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A real number a is first-order definable in the language of set theory, without parameters, if there is a formula φ in the language of set theory, with one free variable, such that a is the unique real number such that φ(a) holds (in the von Neumann universe V). In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ...
Set theory is the mathematical theory of sets, which represent collections of abstract objects. ...
In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation for a place or places in an expression, into which some definite substitution may take place, or with respect to which some operation (summation or quantification, to give two...
In axiomatic set theory and related branches of mathematics, the Von Neumann universe, or Von Neumann hierarchy of sets is the class of all sets, divided into a transfinite hierarchy of individual sets. ...
For the purposes of this article, such reals will be called simply definable numbers. This should not be understood to be standard terminology.
General facts The definable numbers form a field containing all the familiar real numbers such as 0, 1, π, e, et cetera. In particular, it contains all the numbers named in the mathematical constants article, and all algebraic numbers (and therefore all rational numbers). However, most real numbers are not definable: the set of all definable numbers is countably infinite (because the set of all logical formulas is) while the set of real numbers is uncountably infinite (see Cantor's diagonal argument). As a result, most real numbers have no description (in the same sense of "most" as 'most real numbers are not rational'). In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...
A mathematical constant is a quantity, usually a real number or a complex number, that arises naturally in mathematics and does not change. ...
In mathematics, an algebraic number relative to a field is any element of a given field containing such that is a solution of a polynomial equation of the form: anxn + an−1xn−1 + ··· + a1x + a0 = 0 where n is a positive integer called the degree of the polynomial, every coefficient...
In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...
In mathematics, a set can be thought of as any collection of distinct things considered as a whole. ...
In mathematics the term countable set is used to describe the size of a set, e. ...
In mathematics, an uncountable set is a set which is not countable. ...
Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ...
In mathematics, the phrase almost all has a number of specialised uses. ...
The field of definable numbers is not complete; there exist convergent sequences of definable numbers whose limit is not definable (since every real number is the limit of a sequence of rational numbers). However, if the sequence itself is definable in the sense that we can specify a single formula for all its terms, then its limit will necessarily be a definable number. In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For...
This is a page about mathematics. ...
In mathematics, the concept of a limit is used to describe the behavior of a function, as its argument gets close to either some point, or infinity; or the behavior of a sequences elements, as their index approaches infinity. ...
While every computable number is definable, the converse is not true: Chaitin's constant is definable but not computable. In mathematics, theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers, are the subset of the real numbers consisting of the numbers which can be computed by a finite, terminating algorithm. ...
In the computer science subfield of algorithmic information theory the Chaitin constant or halting probability is a construction by Gregory Chaitin which describes the probability that a randomly generated program for a given model of computation or programming language will halt. ...
One may also talk about definable complex numbers: complex numbers which are uniquely defined by a logical formula. A complex number is definable if and only if both its real part and its imaginary part are definable. The definable complex numbers also form a field. Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for one of the square roots of negative one (−1). ...
Wikibooks Algebra has more about this subject: Complex numbers In mathematics, a complex number is an expression of the form a + bi, where a and b are real numbers, and i stands for one of the square roots of negative one (−1). ...
The related concept of "standard" numbers, which can only be defined within a finite time and space, is used to motivate axiomatic internal set theory, and provide a workable formulation for illimited and infinitesimal number. Definitions of the hyper-real line within non-standard analysis (the subject area dealing with such numbers) overwhelmingly include the usual, uncountable set of real numbers as a subset. Internal set theory (IST) is a mathematical theory of sets developed by Edward Nelson which provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. ...
In mathematics, an infinitesimal, or infinitely small number, is a number that is greater in absolute value than zero yet smaller than any positive real number. ...
Notion does not exhaust "unambiguously described" numbers Not every number that we would informally say has been unambiguously described, is definable in the above sense. For example, we can enumerate all such definable numbers by the Gödel numbers of their defining formulas, and then use Cantor's diagonal argument to find a particular real that is not first-order definable in the same language. In formal number theory a Gödel numbering is a function which assigns to each symbol and formula of some formal language a unique natural number called a Gödel number (GN). ...
Cantors diagonal argument is a proof devised by Georg Cantor to demonstrate that the real numbers are not countably infinite. ...
Other notions of definability The notion of definability treated in this article has been chosen primarily for definiteness, not on the grounds that it's more useful or interesting than other notions. Here we treat a few others:
Definability in other languages or structures
Language of arithmetic The language of arithmetic has symbols for 0, 1, the successor operation, addition, and multiplication, intended to be interpreted in the usual way over the natural numbers. Since no variables of this language range over the reals, we cannot simply copy the earlier definition of definability. Rather, we say that a real a is definable in the language of arithmetic (or arithmetical) if its Dedekind cut can be defined as a predicate in that language; that is, if there is a first-order formula φ in the language of arithmetic, with two free variables, such that In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...). The former definition is generally used in number theory, while the latter is preferred in set theory. ...
The word real has many different meanings. ...
In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene hierarchy classifies the set of arithmetic formulas (or arithmetic sets) according to their degree of solvability. ...
In mathematics, a Dedekind cut, named after Richard Dedekind, in a totally ordered set S is a partition of it, (A, B), such that A is closed downwards (meaning that for all a in A, x ≤ a implies that x is in A as well) and B is closed upwards...
In mathematics, a predicate is a relation. ...
2nd-order language of arithmetic The second-order language of arithmetic is the same as the first-order language, except that variables and quantifiers are allowed to range over sets of naturals. A real that is second-order definable in the language of arithmetic is called analytical. In mathematical logic and descriptive set theory, the analytical hierarchy is a second-order analogue of the arithmetical hierarchy. ...
Definability with ordinal parameters Sometimes it is of interest to consider definability with parameters; that is, to give a definition relative to another object that remains undefined. For example, a real a (or for that matter, any set a) is called ordinal definable if there is a first-order formula φ in the language of set theory, with two free variables, and an ordinal γ, such that a is the unique object such that φ(a,γ) holds (in V). Ordinal numbers, or ordinals for short, are numbers used to denote the position in an ordered sequence: first, second, third, fourth, etc. ...
The other sorts of definability thus far considered have only countably many defining formulas, and therefore allow only countably many definable reals. This is not true for ordinal definability, because an ordinal definable real is defined not only by the formula φ, but also by the ordinal γ. In fact it is consistent with ZFC that all reals are ordinal-definable, and therefore that there are uncountably many ordinal-definable reals. However it is also consistent with ZFC that there are only countably many ordinal-definable reals. The Zermelo-Fraenkel axioms of set theory (ZF) are the standard axioms of axiomatic set theory on which, together with the axiom of choice, all of ordinary mathematics is based in modern formulations. ...
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