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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?

Emilio Novati
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1 Answers1

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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.

Asaf Karagila
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  • When the Wikipedia article says "the standard model" I don't think the set theorist's notion of "standard model" is even close to what it intends. I think it's trying to refer to an "intended interpretation" of the language of set theory, that is, an assumed Platonically existing universe of actual sets. There are numerous problems with that concept, but I don't think it illuminates those problems to point to the (more or less unrelated) technical use of the words "standard model". – hmakholm left over Monica Jan 29 '15 at 17:24
  • Henning, that might be; doesn't mean that one should say things like "the standard model of set theory", since unlike the natural numbers or the real numbers, it's nearly impossible for set theorists to decide what is "the intended interpretation" (and most mathematicians simply don't care). – Asaf Karagila Jan 29 '15 at 17:27
  • Sure sure, hence the "numerous problems" I alluded to. – hmakholm left over Monica Jan 29 '15 at 17:30
  • Asaf, do you know an example of a model of ZFC where the collection of definable reals in the model does not belong to the model? –  Jan 29 '15 at 17:46
  • Never mind, the pointwise definable reals in any model always form a definable set in that model. –  Jan 29 '15 at 18:17
  • @OohAah: How do you figure? – Asaf Karagila Jan 29 '15 at 21:37
  • @Asaf The problem is much more complicated than I thought and the indicated links are too hard for me. What I understand is that, given a real number, we can recognize if it's definable ( but this seems an obvious statement), but we can not formally define a set of ''definable'' numbers because there are many "standard models" of set theory that, in some sense, contains different models of real numbers. But, if is so, why not define the model in which all reals are definable the standard model? – Emilio Novati Jan 30 '15 at 13:46
  • @Emilio: It's not that obvious that we can recognize that a real number is definable from within the model (recognizing something is definable from outside a model is indeed trivial); the question is whether or not the property "$r$ is a definable real number" is expressible is trickier though. As for your last question, simply because this is it not enough to characterize a model of set theory, and frankly not that important of a property for it to crown a particular model as the standard model of set theory. Why not actual, important, set theoretic axioms? [...] – Asaf Karagila Jan 30 '15 at 15:12
  • [...] And the answer here is essentially many axioms are valid, and many of them are incompatible. $\sf CH$ is incompatible with $\sf PFA$, which is incompatible with the existence of various $\square$ sequences and trees, and so on and so forth. There are uses and merits for these sort of axioms and for their negations; so there is no real reason to target one specific model as the standard model. Not to mention, you expect the standard model to have some relation with every other model of the theory, which is certainly not the case here. Not even a little. – Asaf Karagila Jan 30 '15 at 15:17
  • @Asaf: thank you for the answers! I fall easily into the trap of the "inside-outside the model"... Finally, can you give me a reference to a model in which all reals are definable? Probably it will be too difficult for me but i'm curious . – Emilio Novati Jan 30 '15 at 16:07
  • @Emilio: You should take a look in Hamkins, Linetsky and Reitz that I referred to (with a link to Hamkins' blog, where you can find additional links). The first part of that paper is also very readable for other general audiences. – Asaf Karagila Jan 30 '15 at 16:42
  • @Asaf, you mentioned that it is known to be consistent that every real number is definable. Wikipedia says that "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". When I read this, I thought it was fallacious, and it seems it truly is. Someone should fix the article. – vhspdfg Mar 04 '16 at 02:01
  • I deleted the fallacious/false sentences on wikipedia. Hopefully it stays deleted. – vhspdfg Mar 04 '16 at 02:52
  • ...And wikipedia reverted the edit. I guess the wikipedia article is just going to stay factually incorrect. – vhspdfg Mar 04 '16 at 03:19
  • @vhspdfg: If I recall correctly, the talk page on that Wikipedia entry is aware of the severe problems on that page. – Asaf Karagila Mar 04 '16 at 07:10