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It's well-known that there exists some real numbers that cannot be defined in a string of finite length (Berry's paradox). However, why can the set of all real numbers be defined?

My gut feelings are

$1$. the definition has some nonconstructive descriptions, like the supremum and infimum principle;

$2$. or we are just talking about computable numbers?

I'm not familiar with real analysis & mathematical logic so there might be something I overlooked.

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    $\begingroup$ That's not what Berry's paradox is about. Berry's paradox is about being very careful with what you choose as a valid way of defining things, not about how there are more real numbers than possible definitions. $\endgroup$
    – Arthur
    Dec 3 '19 at 8:37
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    $\begingroup$ I'm not saying your question is wrong. I'm saying Berry's paradox is the wrong thing to reference here, as it has nothing to do with how many things you can define. $\endgroup$
    – Arthur
    Dec 3 '19 at 8:57
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    $\begingroup$ Which of "the set of all real numbers" have an infinite length? $\endgroup$
    – Asaf Karagila
    Dec 3 '19 at 8:58
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    $\begingroup$ @AsafKaragila The way I read the question, I think the issue here is how you can even say something like "all real numbers" if it is provably impossible to define each of them. $\endgroup$
    – Arthur
    Dec 3 '19 at 9:02
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    $\begingroup$ @ekd123: Short answer is that this is a very subtle argument that is usually overlooked. And unfortunately it's hard to wrap your head around this without knowing some logic and set theory. $\endgroup$
    – Asaf Karagila
    Dec 3 '19 at 9:10

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