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"Real numbers that cannot be defined with language" is not a well-defined set though. Just like "integers that cannot be described in less than 100 words" is no

"Real numbers that cannot be defined with language" is not a well-defined set though. Just like "integers that cannot be described in less than 100 words" is not well-defined (I could reach a contradiction by pointing to the "smallest integer that cannot be described in less than 100 words").

Smaug123 9y ago
A sentence which fails to specify a real uniquely… fails to specify a real uniquely. Borel's talking about numbers which can be defined uniquely: that is, picked out, identified. Your Berry-paradox description doesn't identify an integer.
- zodiac 9y ago
  But if we assume that we can say a sentence either specifies a real number uniquely or does not, the Berry paradox number is uniquely specified.
FabHK 9y ago
True, but the argument that one set is larger still works.
- upquark 9y ago
  Wait, you're agreeing with me that the set in question is ill-defined, but still somehow comparable to other sets? I am not sure I follow. My statement is that you cannot meaningfully talk about "all the numbers that cannot be described with language". If you could, I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with language? I'm sure some sort of paradox similar to the one with integers can be constructed here... In my view, every real number is well-defined and there's nothing controversial about the set of real numbers. If the infinite aspect of it causes some researchers to call it a "mathematical fantasy", so be it, so is literally every other mathematical model we use in our lives.
  - Smaug123 9y ago
    Your example is faulty. > I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with language? The "indescribable numbers" are dense in [0,1], and so (if the set exists) the inf of the set of indescribable numbers which are between 0 and 1 is 0. Perfectly describable.
    - upquark 9y ago
      My example is incomplete, not faulty. I left it as a question (does the inf belong to the set?). If the answer is yes, we reached a contradiction. If the answer is no, we have to continue further zooming in to this interval (or some other construction along those lines). See, I claim that this set is ill-defined, so I can't know its properties like whether or not it's dense, open, closed, Borel-measurable, etc. etc. You have to tell me what its properties are, and I will come up with a concrete proof that the set in question is ill-defined. EDIT: After I RTFA'd, this is actually the paradox in section 2.3 of the linked article
  - FabHK 9y ago
    There are more people on earth than, say, kings. That's true, even though I can't enumerate all non-kings, and even if the set of non-kings is somehow ill-defined.
    - upquark 9y ago
      Still not sure I follow. Set of non-kings is not ill-defined, not in the same way as set of numbers non-describable by any formal system is.
      - mcherm 9y ago
        > Set of non-kings is not ill-defined, not in the same way as set of numbers non-describable by any formal system is. Um... roughly the same. Is Robert Mugabe a king? Since we didn't give a clear and precise definition of "king" you can't really say.

Smaug123 9y ago

A sentence which fails to specify a real uniquely… fails to specify a real uniquely. Borel's talking about numbers which can be defined uniquely: that is, picked out, identified. Your Berry-paradox description doesn't identify an integer.

zodiac 9y ago
But if we assume that we can say a sentence either specifies a real number uniquely or does not, the Berry paradox number is uniquely specified.

FabHK 9y ago

True, but the argument that one set is larger still works.

upquark 9y ago
Wait, you're agreeing with me that the set in question is ill-defined, but still somehow comparable to other sets? I am not sure I follow. My statement is that you cannot meaningfully talk about "all the numbers that cannot be described with language". If you could, I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with language? I'm sure some sort of paradox similar to the one with integers can be constructed here... In my view, every real number is well-defined and there's nothing controversial about the set of real numbers. If the infinite aspect of it causes some researchers to call it a "mathematical fantasy", so be it, so is literally every other mathematical model we use in our lives.
- Smaug123 9y ago
  Your example is faulty. > I'd ask you if this set intersected with [0, 1] has a lower bound / 'inf' that's contained in it, and if so, did I just describe that lower bound with language? The "indescribable numbers" are dense in [0,1], and so (if the set exists) the inf of the set of indescribable numbers which are between 0 and 1 is 0. Perfectly describable.
  - upquark 9y ago
    My example is incomplete, not faulty. I left it as a question (does the inf belong to the set?). If the answer is yes, we reached a contradiction. If the answer is no, we have to continue further zooming in to this interval (or some other construction along those lines). See, I claim that this set is ill-defined, so I can't know its properties like whether or not it's dense, open, closed, Borel-measurable, etc. etc. You have to tell me what its properties are, and I will come up with a concrete proof that the set in question is ill-defined. EDIT: After I RTFA'd, this is actually the paradox in section 2.3 of the linked article
- FabHK 9y ago
  There are more people on earth than, say, kings. That's true, even though I can't enumerate all non-kings, and even if the set of non-kings is somehow ill-defined.
  - upquark 9y ago
    Still not sure I follow. Set of non-kings is not ill-defined, not in the same way as set of numbers non-describable by any formal system is.
    - mcherm 9y ago
      > Set of non-kings is not ill-defined, not in the same way as set of numbers non-describable by any formal system is. Um... roughly the same. Is Robert Mugabe a king? Since we didn't give a clear and precise definition of "king" you can't really say.