# 回复 :[FOM] Nik Weaver's conceptualism and the correctness of the Schu"tte-Feferman analysis

Wed Apr 12 11:11:08 EDT 2006

```Nik Weaver wrote:
>
> SECOND OBJECTION:
> Let us grant that the predicativist can somehow make the disputed
> inference.  Then for each a he has some way to make the deduction
>
> (*)   from  I(a)  and  Prov_{S_a}(A(n)),   infer   A(n)
>
> for any formula A, where I(a) formalizes the assertion that a is an
> ordinal notation (supporting transfinite induction for sets).
>
> Shouldn't he then accept the assertion
>
> (**)   (forall a)(forall n)[I(a) and Prov_{S_a}(A_n)  -->  A(n)]
>
> for any formula A?
>
>

A few comments, informal and perhaps just confusions. But they are to
help me understand the matter.

The "predicativist" in Feferman's sense (an early sense of his) is not
the "rational actor" in Weaver's sense, but rahter an idealized actor
whose mind amounts just to the "autonomous progression". He tries to
start from natural numbers and go as far as he can in a straight way
(that is, new objects are obtainable only through proofs using
previously obtained objects). He just keeps going by using something as
an instance of (*) at each step taken, but as he acually knows no limit
on the way forward, he is not in a position to konw something about ALL
instances of (*), not to mention the general (**), even if he is
potentially capable of going through all instances of (*) to some
extent. Only from outside, from a wider perspective including a general
understanding of some kind of ordinals, can we (but not he) have the
measure that shows he cannot go beyond a certain limit.

An analogy. Achilles, let's suppose his idea of distance is just what
he can cover with his feet, tries to know how long the longest distance
would possibly be. He starts at point 0 and goes forward with each step
respectively being 1/2, 1/4. 1/8, ... long. Being strong enought he can
always manage to take step n+1 fllowing every step n, and therefore he
will keep going without stop. He doesn't know how many steps would be
eventually taken and therefore is not in a position to make assertions
involving quantifications over ALL the steps. In his perspective, there
are no things like the limt 1 which we, enjoying a wider perspective,
will put to his journey. He doesn't know the limit because he has never
been at or beyond that point.

Does this make sense?

Best regards,

Xing Taotao
Philosophy Department
Beijing University

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