[FOM] From theorems of infinity to axioms of infinity (Michael Detlefsen)

[josef at us.es]

Dear Mic Detlefsen, 

there has been a lot of historical and
philosophical work dealing with the point you raise. The whole project
of reducing 19th century math to set theory made the existence of
infinite sets a necessary assumption -- this was not an open question
for almost all mathematicians. 

Between Dedekind and Zermelo, the key
issue is whether the principle is a mathematical axiom, or it can be
reduced to more basic principles: Dedekind believes it is a logical
truth and tries to establish this, while Zermelo (aware of the
difficulties linked with the paradoxes) is content in his 1908 paper to
present it a an axiom of set theory. This of course is more modest, but
his move involved also the realization that all of Dedekind's work goes
through as soon as one grants the existence of infinite sets. 

One can
actually show that the existence of infinite sets, and furthermore the
power set axiom, are intimately linked with the real numbers conceived
e.g. as infinite decimals. I have presented an argument to this effect
in a recent paper, 'On arbitrary sets and ZFC' (Bulletin of Symbolic
Logic), which I hope you will find instructive in this respect. Also, if
you'll excuse self-advertisement, there is a lot of information on this
topic in my book "Labyrinth of Thought". (Incidentally, the axioms in
Hilbert's paper on the real numbers imply that the "system" of real
numbers is infinite, so his approach is not so different from
Zermelo's.) 

Greetings to you and all from a rainy Seville, 

Jose
Ferreiros 

El 11/03/2013 04:46, Michael Detlefsen escribió: 

>
Message: 1
> Date: Sat, 9 Mar 2013 13:21:00 -0500
> From: Michael
Detlefsen <mdetlef1 at nd.edu>
> To: FOM <fom at cs.nyu.edu>
> Subject: [FOM]
>From theorems of infinity to axioms of infinity
> Message-ID:
<FF7A7E32-C085-4ED3-A276-44A5E0EFD7C9 at nd.edu>
> Content-Type:
text/plain; charset=windows-1252
> 
> 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. 
> 
> Best from a dishearteningly
wintry South Bend,
> 
> Mic Detlefsen
 
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