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This question already has an answer here:

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.

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marked as duplicate by Stefan Mesken, Community Mar 4 '16 at 1:59

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  • $\begingroup$ "as truth is undefinable." What? $\endgroup$ – Thomas Andrews Mar 4 '16 at 1:49
  • $\begingroup$ @ThomasAndrews From the context, it's clear that OP talks about a truth predicate / modeling relation inside a given model... $\endgroup$ – Stefan Mesken Mar 4 '16 at 1:53
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    $\begingroup$ #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. $\endgroup$ – vhspdfg Mar 4 '16 at 2:35
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    $\begingroup$ 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? $\endgroup$ – hardmath Mar 4 '16 at 3:19
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    $\begingroup$ @Stefan: There is also karagila.org/2015/name-that-number $\endgroup$ – Asaf Karagila Mar 4 '16 at 8:27