Definable number
Encyclopedia
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 standard model of set theory (see Kunen 1980:153).
For the purposes of this article, such reals will be called simply definable numbers. This should not be understood to be standard terminology.
Note that this definition cannot be expressed in the language of set theory itself.
containing all the familiar real numbers such as 0, 1, π, e, et cetera. In particular, this field contains all the numbers named in the mathematical constant
s article, and all algebraic number
s (and therefore all rational number
s). 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').
The field of definable numbers is not complete
; there exist convergent sequence
s 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.
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#
are examples of numbers that are definable but not computable. Many other such numbers are known.
One may also wish to talk about definable complex number
s: complex numbers which are uniquely defined by a logical formula. However, whether this is possible depends on how the field of complex numbers is derived in the first place: it may not be possible to distinguish a complex number from its conjugate (say, 3+i from 3-i), since it is impossible to find a property of one that is not also a property of the other, without falling back on the underlying set-theoretic definition. Assuming we can define at least one nonreal complex number, however, 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 if they form a set.
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.
s of their defining formulas then we can use Cantor's diagonal argument
to find a particular real that is not first-order definable in the same language. The argument can be made as follows:
Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G(n) is the real number described by the nth description in the sequence. Using the diagonal argument, it is possible to define a real number x, which is not equal to G(n) for any n. This means that there is a language L' that defines x, which is undefinable in L.
s. Since no variables of this language range over the real
s, 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
.
γ, such that a is the unique object such that φ(a,γ) holds (in V).
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.
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
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
Set theory
Set theory is the branch of mathematics that studies sets, which are collections of objects. Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics...
, with one free variable, such that a is the unique real number such that φ(a) holds in the standard model of set theory (see Kunen 1980:153).
For the purposes of this article, such reals will be called simply definable numbers. This should not be understood to be standard terminology.
Note that this definition cannot be expressed in the language of set theory itself.
General facts
Assuming they form a set, the definable numbers form a fieldField (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
containing all the familiar real numbers such as 0, 1, π, e, et cetera. In particular, this field contains all the numbers named in the mathematical constant
Mathematical constant
A mathematical constant is a special number, usually a real number, that is "significantly interesting in some way". Constants arise in many different areas of mathematics, with constants such as and occurring in such diverse contexts as geometry, number theory and calculus.What it means for a...
s article, and all algebraic number
Algebraic number
In mathematics, an algebraic number is a number that is a root of a non-zero polynomial in one variable with rational coefficients. Numbers such as π that are not algebraic are said to be transcendental; almost all real numbers are transcendental...
s (and therefore all rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s). 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
Uncountable set
In mathematics, an uncountable set is an infinite set that contains too many elements to be countable. The uncountability of a set is closely related to its cardinal number: a set is uncountable if its cardinal number is larger than that of the set of all natural numbers.-Characterizations:There...
(see Cantor's diagonal argument
Cantor's diagonal argument
Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural...
). As a result, most
Almost all
In mathematics, the phrase "almost all" has a number of specialised uses."Almost all" is sometimes used synonymously with "all but finitely many" or "all but a countable set" ; see almost....
real numbers have no description (in the same sense of "most" as 'most real numbers are not rational').
The field of definable numbers is not complete
Complete space
In mathematical analysis, a metric space M is called complete if every Cauchy sequence of points in M has a limit that is also in M or, alternatively, if every Cauchy sequence in M converges in M....
; there exist convergent sequence
Sequence
In mathematics, a sequence is an ordered list of objects . Like a set, it contains members , and the number of terms is called the length of the sequence. Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence...
s of definable numbers whose limit
Limit of a sequence
The limit of a sequence is, intuitively, the unique number or point L such that the terms of the sequence become arbitrarily close to L for "large" values of n...
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.
While every computable number
Computable number
In mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm...
is definable, the converse is not true: the numeric representations of the Halting problem
Halting problem
In computability theory, the halting problem can be stated as follows: Given a description of a computer program, decide whether the program finishes running or continues to run forever...
, Chaitin's constant
Chaitin's constant
In the computer science subfield of algorithmic information theory, a Chaitin constant or halting probability is a real number that informally represents the probability that a randomly constructed program will halt...
, the truth set of first order arithmetic, and 0#
Zero sharp
In the mathematical discipline of set theory, 0# is the set of true formulas about indiscernibles in the Gödel constructible universe. It is often encoded as a subset of the integers , or as a subset of the hereditarily finite sets, or as a real number...
are examples of numbers that are definable but not computable. Many other such numbers are known.
