3 ms·

It doesn't contain the floating point representation of that number, which would be infinitely long, but it does contain the UTF8-encoded constructive proof tha

It doesn't contain the floating point representation of that number, which would be infinitely long, but it does contain the UTF8-encoded constructive proof that uniquely identifies that number.

Anderkent 7y ago
Surely not? If there are more numbers that can be represented in any amount of bytes (as shown by the diagonal argument), then you cannot represent a constructive proof for each of them in any amount of bytes.
- Filligree 7y ago
  The argument still applies, yes. Nevertheless the string still contains all finite substrings, which includes all English sentences describing such numbers. The implication is that human minds wouldn't be able to represent it either, at least not with language.
- dmurray 7y ago
  Kind of. A constructive proof by definition means something like you can explain exactly what the number is in a finite number of bytes. The standard diagonalization argument is a constructive proof, identifying a specific number. Normally a constructive proof is preferable. But with a bit of hand waving you can make it into a non-constructive proof that there are "unknowable" numbers that cannot be described in any finite amount of bytes.
  - afiori 7y ago
    In practice constructible for reals mean that you can approximate them with arbitrarily small know precision e.g. a sequence of retional number numbers q_n each no more that 1/n apart from the actual real number. The point is that such a sequence needs to be constructive in the usual sense, so there are still only a countable number of them.
jnbiche 7y ago
Well, it could contain the floating-point representation, since floating-point is a finite representation. But it couldn't contain the real number representation.
paulddraper 7y ago
Incorrect. Most real numbers have infinite-length constructive proofs. Uncountable is uncountable is uncountable. If any correspondence/encoding existed between naturals and reals, the reals would be countable. (But they are not, so it is fool's errand to search for such an encoding.)

It doesn't contain the floating point representation of that number, which would be infinitely long, but it does contain the UTF8-encoded constructive proof that uniquely identifies that number.

Anderkent 7y ago

Surely not? If there are more numbers that can be represented in any amount of bytes (as shown by the diagonal argument), then you cannot represent a constructive proof for each of them in any amount of bytes.

Filligree 7y ago
The argument still applies, yes. Nevertheless the string still contains all finite substrings, which includes all English sentences describing such numbers. The implication is that human minds wouldn't be able to represent it either, at least not with language.
dmurray 7y ago
Kind of. A constructive proof by definition means something like you can explain exactly what the number is in a finite number of bytes. The standard diagonalization argument is a constructive proof, identifying a specific number. Normally a constructive proof is preferable. But with a bit of hand waving you can make it into a non-constructive proof that there are "unknowable" numbers that cannot be described in any finite amount of bytes.
- afiori 7y ago
  In practice constructible for reals mean that you can approximate them with arbitrarily small know precision e.g. a sequence of retional number numbers q_n each no more that 1/n apart from the actual real number. The point is that such a sequence needs to be constructive in the usual sense, so there are still only a countable number of them.

jnbiche 7y ago

Well, it could contain the floating-point representation, since floating-point is a finite representation. But it couldn't contain the real number representation.

paulddraper 7y ago

Incorrect. Most real numbers have infinite-length constructive proofs. Uncountable is uncountable is uncountable. If any correspondence/encoding existed between naturals and reals, the reals would be countable. (But they are not, so it is fool's errand to search for such an encoding.)