[FOM] From theorems of infinity to axioms of infinity

[Arik Hinkis]

Perhaps you will have some interest in looking in Chapter 8 of the following
http://link.springer.com/book/10.1007/978-3-0348-0224-6/page/1.

On Sat, Mar 9, 2013 at 8:21 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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