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Re: Matheology § 157
Posted:
Nov 21, 2012 3:30 PM


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Nelson has withdrawn that paper from his webpage.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



