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https://en.wikipedia.org/wiki/Definable_real_number

Wikipedia defines a "definable real number" as one definable by a parameter-free formula in the language of set theory.

The article says, "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".

This seems fallacious, as truth is undefinable and thus the bijection between $\omega$ and the "set" of definable reals need not exist in the model (and indeed, there not even need be a set of definable reals).

Question 1: Is this claim on wikipedia as fallacious as I think it is?

Question 2: If so, is there a model of ZFC in which every real number is definable?

An obvious candidate for such a model is $L_\alpha$, where $\alpha$ is the least ordinal such that $L_\alpha \models $ ZFC. Note that by Gödel's condensation lemma, this model is countable.

Edit: I edited the wikipedia to delete the false sentences. Hopefully it stays deleted.

vhspdfg
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  • "as truth is undefinable." What? – Thomas Andrews Mar 04 '16 at 01:49
  • @ThomasAndrews From the context, it's clear that OP talks about a truth predicate / modeling relation inside a given model... – Stefan Mesken Mar 04 '16 at 01:53
  • Stefan, I thought I searched, but apparently not hard enough. Asaf there answers my question. Thanks. – vhspdfg Mar 04 '16 at 01:59
  • Just real numbers? Isn't there a model in which every set is definable? – bof Mar 04 '16 at 01:59
  • Ok so, according to the Q&A linked by Stephen, wikipedia is indeed not only fallacious, but wrong. And there is a model in which every real is definable.

    Someone really needs to fix that wikipedia article.

     – vhspdfg Mar 04 '16 at 02:10
  • @vhspdfg The beginning of Joel Hamkins' blogpost indicates that many people get this wrong - maybe we should begin to call it the "HLR paradox" in honor of Hamkins, Linetsky and Reitz? Anyways, you deserved your $+1$ for spotting this mistake! – Stefan Mesken Mar 04 '16 at 02:25
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    #Stephen I approve; it is indeed an ostensible paradox that although there are only countably many parameter-free definitions, it is possible for every real number (indeed, every set) to be definable. I think it's as noteworthy as the Skolem paradox. – vhspdfg Mar 04 '16 at 02:35
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    So truth may be undefinable, but you are enough of an expert to delete stuff on Wikipedia that you don't understand, because it must be "fallacious" if you say so? – hardmath Mar 04 '16 at 03:19
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    @Stefan: There is also http://karagila.org/2015/name-that-number/ – Asaf Karagila Mar 04 '16 at 08:27
  • @hardmath if you read the comments, you'd see that the source by the well-respected logician Hamkins corroborated what I said. – vhspdfg Mar 04 '16 at 17:44

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