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I don't think that the expansion being sometimes infinite is a minor difficulty. In fact it sounds pretty major, similiar to the halting problem. Unless you can

I don't think that the expansion being sometimes infinite is a minor difficulty. In fact it sounds pretty major, similiar to the halting problem. Unless you can prove if it's infinite or not a priori. I'm not knowledgeable about this topic but the quote suggests you can't prove that.

dllthomas 11y ago
"In fact it sounds pretty major, similiar to the halting problem." I think it's identical to the halting problem. The issue is that, for irrational numbers, there is never a point at which you can stop and say "okay, we know it is irrational now". In fact, if we had an algorithm that could always tell us whether a sequence converged to a rational number, we could solve the Halting Problem: For any Turing Machine you wish to check for halting-ness, produce a sequence S by successively appending a full encoding (any will do) of the Turing Machine state at every step to the decimal expansion of the previous value in S. Apply your algorithm to S. If S is irrational, then we know it never halts. If S is rational, then we know it halts or repeats after a finite number of digits - so we can simply run it until it does one or the other (this may take arbitrarily long, and an arbitrarily large amount of storage, but finitely in both cases).
- ScottBurson 11y ago
  Everyone has gotten distracted by the infinite expansion and has missed the point of my question. Let me try again, with no infinite expansion. The constructivist position is to refuse to accept the statement "every real number is either rational or irrational" because they're not in possession of proofs, for each possible real-valued expression, that its value is rational or, alternatively, a proof that its value is irrational. And yet, according to the quoted paragraph, they accept the statement "every integer is either even or odd", quote, "because we have an effective means of determining which alternative holds for any integer". My argument is that they should reject this statement on the same grounds: I named an integer whose value is not known, and therefore it is not known whether it is even or odd. There is not, in fact, an effective procedure for determining the evenness or oddness of the value of an arbitrary integer-valued expression. It's only after one actually knows the value that answering the question becomes easy.
  - dllthomas 11y ago
    IANAM, but I think the difference comes down to the fact that we know the integer you are producing is one we possess a proof for. We do not know that for reals.
  - plonh 11y ago
    Constructivists reject the notion of an integer whose value is not known (more precisely, whose value cannot be computed in finite time) so there is no room to even ask of "it" is odd. Infinite vs finite is an important distinction. A fact that takes infinite time to prove is not constructive and is likewise not computable.
  - pavpanchekha 11y ago
    Constructivists don't accept that you've named an integer. They would say: an integer is any member of the smallest constructive set that contains 0 and one more than any member of that set. Since the set is a constructive set, if you have anything you claim to be an integer, you must be able to either prove that it is 0 or one more than an integer. The number you named isn't zero, so presumably it must be more than some integer ("one less than the number you named"). But if I then ask you whether that number is one more than an integer, and then the same about that one, and so on, at some point you will be unable to answer me. Since you can't provide the evidence that qualifies something as an integer, there's no reason for me to think it is an integer.
    - ScottBurson 11y ago
      I see. The difference, then, between that situation and the situation where I provide a real number named as the sum of an infinite series is that in the latter case I may be able to provide also the modulus of convergence, thus proving that the number is a constructive real number [0]. This is different from a situation where I have a number defined non-constructively and no series is known that converges on the number. [0] https://en.wikipedia.org/wiki/Constructivism_%28mathematics%29 https://en.wikipedia.org/wiki/Constructivism_%28mathematics%...

dllthomas 11y ago

"In fact it sounds pretty major, similiar to the halting problem." I think it's identical to the halting problem. The issue is that, for irrational numbers, there is never a point at which you can stop and say "okay, we know it is irrational now". In fact, if we had an algorithm that could always tell us whether a sequence converged to a rational number, we could solve the Halting Problem: For any Turing Machine you wish to check for halting-ness, produce a sequence S by successively appending a full encoding (any will do) of the Turing Machine state at every step to the decimal expansion of the previous value in S. Apply your algorithm to S. If S is irrational, then we know it never halts. If S is rational, then we know it halts or repeats after a finite number of digits - so we can simply run it until it does one or the other (this may take arbitrarily long, and an arbitrarily large amount of storage, but finitely in both cases).

ScottBurson 11y ago
Everyone has gotten distracted by the infinite expansion and has missed the point of my question. Let me try again, with no infinite expansion. The constructivist position is to refuse to accept the statement "every real number is either rational or irrational" because they're not in possession of proofs, for each possible real-valued expression, that its value is rational or, alternatively, a proof that its value is irrational. And yet, according to the quoted paragraph, they accept the statement "every integer is either even or odd", quote, "because we have an effective means of determining which alternative holds for any integer". My argument is that they should reject this statement on the same grounds: I named an integer whose value is not known, and therefore it is not known whether it is even or odd. There is not, in fact, an effective procedure for determining the evenness or oddness of the value of an arbitrary integer-valued expression. It's only after one actually knows the value that answering the question becomes easy.
- dllthomas 11y ago
  IANAM, but I think the difference comes down to the fact that we know the integer you are producing is one we possess a proof for. We do not know that for reals.
- plonh 11y ago
  Constructivists reject the notion of an integer whose value is not known (more precisely, whose value cannot be computed in finite time) so there is no room to even ask of "it" is odd. Infinite vs finite is an important distinction. A fact that takes infinite time to prove is not constructive and is likewise not computable.
- pavpanchekha 11y ago
  Constructivists don't accept that you've named an integer. They would say: an integer is any member of the smallest constructive set that contains 0 and one more than any member of that set. Since the set is a constructive set, if you have anything you claim to be an integer, you must be able to either prove that it is 0 or one more than an integer. The number you named isn't zero, so presumably it must be more than some integer ("one less than the number you named"). But if I then ask you whether that number is one more than an integer, and then the same about that one, and so on, at some point you will be unable to answer me. Since you can't provide the evidence that qualifies something as an integer, there's no reason for me to think it is an integer.
  - ScottBurson 11y ago
    I see. The difference, then, between that situation and the situation where I provide a real number named as the sum of an infinite series is that in the latter case I may be able to provide also the modulus of convergence, thus proving that the number is a constructive real number [0]. This is different from a situation where I have a number defined non-constructively and no series is known that converges on the number. [0] https://en.wikipedia.org/wiki/Constructivism_%28mathematics%29 https://en.wikipedia.org/wiki/Constructivism_%28mathematics%...