Mathematicians Bridge Finite-Infinite Divide

Let Y = some ridiculously huge finite number. The notion that ∞ = Y + 1 has to be one of the most insidious defects in reasoning. Even to acknowledge the defect by dismissing it is almost like saying things could have turned out that way when they _never_ could.

Finite means limited or bounded or quantifiable. Infinite means unlimited or unbounded or unquantifiable. They are two wholly separate classes of things. Another way to say this is discrete versus continuous (in nature).

Imagine I said that I could bridge the coloured/non-coloured divide. What could I possibly mean by that. It would mean that I have found a way to talk about both coloured things and non-coloured things that applies to both class of things.

Put another way. Imagine I separate coloured things into green ones and not green ones. Both subclasses belong to the class of coloured things. What class then do the subclasses of finite things and non-finite things belong to? Negation constructs the distinction, negation _is_ the bridge in a way and all the could be said of both subclasses of things is that they are both things. (Which, to be clear, is not saying very much at all, is it?) If those things are numbers then all we are saying is that both things are numbers, if both things are procedures then all we are saying is that both things are procedures. And so on.

If I'm thinking about this properly then the paper is trying to more precisely define the term "countably". That's as most as it can do. To say otherwise is not just vastly overstating what the paper is about but outright misleading.

I think it is quite cool that people are still hunting down the various different incarnations of infinity and are able to prove deep results like this.

On the other hand, I don't think that these questions are as essential as they are made out to be. How do we know that everything finite is on unshakable foundations? Doesn't even thinking about what that statement means involve reasoning about infinities? Let's just accept that nothing ever is really unshakable and let's use the tools that will get the job at hand done.

Unfortunately, while SOME such formulas worked (the ones you learned in calculus - stuff like the chain rule), other formulas generated using the same kind of reasoning ("just imagine that d becomes infinitely small") come up with answers that are nonsensical or just wrong. How can we know when it's OK to just say "let things get infinitely small" and when it gives bogus answers?

The solution, which is taught in high-school calculus, was the "epsilon-delta" formulation. Instead of saying "let d become infinitely small" (a statement that may just be nonsense), say "I will prove that for ANY small epsilon (greater than 0) I can find a delta > 0 such that for values of x closer than delta, the error will be less than epsilon". That statement doesn't require an infinity to exist anywhere -- it is just a statement about particular finite numbers. And we can build calculus on such principles.

This isn't an EXACT analogy to your question about real numbers, but it uses the same type of reasoning, and I'm hoping the analogy is in terms of mathematical reasoning you are already familiar with.