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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.

Andrés E. Caicedo
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ekd123
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    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. – Arthur Dec 03 '19 at 08:37
  • @Arthur I don't think that will change the conclusion here. As long as the length of a definition (in any language $L$) is finite, then the set of definable numbers is countable, and there will always be some real numbers that cannot be defined, because $\mathbb{R}$ is uncountable. – ekd123 Dec 03 '19 at 08:53
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    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. – Arthur Dec 03 '19 at 08:57
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    Which of "the set of all real numbers" have an infinite length? – Asaf Karagila Dec 03 '19 at 08:58
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    @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. – Arthur Dec 03 '19 at 09:02
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    @Arthur: It is??? – Asaf Karagila Dec 03 '19 at 09:03
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    @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. – Asaf Karagila Dec 03 '19 at 09:10
  • @AsafKaragila At least I think it is. If you can find a list of definitions that defines each real number, one by one, let me know. – Arthur Dec 03 '19 at 09:22
  • @Arthur: http://karagila.org/2015/name-that-number/ – Asaf Karagila Dec 03 '19 at 09:25
  • @AsafKaragila Ok, fair enough. You can't apply the same counting to all possible definitions of real numbers as you can to the set of real numbers because the two concepts necessarily live in different domains of your theory. Therefore you can't in any fair way compare them. Is that a somewhat sensible summary of your blog post? – Arthur Dec 03 '19 at 09:41
  • @Arthur: Yes. And to the "let me know when you are done enumerating the definitions", let me start with $0,1,2,3,4,\dots$ just to get these out of the way. If you want me to skip, I'll start skipping to $\frac1n,1+\frac1n$, etc. At some point we can skip those. Let me know when you want me to skip to a number without a name. Or, to quote Hugh Woodin's nice retort: if the real numbers cannot be well-ordered, then there is a real number which is not ordinal definable, can you give me an example of such number? – Asaf Karagila Dec 03 '19 at 09:44

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