Is there any philosophical significance to the arithmetization of infinity?

Question

There are two arithmetics of infinity, ordinal & cardinal. I'm going to focus on the cardinal arithmetic as it requires less structure, that is they need less (i.e., ordinals require the idea of ordering whereas cardinals do not).

Cardinal arithmatic recognises that counting can be characterised in two ways, by using numbers, for example 3 means 1+1+1, but there is another way which simply relies on the idea of matching. If I want to see if two bags of beans contains the same number, rather than counting them, I can match them (i.e., one bean from the first matches one bean from the other, and so on), this does not rely on the idea of number and so is more fundamental.

Counting requires number and so can only measure the finite, but matching does without and so go further, and in fact it can measure the infinite, and more it can actually construct an arithmetic. This was first recognised by Cantor one of the inventors of Set Theory.

Does this idea of infinite have any philosophical significance?

I know that Badiou uses Set Theory in his attempt to ground continental philosophy away from post-modern excess (not that I understand how he does it), and he does use the idea of the infinite in this way in his book Being & Event.

Are there any other examples?

Are there arguments against seeing any real significance in this idea of the infinite. For example, Hegel talks of the absolute, and my impression is that in this context, the set theory infinite has no purchase on it.

Answer 1

The definition of "ordinal" and "cardinal" you give is not optimal, because the concept of ordinal is richer than the concept of cardinal, and more subtle. The concept of ordinal is best phrased as a linearly ordered discrete collection, you should think of points on an interval. The points are discrete, and in order, so they always go up, but they can have accumulation points. To make them an ordinal, the accumulation point is always in the set, and the accumulation is always from the left, so that the accumulation point is always on the right, and not on the left.

Then there is a point furthest left, and this is given the name "0", then there is a next point to the right, and this is 1, and then you go on, and you find 2,3,4. When you hit the first accumulation point, you have passed all the integers, and this is the first infinite ordinal $\omega$. After $\omega$, the next point is $\omega+1$, then $\omega+2$, and so on until the next accumulation point, which is $\omega+\omega$. If the accumulation points accumulate themselves, you get to $2\omega$, and so on through the cantor normal forms (powers of $\omega$).

The ordinals defined this way are the countable ordinals, but this is not really a restriction.

The cardinals are the matching definition of number, and this is easier to explain than the ordinals, but it is also less well defined. The properties of uncountable sets are always vague, since an axiomatic system can only pin-down the properties of countable collections (and even then, only asymptotically, so that describing the integers requires ever growing axiom systems)

You can't ground philosophy in set theory, because set theory itself is not well founded really. The foundation is logic and computation, and the set theory is a sophisticated set of axioms on top of the logic. The grounding of philosophy in logic is the basic idea of the logical positivists.

Answer 2

First what you call matching requires counting too. Matching of countable sets is in general done by bijecting one of them with the set of natural numbers. And even if the natural numbers are not involved, the bijection works with ordered sets, i.e., sequences. But also for uncountable sets order and therefore counting in an extended sense is required. A "mush-set" of unordered numbers could not be counted. That's why the axiom of choice is so important. It allows to well-order every set. The necessity of order my become clear by the following quotes of two forbears of set theory:

"These are Cantor's first transfinite numbers, the numbers of the second number class as Cantor calls them. We get to them simply by counting beyond the normal countable infinite, i.e., in a very natural and uniquely defined consistent continuation of the normal counting in the finite." [D. Hilbert: "Über das Unendliche", Math. Annalen 95 (1925) p. 169]

All well-ordered sets can be compared. They have the same number if they, by preserving their order, can be uniquely mapped or counted onto each other.

"Therefore all sets are 'countable' in an extended sense, in particular all 'continua'." [G. Cantor, letter to R. Dedekind (3 Aug 1899)]

Finally, to answer your last question, it has turned out that set theory is in contradiction with mathematics and therefore is insignificant. It cannot serve as as a basis for any scientific or realistic endeavour. One of several proofs collected in https://www.hs-augsburg.de/~mueckenh/Transfinity/Transfinity/pdf is this: Scrooge Mc Duck earns daily 10 enumerated dollars and spends daily 1 $. If he always spends that one with the lowest number, he will get bankrupt according to set theory, because the set theoretical limit of not spent dollars is empty. According to mathematics he will get increasingly wealthy.