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

[邢滔滔]

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