Definable real numbers

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?

Solution 1:

There are several problems here:

1. There is not "the standard model of set theory". There are notions of "standard models" (note the plural), but there is no "the standard model". With respect to the real numbers there are several possible scenarios:

   • It might be the case that there is a standard model containing all the reals. This model, if so, has to be uncountable.

   • It might be that every real is a real number of some standard model, but there is no standard model containing all the reals.

   • It might be that there are real numbers which cannot be members of any standard model, and some that can be.

   • It might be that there are no standard models at all.

   So this is really a delicate issue here. But in any case, one shouldn't qualify "standard model of set theory" with "the". At all.

2. The notion of "definable real number" often means definable over $\Bbb R$ as a real number in a language augmented by all sort of things we are used to have in mathematics, integrals, sines and cosines, etc. In that case, there are generally only countably many definable reals, since there are only countably many formulas to define reals with.

   Once you add the rest of the set theoretic universe into play, you can have that every real number is definable. This is a delicate issue, and known to be consistent, see Joel Hamkins, David Linetsky and Jonas Reitz's paper "Pointwise Definable Models of Set Theory" (and Joel Hamkins' blog post on the paper which has a nice discussion on the topic).

3. And this brings us to the problem at hand. It might be the case that the collection of all definable reals is not itself definable internally. Namely, we can recognize whether a real number is definable or not; but there is no formula whose content is "$x$ is a definable real number". This can be the case because we cannot match a real number to its definition, and we cannot really quantify over formulas to say "There exists a definition".

   But sometimes we are in a case where we can in fact identify the definable real numbers, either we know that they form a set (which was defined using some other formula) or that we managed to circumvent the inability to match a real to its definition by adding further assumptions that make things like that possible. And in those cases the set of definable reals, the Wikipedia article states, is a subfield of $\Bbb R$ of that model of set theory.