Definable real number on arvau

Wikipedia / Definable real number

Informally, a definable real number is a real number that can be uniquely specified by its description. The description may be expressed as a construction or as a formula of a formal language, and it can be constructed with a compass and straightedge.

Different choices of a formal language or its interpretation give rise to different notions of definability. Specific varieties of definable numbers include the constructible numbers of geometry, the algebraic numbers, and the computable numbers.

Constructible numbers

using a compass and straightedge, beginning with a fixed line segment of length 1.

Each positive integer, and each positive rational number, is constructible. The positive square root of 2 is constructible. However, the cube root of 2 is not constructible; this is related to the impossibility of doubling the cube.

Real algebraic numbers

is called a real algebraic number if there is a polynomial is zero.

All rational numbers are constructible, and all constructible numbers are algebraic. There are numbers such as the cube root of 2 which are algebraic but not constructible.

The real algebraic numbers form a subfield.

that are real numbers are also algebraic.

There are only countably many algebraic numbers, but there are uncountably many real numbers, so in the sense of cardinality most real numbers are not algebraic. This nonconstructive proof that not all real numbers are algebraic was first published by Georg Cantor in his 1874 paper "On a Property of the Collection of All Real Algebraic Numbers".

Non-algebraic numbers are called transcendental numbers. The best known transcendental numbers are and .

Computable real numbers

A real number is a computable number decimal places. This notion was introduced by Alan Turing in 1936.

th roots for each

Not all real numbers are computable. Specific examples of noncomputable real numbers include the limits of Specker sequences, and algorithmically random real numbers such as Chaitin's Ω numbers.

Definability in arithmetic

Another notion of definability comes from the formal theories of arithmetic, such as Peano 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. Because no variables of this language range over the real numbers is definable in the language of arithmetic (or arithmetical) if its Dedekind cut can be defined as a predicate Here m, n, and p range over nonnegative integers.

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.

Every computable real number is arithmetical, and the arithmetical numbers form a subfield of the reals, as do the analytical numbers. Every arithmetical number is analytical, but not every analytical number is arithmetical. Because there are only countably many analytical numbers, most real numbers are not analytical, and thus also not arithmetical.

Every computable number is arithmetical, but not every arithmetical number is computable. For example, the limit of a Specker sequence is an arithmetical number that is not computable.

The definitions of arithmetical and analytical reals can be stratified into the arithmetical hierarchy and analytical hierarchy of the arithmetical hierarchy, one of the lowest levels. Similarly, the reals with arithmetical Dedekind cuts form the lowest level of the analytical hierarchy.

Definability in models of ZFC

is first-order definable in the language of set theory, without parameters, relative to a particular model of ZFC in the language of set theory, with one free variable holds. This gives different notions of definability for different models, which cannot be characterized by a formula in the language of set theory.

All analytical numbers, and in particular all computable numbers, are definable in the language of set theory. Thus the real numbers definable in the language of set theory include all familiar real numbers such as 0, 1, et cetera, along with all algebraic numbers. Assuming that they form a set in the model, the real numbers definable in the language of set theory form a field.

This argument becomes more problematic if it is applied to class models of ZFC, such as the von Neumann universe is definable over the class" cannot be expressed as a formula of ZFC. Similarly, the question of whether the von Neumann universe contains real numbers that it cannot define cannot be expressed as a sentence in the language of ZFC. Moreover, there are countable models of ZFC in which all real numbers, all sets of real numbers, functions on the reals, etc. are definable.