[FOM] From theorems of infinity to axioms of infinity

[Giorgio VENTURI]

Dear Professor Detlefsen,

the answer is to be found in Hilbert attitude toward the 
foundation of a theory and his claim that consistency entails 
existence. Zermelo axiomatisation of set theory is in the 
context of Hilbert's school, if we can call it so . Zermelo 
says explicitly that he was trying to give a foundation in 
order to avoid paradoxes, but the way in which he thought of a 
foundation is not in terms of looking for the the foundamental 
principle that are able to characterize the essence of the 
subject matter of a theory, as Frege would have done. Zermelo 
tried to "deepen the foundation", in Hilbert's terms, in order 
to find which were the propositions that made possibile to 
derive all the relevant theorems of set theory (as Hilbert did 
with Euclidean geoemtry). It was not a foundation that aimed 
to explain the concept of set, but a foundation build up in 
order to secure the use of set-theoretical methods. Zermelo's 
work was trying to aswer the question: what do we need in 
order to develop the relevant part of set theory that we need 
in our mathematical work? Then the existence of a infinite set 
is something that is necessary for the application of 
set-theoretical methods. And the axiom of infinity grants his 
importance.
Then, from this foundational point of view, the axiom of 
infinity solely is not sufficient in order to prove the real 
existence of an infiite set, because still a consistency proof 
is needed. Indeed Zermelo says explicitly that he regrets not 
to have been able to prove the consistency of his axiom 
system.
What is interesting in this story is to understand what 
Zermelo thought a consistency proof for set theory was. Indeed 
in the Thirties he gave one in an old fashioned way: he just 
exibited a model. Is there a change in his view of a 
consistency proof? is it related to his abandon of Hilbert's 
circle?
Then after Zermelo's axiomatization of set theory it become 
common to pose the existence of an infinite set as an axiom. 
The reasons for this, again, are maybe to be found, as for 
Zermelo, in the role that Hilbert's ideas played in the later 
development of logic.

Best,

Giorgio Venturi



On Sat, 9 Mar 2013 13:21:00 -0500
  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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