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Matheology § 157
Posted:
Nov 21, 2012 1:55 AM


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



