[FOM] From theorems of infinity to axioms of infinity

[Joe Shipman]

I think you are making this more mysterious than it is.

It is an inevitable consequence of the process of axiomatic formalization that informally understood concepts for which informal and intuitive "arguments" were offered will be replaced by precise statements in a formal language which will in turn be deduced from axioms chosen parsimoniously.

There was a general understanding by 1900 that various types of infinite collections needed to be better understood and put on a firmer basis, and it would not have been surprising to most mathematicians that the infinite collections might require ontological postulates in the context of a formalized axiomatic system. The example of non-Euclidean geometry had made it clear that finding an adequate and parsimonious axiomatic foundation for a system was an empirical process and that it was often highly non-obvious which axioms followed from which other axioms.

Zermelo observed that one could start with a simple infinite set equivalent to the integers and get many higher levels of infinity using his axiomatization, and Frankel noted that with the Replacement axiom the infinite sets that could be shown to exist sufficed for  everything in mathematical practice. That a special axiom of Infinity was needed to get started was simply a consequence of the increasing rigor of the formalizations of mathematical reasoning--whatever philosophical principles has been used to "prove" that infinite sets existed by Bolzano and Dedekind were themselves problematic to formalize, and Zermelo chose to postulate an infinite set in the way he did because it fit in with his system better than trying to formalize the principles used by Bolzano and Dedekind.

There are other ways to formalize the development of mathematics, for example by using second-order logic instead of first-order logic, or a system like Russell's, in which the existence of an infinite collection follows without needing a specific "axiom of infinity". But since the hereditarily finite sets  satisfy all the ZF axioms except Infinity, it makes no sense to criticize Zermelo for having an Axiom of  Infinity *in his particular system* since it is clearly necessary there.

Why is this bewildering?

I should also point out that there are many alternate forms Zermelo could have used; any sentence in first order logic which has only infinite models would do (so you could assume there was a dense linear ordering, or a non-commutative division ring, etc.; since Zermelo was working in a language with only the membership relation as a non-logical symbol his version is hard to improve upon).

-- JS

Sent from my iPhone

On Mar 9, 2013, at 1:21 PM, Michael Detlefsen <mdetlef1 at nd.edu> wrote:

I'd like to understand what were the forces underlying the transition from treating existence claims for infinite collections
as theorems (i.e. propositions that require proof) to propositions that can be admitted as axioms.

In the latter half of the nineteenth century, both Bolzano (Paradoxes of the Infinite (1851), sections 13, 14) and Dedekind (Theorem 66
of "Was sind …" (1888)) offered proofs of the existence of infinite collections (using similar arguments).

By Zermelo's 1908 paper, it had become an axiom (Axiom VII). Zermelo remarked that he found Dedekind's proof unsatisfying because
it appealed to a "set of everything thinkable", and, in his view, such a collection could not properly form a set.

Instead of jettisoning the assertion of an infinite collection, though, this led Zermelo to make it an axiom. Seen one way, this is essentially
to have reasoned along the following lines: 

Problem: Dedekind's "proof" of the assertion of the existence of an infinite collection is flawed, perhaps
fatally so. 

Solution: Make the proposition purportedly proved by Dedekind's flawed proof an axiom!

I'm guessing I'm not the only one who finds this a little funny, and a little bewildering.

More than this, though, I'm wondering what rational forces there might have been that would have made such a move serious and plausible enough
to sustain the weight that an "axiom" in a foundation of set theory (and, eventually, of mathematics) would seemingly have to bear.

Zermelo seemed to have much the same confidence that Bolzano had in the bare existence of infinite collections. By this I mean that,
just as Bolzano, Zermelo seems to have believed (or to have assumed) that asserting the mere existence of infinite collections should 
not itself engender paradox.

But this confidence did not lead Bolzano to make the existence of infinite collections an axiom.

Still less would it have tempted Dedekind to do so. He, remember, is the guy who both so famously wrote that nothing in mathematics is more dangerous
than to accept existence without sufficient proof of it and a guy who then undertook to prove the existence of infinite systems.

Soooo …

(Focal Question): How should we understand the transition from infinity a la Bolzano and Dedekind to infinity a la Zermelo?

Some may be tempted to bring Cantor in here … specifically the Cantor who so emphasized the distinction between immanent and transient reality and argued that immanent reality is the type
of reality that figures in pure mathematics. But there's little indication of a groundswell of acceptance of Cantor's distinction … not even in Göttingen. So it doesn't seem
to have been determinative.

Neither does it seem plausible to say that it was simply (or even primarily) the more general shift away from the classical view of axioms (as evident or self-evident truths) to a more "hypotheticist"
conception of them. Zermelo's axioms, after all, were supposed to form a "foundation" for set theory … that is (at least) a basis for it (them) that is secure from threat of
further paradox. (Neither is there much indication that Zermelo viewed the justification of his axioms in essentially the same "regressive" or "inductive" way that
Russell viewed the principles of PM.)

But what then is the answer? 

Is it perhaps that there is no answer … that is to say, is it that the only "answer" is "expediency" … that Zermelo needed infinity
to give him the type of theory of sets he was looking for, and he saw no way to provide for the existence of infinity save that of making it
an axiom?

This isn't a very satisfactory "answer" to me. 

Am I underestimating its virtues? 

Are there other, more satisfying answers?

Best from a dishearteningly wintry South Bend,

Mic Detlefsen
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