[FOM] Predicativism and natural numbers

Charles Parsons 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

Some references

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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