A real number a is firstorder 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.
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General facts
Assuming they form a set, the definable numbers form a field 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 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').
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.
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 numbers: 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 3i), 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 settheoretic 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.
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