TITLE: what is actually a "definable" real number?
QUESTION [9 upvotes]: Wikipedia states that a definable (real) number is a number "that can be uniquely specified by its description". I naïvely thought that this was the same as "a number which may be computed", but elsewhere in math.SE I found out that it is not the case, because for example numbers related to Chaitin's $\Omega$ are definable but not computable. Could someone explain me what a specification is, then? I always thought that $\Omega$ was a probability value, which was not a real specification.

REPLY [2 votes]: "Definable" as a very precise meaning. Given a structure $M$ on a first-order language $\mathcal L$, an element $a\in M$ is definable if and only if there exists a formula $\varphi(x)$ in one variable such that
$$ M \models \forall x, \varphi(x) \leftrightarrow (x=a)$$
In particular "definable" is always relative to the language of specification. For example, given $M$ as before, an element $a\in M$ is always definable in (the induced structure on) $M$ over the language $\mathcal L \sqcup \{c_a\}$ where the new constant $c_a$ is interpreted as $a$ (the wanted $\varphi(x)$ is just "$x=c_a$"). Even if $a$ was not definable over $\mathcal L$.

When saying "Is the real number $a$ definable?", it is probable that people mean: given a model $\mathcal S$ of ZFC, is (the set corresponding to) the real number $a$ (in the usual encoding of reals in ZFC) definable over the language $\{{\in}\}$?
As nobody know an explicit model of ZFC, there is a good chance that "the real number $a$" is actually already a shortcut for the formula $\varphi(x)$. To go back to your question, when people say "Chaitin's constant", they do not refer to a specific element of a God-given model of ZFC, they refer to the construction you probably know: enumerates the countably infinite Turing's machines as $m_1,\dots$ and affect the value $t_i=0$ if $m_i$ terminates or $t_i=1$ otherwise then creates the number $\sum_{i=1}^\infty t_i10^{-i}$. All of that is completely expressable in the language $\{{\in}\}$ under the axioms of ZFC. In particular, what is bothering you is the "if $m_i$ terminates", but termination of a Turing machine is a property that can be encoded into ZFC once the notion of Turing machine is encoded into ZFC. (ZFC would be making a poor job as a foundation of mathematics if such things was not expressable.)