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> It's poor because it's undecidable. We would ideally want to work with the real numbers, but no computational model can represent all real numbers (by a trivi
by ogogmad 6y ago
> It's poor because it's undecidable. We would ideally want to work with the real numbers, but no computational model can represent all real numbers (by a trivial cardinality argument).
This is a fallacy. The standard model for Computable Real Analysis, Type Two Effectivity, supports computation over all real numbers. See: https://en.wikipedia.org/wiki/Computable_analysis#Real_numbers https://en.wikipedia.org/wiki/Computable_analysis#Real_numbe...
Essentially, in TTE, a real number is an arbitrary stream of digits that can come from any source. Therefore any real number can be computed on.
Additionally, the ability to decide equality of floating point numbers is an illusion. Rounding errors can make numbers that ought to equal each other not equal each other.
- dllthomas 6y agoYou can compute on any real number you're given, but IIUC that's not what "computability" means. By the cardinality argument above (there are more infinite streams of digits than finite streams of digits, in an important sense) there will exist streams of digits that you cannot produce by any program (regardless of how you encode it).
- ogogmad 6y agoYou can produce an arbitrary infinite stream using a random number generator, for instance.
- karatinversion 6y agoThe probability of generating any given infinite stream is 0, though.
- qsort 6y agoWhere is the fallacy? I don't doubt type-2 effectivity is used in the way you mention, but that wasn't what I was talking about. See my other reply.