Re: Impredicativity in Rosennean parlance

[Calvin Ostrum <***>]

On 8/10/05, Torkel Franzen <***> wrote:

>   Reading these excerpts suggests to me that in reading Rosen, it's
> better not to take too seriously his apparent references to formal
> logic, but concentrate instead on his own thinking.

Perhaps, but it seems that some of Rosen's claims could in
fact be stated more clearly and precisely in terms of computability 
and related notions as they are understood at large.   However,
as far as I can see, this has not yet been done.  
 
In fact, I think one could do this totally within classical recursion theory,
by giving a formal definition of what it means for one set in the recursive
hierarchy to model another one, and then show the existence of sets
that have no "largest" model under some rigorous but intuitively
motivated ordering.   It almost sounds like the sort of thing people
studying the structure of the Turing Degrees might end up doing,
since "modelling" sounds like some kind of reducibility relation.

> >In this sense, syntactic or algorithmic systems, or formalizable
> >systems, are extremely weak in entailment: almost everything about
> >them must be posited, and very little is actually entailed within such
> >a system. That is, almost every why question about what is in the
> >system cannot be answered from within the system, and thus their
> >answers take us outside the system. In this sense, the objective
> >appearance of formalizable systems, their apparent rigor, is an
> >illusion.
> 
>   Rosen suggests no justification for this description, which is on
> the face of it arbitrary. Indeed it is unclear what is claimed
> ("very little", "almost every why question"). 

Part of the problem is that there seems to be a confusing
pushing together of causality and inference, with both of them 
being referred to with the unclear term "entailment".   The
kind of question being talked about here is not really a
question of inference, but of causality.   If you asked why
a formal system existed in the causal sense, you would
have to refer to the people who thought it up and produced
various physical tokens relating to it.   Of course a formal
system cannot "entail" itself in this sense.   But this is
the sense in which parts of an organism "entail" other parts.
An organism continuously constructs itself over time.
No formal system can do this.  But I don't
see why any of this means that the various parts of
an organism cannot be specified as mechanisms, as for
example Von Neumann did, and as Rosen seems to
think impossible (and for some reason apparently related
to Goedel's Theorem).  

> I would guess that
> to anybody familiar with the formal systems commonly used or
> studied in logic, what is striking is rather how *much*
> is provable within these systems, including statements about the
> systems themselves.

Indeed.  And  when it comes to "real" mathematics, it 
should be pointed out that as a statement of number 
theory, the Goedel sentence for a given system
is generally of absolutely no mathematical interest, and it
took a long time for anyone to find mathematically interesting
statements of number theory that were not provable in PA
(and I believe even those are somewhat contrived variants
of the usual more interesting results).  So Rosen's remarks, 
such as the one quoted below that "syntactic rules 
captured only an infinitesimal part of "real" mathematics"
thus seem to be incorrect.   PA "captures" a great deal of 
number theory.  ZFC  "captures" almost all useful 
mathematics.