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    $\begingroup$ @Anixx: No, this is not what Joel was saying. He did not say that it is consistent to "postulate in ZFC that undefinable numbers do not exist". What he was saying was that ZFC cannot even express the notion "is definable in ZFC". And no, this has absolutely nothing to do with constructivism (also please note that even in constructivism uncountable means "not countable", whereas you stated that it means "no practical enumeration" whatever that might mean). $\endgroup$ Commented Oct 29, 2010 at 14:50
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    $\begingroup$ Joel made a very fine answer, please study it carefully. Joel states that there are models of ZFC such that every element of the model is definable. This does not mean that inside the model the statement "every element is definable" is valid. The statement is valid externally, as a meta-statement about the model. Internally, inside the model, we cannot even express the statement. $\endgroup$ Commented Oct 29, 2010 at 15:13
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    $\begingroup$ This is off-topic, but: it makes no sense to claim that "constructivist continuum is countable in ZFC sense". What might be the case is that there is a model of constructive mathematics in ZFC such that the continuum is interpreted by a countable set. Indeed, we can find such a model, but we can also find a model in which this is not the case. Moreover, any model of ZFC is a model of constructive set theory. You see, constructive mathematics is more general than classical mathematics, and so in particular anything that is constructively valid is also classically valid. $\endgroup$ Commented Oct 29, 2010 at 15:15
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    $\begingroup$ A minor technical comment on the first bullet point in Joel's answer: To use the minimal transitive model, one needs to assume that ZFC has well-founded models, not just that it's consistent. The main claim there, that there is a pointwise definable model of ZFC, is nevertheless correct on the basis of mere consistency, essentially by the second bullet point plus the consistency of V=HOD relative to ZFC. $\endgroup$ Commented Oct 29, 2010 at 16:44
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    $\begingroup$ Following the comment of Andreas got me thinking: as a topos theorist (which I am not) I would look at the syntactic model of ZFC (the "Lindenabaum algebra") in order to get to both the minimal model and the one in which every set is definable. But I suppose set theorists don't like that kind of model too much because they prefer transitive models that are "really made of sets". Is that so? Historically, where does this tendency come from? $\endgroup$ Commented Oct 29, 2010 at 21:02