Is there any philosophical significance to the arithmetization of infinity?

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.

• why should counting only be able to measure the finite if you have infinite numbers? Badiou is in touch with Lacanian psychoanalysis (and Lacan may sound really obscure most time)... I haven't read much Badiou. Maybe you could point out some especific example, if it is the case? Jun 21 '12 at 17:13
• @Tames: True, but historically finite numbers came first (and by a long way), and thats what I'm talking about when I say number. The finite realm is very different in character from the infinite. And to be honest, only certain set theorists deal with them seriously, they haven't had (except one I can think of in Category Theory) any wider significance in mathematics, charming as they are. I find Lacan obscure. What kind of example are you after? I don't know if I can help you, I've tried reading Badiou, and only the introduction makes sense to me :). But one keeps trying... Jun 21 '12 at 17:46
• by examples I meant, of Badiou uses. I guess you are after similar uses? Yes.. Lacan is the Heraclitus of our age! I've only read a few things by Badiou, here and there, not any major work like the one you cited. Jun 21 '12 at 18:06
• possibly Granger's 'Philosophy of style' could be of interest to you. Jun 22 '12 at 15:42
• @Beaudrap: and theres more: the surreal numbers & the infinitesimals used in synthetic smooth geometry, but here they don't have a measure of the infinitely large. Jun 26 '12 at 16:10

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.

• I didn't define either the Ordinals or the Cardinals deliberately, and I only sketched the idea behind the the Cardinals. It didn't seem neccessary to go into details in what is a philosophical question. No axiomatic system is well-founded. I don't see how logic & computation can escape that either. Topos theory is an alternative foundation to the usual ZFC set theory for mathematics, and it combines in one logic, geometry and set theory; they're not one built on another, they're three different perspectives. Jun 21 '12 at 17:01
• So here, logic is another perspective on set theory (its called the internal language of the topos). I don't see how one can ground philosophy in Set Theory either, its attempting to put the larger into the smaller, I'm sure Badiou is aware of this sense of ground, so what he's doing is different from this, what, I don't know; and I make this clear in the question. Jun 21 '12 at 17:09
• @MoziburUllah maybe what Badiou is doing is similar to Lacan (and Badiou is lacanian to some extent): as you try to ground philosophy in set theory, you find out that there's something left out; this point to the presence of subjectivity problems, so he would be indicating the limits of it. That is why Lacan is fond of Godel's incompleteness theorem, because it indicate that there's always a hole in formalization, and subjectivity is drained by it. Jun 21 '12 at 17:20
• @Tames: That makes some sense, I didn't know that Lacan was aware of Godel. It interests me how formalisation creates seriousness. Godel was quite specific in how he adopted the liars paradox for his theorem, and one could say that this paradox already hints at the fallibility of the formalisation project. Jun 21 '12 at 17:58
• @MoziburUllah yes... Aristotelic logic, set theory, Russell's paradox, Godel's theorem... you may find all of it in the 16th seminar. Lacan's attempt is to formalize psychoanalysis to a point where you see where the "hole" is (it would be what he calls 'object a', this is what 'real' means to Lacan - an impossibility, a gap..) Jun 21 '12 at 18:09

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.

• That "proof" does not show any contradiction---there's a tremendous difference between a bijection and a limit. The limit on Scrooge's money does not exist, but his income is in bijection with his spending. I've looked at the pdf you linked but it brought to mind a different duck: Daffy. Feb 14 '18 at 19:45
• @Daffy: The limit in this example does exist according to set theory. You may look at set-theoretical limits of sequences of sets on p. 55 (if you don't trust the PDF try Wikipedia). Without this limit the completeness of the bijection would not be provable. Neither would the diagonal argument. By the way the story is also mentioned by Fraenkel. Feb 15 '18 at 14:02