurt Gödel.

Moreover, for any such set there is a computable enumeration of Gödel numbers of basic open sets whose union is the original set.

For example, supposing there are K basic symbols, an alternative Gödel numbering could be constructed by invertibly mapping this set of symbols ( through, say, an invertible function h ) to the set of digits of a bijective base-K numeral system.

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

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.

Similarly, although a subset of the real numbers that is not Lebesgue measurable can be proven to exist using the axiom of choice, it is consistent that no such set is definable.

** If the set A is infinite, then there exists an injection from the natural numbers N to A ( see Dedekind infinite ).

For example, if we abbreviate by BP the claim that every set of real numbers has the property of Baire, then BP is stronger than ¬ AC, which asserts the nonexistence of any choice function on perhaps only a single set of nonempty sets.

The standard structure is where is the set of natural numbers, is the successor function and is naturally interpreted as the number 0.

The real numbers are uniquely picked out ( up to isomorphism ) by the properties of a Dedekind complete ordered field, meaning that any nonempty set of real numbers with an upper bound has a least upper bound.

* The set of algebraic numbers is countable ( enumerable ).

* Hence, the set of algebraic numbers has Lebesgue measure zero ( as a subset of the complex numbers ), i. e. " almost all " complex numbers are not algebraic.

* The set of real algebraic numbers is linearly ordered, countable, densely ordered, and without first or last element, so is order-isomorphic to the set of rational numbers.

Each orbital is defined by a different set of quantum numbers ( n, l, and m ), and contains a maximum of two electrons each with their own spin quantum number.

These states are labeled by a set of quantum numbers summarized in the term symbol and usually associated with particular electron configurations, i. e., by occupation schemes of atomic orbitals ( e. g., 1s < sup > 2 </ sup > 2s < sup > 2 </ sup > 2p < sup > 6 </ sup > for the ground state of neon -- term symbol: < sup > 1 </ sup > S < sub > 0 </ sub >).

Because of the quantum mechanical nature of the electrons around a nucleus, atomic orbitals can be uniquely defined by a set of integers known as quantum numbers.

Area can be defined through the use of axioms, defining it as a function of a collection of certain plane figures to the set of real numbers.

Area can be defined as a function from a collection M of special kind of plane figures ( termed measurable sets ) to the set of real numbers which satisfies the following properties:

Some large telephone companies have toll-free numbers set up.

These numbers are set up by a company offering low charge calls in the UK, these numbers are meant to be used as a sort of operator that you go through in order to qualify for these cheap calls.

Sometimes slightly stronger theories such as Morse-Kelley set theory or set theory with a strongly inaccessible cardinal allowing the use of a Grothendieck universe are used, but in fact most mathematicians can actually prove all they need in systems weaker than ZFC, such as second-order arithmetic.

Note that " completeness " has a different meaning here than it does in the context of Gödel's first incompleteness theorem, which states that no recursive, consistent set of non-logical axioms of the Theory of Arithmetic is complete, in the sense that there will always exist an arithmetic statement such that neither nor can be proved from the given set of axioms.

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

In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function ƒ ( n ) defined on the set of natural numbers ( i. e. positive integers ) that " expresses some arithmetical property of n ."

If, for instance, an addition operation was requested, the arithmetic logic unit ( ALU ) will be connected to a set of inputs and a set of outputs.

If the addition operation produces a result too large for the CPU to handle, an arithmetic overflow flag in a flags register may also be set.

An arithmetic calendar is one that is based on a strict set of rules ; an example is the current Jewish calendar.

* Midrange – the arithmetic mean of the maximum and minimum values of a data set.

A complex instruction set computer ( CISC, ) is a computer where single instructions can execute several low-level operations ( such as a load from memory, an arithmetic operation, and a memory store ) and / or are capable of multi-step operations or addressing modes within single instructions.

While every computable number is definable, the converse is not true: the numeric representations of the Halting problem, Chaitin's constant, the truth set of first order arithmetic, and 0 < sup >#</ sup > are examples of numbers that are definable but not computable.

Many common axiomatic systems, such as first-order Peano arithmetic and axiomatic set theory, including the canonical Zermelo – Fraenkel set theory ( ZF ), can be formalized as first-order theories.

The arithmetic is defined as a set of actions on the representation that simulate classical arithmetic operations.

The geometric mean is similar to the arithmetic mean, except that the numbers are multiplied and then the nth root ( where n is the count of numbers in the set ) of the resulting product is taken.

The geometric mean of a data set is less than the data set's arithmetic mean unless all members of the data set are equal, in which case the geometric and arithmetic means are equal.