[FOM] Predicativism and natural numbers
parsons2 at fas.harvard.edu
Mon Jan 2 15:18:45 EST 2006
At 4:57 AM +0100 12/27/05, Giovanni Lagnese wrote:
>How do most predicativists justify their acceptance of an impredicative
>definition of the set of natural numbers?
>Is there a philosophical (not practical) argument that justifies this
I think it's a little misleading to talk about
traditional predicativism as accepting an
impredicative _definition_ of the natural
numbers. What one finds in, say, Poincaré and
Weyl are reasons for _assuming_ the natural
numbers. So that predicativity in most later
work, in particular the analyses of Feferman and
Schuette, is predicativity _given_ the natural
numers. I think they are quite clear about that.
Several people, myself included, have argued that
the concept of natural number is impredicative,
apart from the question of a definition.
I'm not sure why you say the predicativist
conception of _set_ is impredicative, if one
allows the assumption of the natural numbers as
the tradition does.
One can see whether one can develop arithmetic in
a strictly predicative way, without making the
kind of assumption I am attributing to the
tradition. The paper by Burgess and Hazen noted
below gives an idea of how far one can go in such
a program, which was inaugurated by Edward
Nelson. (Burgess and Hazen mention earlier
relevant results of Skolem.)
Charles Parsons, "The impredicativity of
induction," in Michael Detlefsen (ed.), _Proof,
Logic, and Formalization_ (London: Routledge,
1992), pp. 139-161. (Expanded version of a paper
published in 1983.)
Edward Nelson, _Predicative Arithmetic_. Princeton University Press, 1986.
John P. Burgess and Allen Hazen, "Predicative
logic and formal arithmetic," _Notre Dame Journal
of Formal Logic_ 39 (1998), 1-17.
Charles Parsons, "Strict predicativity," _11th
International Congress of Logic, Methodology, and
Philosophy of Science, Cracow 1999_. Volume of
Abstracts, p. 278. (Section 9.)
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