One may also wish to talk about definable complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
s: complex numbers which are uniquely defined by a logical formula. However, whether this is possible depends on how the field of complex numbers is derived in the first place: it may not be possible to distinguish a complex number from its conjugate (say, 3+i from 3-i), since it is impossible to find a property of one that is not also a property of the other, without falling back on the underlying set-theoretic definition. Assuming we can define at least one nonreal complex number, however, a complex number
Complex number
A complex number is a number consisting of a real part and an imaginary part. Complex numbers extend the idea of the one-dimensional number line to the two-dimensional complex plane by using the number line for the real part and adding a vertical axis to plot the imaginary part...
is definable if and only if both its real part and its imaginary part are definable. The definable complex numbers also form a field if they form a set.
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
Internal set theory
Internal set theory is a mathematical theory of sets developed by Edward Nelson that provides an axiomatic basis for a portion of the non-standard analysis introduced by Abraham Robinson. Instead of adding new elements to the real numbers, the axioms introduce a new term, "standard", which can be...
, 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.
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, if we can enumerate all such definable numbers by the Gödel numberGödel number
In mathematical logic, a Gödel numbering is a function that assigns to each symbol and well-formed formula of some formal language a unique natural number, called its Gödel number. The concept was famously used by Kurt Gödel for the proof of his incompleteness theorems...
s of their defining formulas then we can use Cantor's diagonal argument
Cantor's diagonal argument
Cantor's diagonal argument, also called the diagonalisation argument, the diagonal slash argument or the diagonal method, was published in 1891 by Georg Cantor as a mathematical proof that there are infinite sets which cannot be put into one-to-one correspondence with the infinite set of natural...
to find a particular real that is not first-order definable in the same language. The argument can be made as follows:
Suppose that in a mathematical language L, it is possible to enumerate all of the defined numbers in L. Let this enumeration be defined by the function G: W → R, where G(n) is the real number described by the nth description in the sequence. Using the diagonal argument, it is possible to define a real number x, which is not equal to G(n) for any n. This means that there is a language L' that defines x, which is undefinable in L.
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: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 numberNatural number
In mathematics, the natural numbers are the ordinary whole numbers used for counting and ordering . These purposes are related to the linguistic notions of cardinal and ordinal numbers, respectively...
s. Since no variables of this language range over the real
Real
Real may also refer to:* Reality, the state of things as they actually exist, rather than as they may appear or may be thought to be.-Finance:* Inflation adjusted amountsCurrencies:* Brazilian realFormer currencies:* Mexican real* Portuguese real...
s, 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
Arithmetical hierarchy
In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene-Mostowski hierarchy classifies certain sets based on the complexity of formulas that define them...
) if its Dedekind cut
Dedekind cut
In mathematics, a Dedekind cut, named after Richard Dedekind, is a partition of the rationals into two non-empty parts A and B, such that all elements of A are less than all elements of B, and A contains no greatest element....
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
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 analyticalAnalytical hierarchy
In mathematical logic and descriptive set theory, the analytical hierarchy is a higher type analogue of the arithmetical hierarchy. It thus continues the classification of sets by the formulas that define them.-The analytical hierarchy of formulas:...
.
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 ordinalOrdinal number
In set theory, an ordinal number, or just ordinal, is the order type of a well-ordered set. They are usually identified with hereditarily transitive sets. Ordinals are an extension of the natural numbers different from integers and from cardinals...
γ, such that a is the unique object such that φ(a,γ) holds (in V).
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.
See also
- Berry paradoxBerry paradoxThe Berry paradox is a self-referential paradox arising from the expression "the smallest possible integer not definable by a given number of words". Bertrand Russell, the first to discuss the paradox in print, attributed it to G. G...
- Computable numberComputable numberIn mathematics, particularly theoretical computer science and mathematical logic, the computable numbers, also known as the recursive numbers or the computable reals, are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm...
- Constructible numberConstructible numberA point in the Euclidean plane is a constructible point if, given a fixed coordinate system , the point can be constructed with unruled straightedge and compass...
- Constructible universeConstructible universeIn mathematics, the constructible universe , denoted L, is a particular class of sets which can be described entirely in terms of simpler sets. It was introduced by Kurt Gödel in his 1938 paper "The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis"...
- EntscheidungsproblemEntscheidungsproblemIn mathematics, the is a challenge posed by David Hilbert in 1928. The asks for an algorithm that will take as input a description of a formal language and a mathematical statement in the language and produce as output either "True" or "False" according to whether the statement is true or false...