Reading this Wikipedia page I found this definition:
A real number $a$ is first-order definable in the language of set theory, without parameters, if there is a formula $\phi$ in the language of set theory, with one free variable, such that $a$ is the unique real number such that $\phi(a)$ holds in the standard model of set theory.
A few lines later we find the statement:
Assuming they form a set, the definable numbers form a field....
But, since they are a subset of the set of real numbers, why shouldn't they be a set?
Coming back from this question to the definition, I've another doubt: if ZFC is consistent this does not means that every set-theoretic object (and so any real number) is definable in some model?
Reading the whole article does not lessen my confusion .... and the ''talk'' is too difficult for me and it does not help.
More generally, this Wikipedia article is "disputed" ad has many "!" So I doubt that it is not reliable.
A brief surf on the web give me many pages on this subject but I've found nothing that I can understand and give a response to the question: we can well define what is a definable real number?