Mathematicians who are finitists.

[polymath257]

I find the whole thing to be semantics. If we're dealing with reality, then of course we can have a potentially unbounded and growing value, which is always finite but "potentially infinite". However, in a purely abstract mathematical setting, it’s easy to create infinite sets using particular rules. One can certainly analyse whether the results are internally consistent or not according to those rules, but the set of elements is infinite regardless.

It can be proved that the size of the set is larger than any number, and so it is infinite. The set doesn’t have to "grow" to produce all the elements, they are simply defined into abstract "existence". I suppose you could get into semantics about how exactly the elements are defined, and if that’s done inductively, then you could say the set "never finishes". Well, of course, but this is imposing an artificial limit on a process which doesn’t involve time passing in the first place. You have defined every element in the set, and there are an infinite number of them.

It is fascinating about relative infinities, as has been mentioned with the real and natural (or rational) numbers. I remember being told at uni about how if you compare the density of all the real and rational numbers in any particular interval (say between 0 and 1), the real numbers actually take up all the space. They are infinitely more numerous (thus a higher cardinality). There is no way of placing the real numbers in a list so they can be counted. You will always be missing elements between any list entrants.

We can go further, but some care is required.

One problem in set theory is that the collection of *all* sets cannot be a set. It is what is known as a proper class. In a sense, it is 'too big' to be a set.

So, not only is the set of real numbers a larger infinity than the set of natural numbers (which is, in a sense, the smallest possible infinity), there is a whole hierarchy of sizes of infinite sets.

But not only is the number of possible cardinalities infinite, even uncountably so, it turns out that the collection of all cardinalities is a proper class: it is larger than any set.

There is an analogy between proper classes in standard set theory and infinite sets in finitistic mathematics. In both cases, the collections are 'too big' to fit into the axioms chosen. And,, in both cases, it is possible to extend the axiom system to allow discussion of these large collections. So, ordinary set theory is an extension of finitistic mathematics where infinite sets are legitimate objects of study. And, there are axiom systems (Godel-Bernays, for example) that allow discussion of proper classes as objects of the system.

The problem is that this only forms a new hierarchy: once we add classes, we want to take collections of classes and then collections of those objects, etc.

This ultimately lead to Russell's theory of types. Most mathematicians do not go this far, being content with sets or, in some cases, classes.

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I'm curious- what sort of insights are gained by using finitist logic?

Well, there are a lot of issues surrounding proof theory and Godel's incompleteness results (and issues such as Turing machines) that reside naturally in a finitistic situation. It is of some interest to see just how few assumptions are required to get the Godel results, for example. And, it is known that systems much weaker than the natural numbers are enough (Godel proves his results about the natural numbers). For example, the 'Robinson naturals' is a weaker system of axioms that turns out to be enough to get most of Godel's version of model theory to go through.

For most mathematicians, though, the finitistic realm is far too constraining and the possibility to talk about infinite sets underlies a lot of calculus and thereby most of modern mathematics.