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Comment by btilly | Hacker News Reader

According to classical mathematics, only a countable number of finite definitions of numbers exist. And an uncountable number of real numbers exist. Therefore almost all real numbers that exist do not correspond to any possible finite definition of a number.

Tell me. In what sense does an abstract concept exist that has no possible definition or unique description?

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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 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!

The 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_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?

So you're arguing that a set can exist even if its elements don't? Because we can surely give a unique description of the set of real numbers (the unique complete ordered field up to iso) and this description forces it to be uncountable

No, I am not.

Also the definition that you gave is complete nonsense to a Constructivist. And the reasoning that forces it to be uncountable in classical set theory requires reasoning that also makes implicit assumptions you are probably not aware of.

You said "acccording to classical mathematics", which is not constructive so I assumed you were working in ZFC or some similar set theory.

Also if we want to be precise I'd like to hear your definition of "finite definition", the only definition of "definability" I'm familiar with is relative to a model of a theory and if ZFC has models at all then it has countable pointwise definable models, but I guess that won't satisfy your idea of "finite definition"

When I said "according to classical mathematics" I did mean according to the normal orthodoxy, which does indeed mean ZFC. By "finite definition" I mean a finite sequence of statements in first or second order logic which can be proven to define a unique real number. (Even if, as with Chaitin's Constant, we can't figure out what that number is.)

The fact that this may not be a definition from my point of view is irrelevant - I was making a statement about what classical mathematics implies, and from the point of view of classical mathematics this is a perfectly reasonable definition.

Since the number of such statements is countable, and only some of them define real numbers, the set of such real numbers is countable. Being countable it is a set of measure 0, and therefore the "almost all" that I stated follows immediately.

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