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The amount of paper in the universe is finite, not countably infinite, which means almost all natural numbers can't be written down either. But despite its phi
by panic 8y ago
The amount of paper in the universe is finite, not countably infinite, which means almost all natural numbers can't be written down either. But despite its philosophical dubiousness, the set of natural numbers is still useful. If we can prove things about all natural numbers, it doesn't matter how much paper we have; the things we prove will still be true for any individual natural number we can write.
It's the same idea for the reals; we don't actually care about the vast majority of the real numbers, but since it's hard to know ahead of time which ones we will care about, we might as well prove things about all of them!
- deleted 8y ago[deleted]
- btilly 8y agoThe thing is that the useful things that we can prove about natural numbers are ones that can be proven about practical ones. We can, at least in principle, follow the construction and wind up with whatever exists. Now compare with the kinds of results that classical mathematics gives us. The Robertson-Seymour theorem (see https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theorem https://en.wikipedia.org/wiki/Robertson%E2%80%93Seymour_theo... for the theorem) says that certain classes of graphs are characterized by a finite forbidden set. This means that membership can be tested by a polynomial time algorithm. However the construction provides no way to actually find that finite set. It also provides no way to find how many members it has. It not only provides no way to prove that you actually have all of them for a given class of graphs, but there are classes of graphs which it is impossible for us to prove that we actually have a complete list. Not only in practice, but in principle. So the theorem asserts the existence of a finite set. But in what meaningful way does it exist, or is it finite?