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From: "Jesse F. Hughes" <je...@phiwumbda.org>
Newsgroups: sci.logic,sci.math
Subject: Re: Matheology S 116
Date: Fri, 12 Oct 2012 15:47:49 -0400
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FredJeffries <fredjeffr...@gmail.com> writes:

> On Oct 12, 7:31=A0am, WM <mueck...@rz.fh-augsburg.de> wrote:
>> Matheology =A7 116
>>
>> How can the assumption of the infinite be justified?
>> =A0 =A0Could not just this seemingly so fruitful hypothsesis of the
>> infinite have introduced straigth contradictions into mathematics,
>> thereby destroying the basic nature of this science that is so proud
>> upon its consistency?
>> [On the hypothesis of the infinite, Ernst Zermelo's Warsaw notes W4
>> (p. 171), reported in H.-D. Ebbinghaus, V. Peckhaus: "Ernst Zermelo,
>> An Approach to His Life and Work", Springer (2007) p. 292.]
>> For German original texts see: Das Kalenderblatt 100322http://www.hs-a=
ugsburg.de/~mueckenh/KB/KB%20201-400.pdf
>
> The entirety of W4:
>
> W4. How can the assumption of the infinite be justified?
>
> Arithmetic--like basically any other mathematical discipline--consists,=

> generally
> speaking, of propositions comprising infinite multitudes of particular
> assertions. Its axioms are concerned with domains of things that can
> be
> mapped one-to-one (or one-to-many) onto proper partial domains and
> hence
> are really "actually infinite". To assume such infinite domains (which
> do not
> always have to be "sets" in the set-theoretic sense of the word!) is
> therefore
> to make the basic assumption underlying all mathematics, which, as
> such,
> certainly requires some justification as well (in accordance with the
> principle
> of sufficient reason!). A "proof", in the proper sense of the word, is
> of course
> not possible where an axiomatic assumption is concerned, as is the
> case here.
> But equally impossible is the realization by means of an explicitly
> specified
> and ready-made model since the infinite as such defies, after all, all
> attempts
> at making it manifest. Such an assumption is capable of justification
> solely
> by its success, by the fact that it (and it alone!) has made possible
> the creation
> and development of all extant arithmetic, which is, in essence, simply
> a science of the infinite. But is the existence of this science
> justified? Could
> it not be the case that this seemingly so fruitful hypothesis of the
> infinite
> carries contradictions into mathematics, thereby utterly destroying
> the real
> essence of this science, which prides itself so much on the
> correctness of its
> inferences? As paradoxical as it may seem in the case of a science
> that has
> achieved the greatest successes in the two thousand years of its
> development,
> we cannot immediately dismiss without closer consideration the
> possibility
> that our mathematics rests on contradictions. But, now, can the
> consistency
> of ("infinitistic") arithmetic itself be proved by logico-mathematical
> means?
> To provide a "consistency proof" by means of realization in a model
> capable
> of exhibition or by embedding the "infinitistic" arithmetic in a
> "finitistic"
> one is, according to what was said above, out of the question. What
> therefore
> remains is the possibility of showing that contradictions detectable
> by
> formal-logical means are not derivable from the arithmetical axioms.
> Such
> a demonstration, if it were possible, would have to rest on a thorough
> and
> complete formalization of all the logic relevant to mathematics. Any
> "incompleteness"
> of the underlying "proof theory" such as a neglected possible
> inference would jeopardize the entire proof. But, now, since such
> "completeness"
> can obviously never be guaranteed, it is,is, in my opinion, not
> possible to
> furnish a formal proof of the consistency.
>
> From Ernst Zermelo - Collected Works/Gesammelte Werke, pp 383-385
>

Thanks, Fred!

-- =

Jesse F. Hughes
"If anything is true in general about Usenet, it's that people can go
on and on about just about anything."  -- James Harris speaks the
truth.