3 ms·
Another argument that's not completely non-constructive. The real numbers have to be constructed. Typically, a number is represented by a Cauchy sequence or a
by solinent 7y ago
Another argument that's not completely non-constructive.
The real numbers have to be constructed. Typically, a number is represented by a Cauchy sequence or a Dedekind cut.
To determine if a real number is representable symbolically, we simply need a finite sequence of symbols which stands for this Cauchy sequence, lets say.
Theroem: The real numbers and definable numbers are the same set.
Assume a real number exists but is not definable. This means at the very least we have a mathematical statement saying there exists a number such that some logical predicate is valid (we may not even have a construction in ZFC), which can also be constructed using a Cauchy sequence. This mathematical statement is embedded in ZFC, and since we are humans it must be finite. In fact, you could come up with a binary representation for such a statement using methods from Godel, by mapping each symbol to some binary representation. Therefore, this number can be represented as a sequence of zeros or ones, a contradiction.
QED
- OscarCunningham 7y agoI don't understand that argument. But in any case, Cantor's argument is very constructive. It literally gives you the decimal expansion of the new number not in your set.
- solinent 7y agoI didn't do it that much justice because I was discovering it independently, my arguments can easily be made rigorous, but you'd need a background in pure mathematics to understand it. However, there's a section on Wiki: https://en.wikipedia.org/wiki/Definable_real_number#Definability_in_models_of_ZFC https://en.wikipedia.org/wiki/Definable_real_number#Definabi... They start with a stronger definition of a definable number, so they find that they do exist. I think given the argument above there must be a hole in my own argument, I'd have to go beyond ZFC.