[FOM] Predicativism and natural numbers

Arnon Avron aa at tau.ac.il
Sun Jan 15 17:03:36 EST 2006

On Wed, Jan 11, 2006 at 08:27:28AM +0100, Giovanni Lagnese wrote:
> Arnon Avron wrote:
> > The basic intuition behind the construction
> > of the concept of natural number is "and so on"
> > (or the idea of "..."). The natural numbers
> > consist of 0, 0', 0'', and so on (or 0, 0', 0'', ... ).
> By this definition, how can you infer that omega+omega is not a natural 
> number?

I have to repeat. This is *not* a definition, but an intuitive explnation.
So you cannot infer things from it in the usual sense of "infer". Either
you understand what most people mean when they say "and so on" (and
what theyt mean does not include omega+omega), or you dont. If you
dont, my explnation would not help you (but I dont really suspect
you dont understand. I am sure you do).

> Yes, because TC *is* the concept of natural number.
> I merely said that it's an impredicative concept. You must take it as 
> primitive.

Of course I take it as a primitive - exactly as I take negation
and the universal quantifier as primitive. They all cannot be defined
using better understood concepts. But the fact that it (or somethging
close or equivalent) does not make it "impredicative", at least not 
in the sense I mean when I classified myself as a predicativist.

> But one can take as primitive also the concept of powerset. So there must be 
> a philosophical reason for taking as primitive the concept of natural 
> numbers but not the concept of powerset. And this reason can not be the 
> predicativism.

My main reason for taking TC as primitive is that for me TC is a crystal 
clear concept, whose  meaning is absolute - while that of the powerset
is not. Again: taking concepts is not something to be done arbitrarily.

There are two other differences between taking TC as primitive and
taking powerset as such, with are more objective. The first is that
it is clearly understood by ALL people, as indicated by the fact that 
the notion of  ancestor exists in all natural languages since ancient
times (which one can hardly say about powerset...). The second is
that along with the basic logical notions, the notion of TC (or
something equivalent) is essential for understanding the notions
of well-formed formula, formal proof etc - which are inductively 
defined in all textbook. So without a prior understanding of 
TC (or something equivalent) you cannot define or prove anything-
you cannot even understand what a definition is. 
TC (or something equivalent)  is a part of what is absoltely necessary to start
develop logic and reasoning. Hence it should be taken as primitive.

  Well - if this would not convince you, then nothing will. Moreover:
It seems to me that if this discussion is continued we shall simpply
repeat our arguments (perhaps using other words). So unless something
really new is said, I finish here my postings on this subject.

Arnon Avron
> GL 
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