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Topic: Matheology § 157
Replies: 2   Last Post: Nov 21, 2012 3:30 PM

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Ralf Bader

Posts: 437
Registered: 7/4/05
Re: Matheology § 157
Posted: Nov 21, 2012 3:30 PM
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WM wrote:

> Matheology § 157
>
>
> Finitism is usually regarded as the most conservative standpoint for
> the foundations of mathematics. Induction is justified by appeal to
> the finitary credo: for every number x
> there exists a numeral d such that x is d. It is necessary to make
> this precise. We cannot
> express it as a formula of arithmetic because "there exists" in "there
> exists a numeral d"
> is a metamathematical existence assertion, not an arithmetical formula
> beginning with ?.
> The finitary credo can be formulated precisely using the concept of
> the standard model
> of arithmetic: for every element xi of |N there exists a numeral d
> such that it can be proved
> that d is equal to the name of xi, but this brings us into set theory.
> The finitary credo has
> an infinitary foundation.
> The use of induction goes far beyond the application to numerals.
> It is used to create
> new kinds of numbers (exponential, superexponential, and so forth) in
> the belief that they
> already exist in a completed infinity. If there were a completed
> infinity |N consisting of all
> numbers, then the axioms of {{PA}} would be valid assertions about
> numbers and {{PA}} would be consistent.
> [E. Nelson: "Outline, Against finitism"]
> http://www.math.princeton.edu/~nelson/papers/outline.pdf
>
> Regards, WM


Nelson has withdrawn that paper from his webpage. The reasons can be seen
here:
http://golem.ph.utexas.edu/category/2011/09/the_inconsistency_of_arithmeti.html
On that page there also appears a comment by Mückenheim himself, of usual
idiocy. So Mückenheim knows the story.

--
"Die Natur hat schon häufig natürliche Zahlen zerlegt, zum Beispiel...die
acht Beine einer Spinne in die vier Himmelsrichtungen." Prof. Dr. W.
Mückenheim, Mathematikkoryphäe der "Hochschule Augsburg", am 01.10.09 in
de.sci.mathematik



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