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