> Zeilberger identifies as an ultrafinitist.

So, um, does that mean there's a "biggest number"? One you can't +1?

IIRC that is one of Zeilberger's opinions. For myself, I don't think "biggest number" makes sense per se in mathematics, but it is a fact that no matter what resources we muster we're left with some numbers equaling things other than their successor when incremented.

My sense is that infinitary axioms like the existence of N, successor, induction, etc., basically allow mathematicians to write shorter and more straightforward proofs. They are likely to be sound until the cows come home, and moreover they represent what mathematics is essentially all about - finding some shortcut through the calculations we need to make, understanding what part of some structure or procedure is irrelevant.

Within certain parameters, that's right.

And that probably makes as much sense (if not more) as the alternative for all I know. There is a way to think of mathematics as corresponding to computation in a way where anything that cannot be computed cannot be proven. If you cannot "construct" or "compute" infinity, then you cannot say that it exists.

I think what I am describing corresponds to the "constructivist" view of mathematics, but tbh I don't know enough about this to say whether the framework allows for infinities.

<sotto voce>These people are crazy

What do you mean?

At a very pop sci-fi level, think of writing down the biggest rational number that you can. As many 9s. Eventually, you'll run out of space in the universe to store 9s.

In a computer, this text is stored in binary format. That's a base2 number. Arbitrary concepts can be encoded into a form that can then be represented and worked with by mathematical operations.

If you want to get really cafe philosophical about it, the universe can be broken down into its smallest elements, quantified, and represented by numerical values. And thus the universe is a number. Of course, there's all sort of "gotchas" and "whatifs" and this is a wildly over-simplified version.

Yeah! Programmers are quite familiar with this universe. In many programming languages, you can't represent 2^64 with a built-in integral datatype. While it's certainly possible to implement data structures and algorithms to support arbitrary-size integer arithmetic, most programmers don't! They just pretend that numbers larger than 2^64-1 do not exist! And, that's a fairly safe assumption for many real-world applications.

But even with that arbitrary-size integer arithmetic, your mathematical universe is still bounded by your computational environment's resources.

Discussion with an ultrafinitist:

-Do you believe in 1?

-Yes.

-Do you believe in 2?

-Yes.

-Do you believe in 4?

-[slight pause] Yes.

-Do you believe in 8?

-[slightly longer pause] Yes.

Eventually you realize each question takes him twice as long to answer as the last.

That's funny. There is in fact a quasi-finitist number scheme called the counting numbers, closed under +1 and addition/multiplication generally but not exponentiation, so unbounded but not too "problematic" as they have limited ability to represent propositions (or machine states, etc) arithmetically.

I suppose a log-finitist takes O(1/n) to answer for 2*n? :-)

Yes. It's like writing code that's aware of MAX_INT. The code doesn't need to know the value of MAX_INT. It just has to handle special cases near that number.

> philosophically extravagant mathematics of transfinite set theory

And poorly-founded, I would add. Cantor's diagonal proof is, quite literally, question begging. Sure, you can construct a number not in the infinite set of natural numbers via that diagonalization method, but only if you're at infinity, or if you make very liberal use of "..." -- otherwise, you cannot.

And Skolem was quick to discredit Cantor's extravagances, on grounds that tend to support finitism generally.

I don't think I agree, but this is difficult to communicate in words.

Cantor's diagonalization argument doesn't rely on the contents of an infinite list of numbers, or on numbers themselves. You don't need to know anything about what "..." represents and you don't need to be "at" anywhere. There is no need to compute anything with concrete numbers.

The only thing the argument relies upon is whether the fundamental structure of numbers allows for defining a bijection. The proof/computation is an abstract relation, not a concrete one with actual number objects.

> The only thing the argument relies upon is whether the fundamental structure of numbers allows for defining a bijection.

That there is no bijection must be the conclusion, not an assumption, otherwise the argument would simply be question begging. If you can't define a new diagonal number in the first place, you don't have an argument why you can't have a bijection.

With finitism it doesn't seem like you can define a diagonal number, because then Cantor's infinite table is merely potentially infinite in width and height. For the diagonal to be finished, the table has to be at least as wide as it is tall. But if you list numbers by their lexicographic order, the list grows much faster in height than in width. For base 10, for n columns (digits), you get 10^n rows (numbers). Then you can't ever finish a diagonal that isn't on the list. The number of digits doesn't grow fast enough to cover the entire list. You don't get the diagonal in the limit.

Now if you assume infinitism, you can simply declare that the height and width of the list is the same (or even that the width is larger), and therefore the diagonal can be constructed. But that's not a convincing assumption for a finitist (or ultrafinitist).