Re: Some thoughts on formalization

[Calvin Ostrum <***>]

On 8/21/05, Tim Gwinn <***> wrote:

Wow, I give up.  Obviously I am not as fast a learner as Torkel 
is.  Or Glen, who before leaving noted the similarity of Rosen 
disciples to Randoids.  

Just to take one example:

> > In fact, on the very page you cite, Rosen quotes Kleene
> > as giving an adequate characterization of the
> > notion of formalizability he is concerned with:
> >
> > "[Formalization [Rosen's interpolation]] will not be
> > finished until all the properties of undefined terms
> > which matter for the deduction of theorems have
> > been expressed by axioms".
>  
> TG: Again, you misstate the facts. Rosen does NOT quote "Kleene as giving an
> adequate characterization of the notion of formalizability he is concerned
> with", nor is that the full quoted text. Instead Rosen says, "The best
> statement I have seen regarding the formalistic program is that given by
> Kleene; it does no harm to quote it again:".[LI p.6]

You are truly grasping at straws.  Rosen gives Kleene's
quote, with it referring to "formalization", right after he uses his
own sense of the term, and right after talking about Goedel's
constructible hierarchy. 

He then goes on immediately, on the page you cited, to say

"The status of all these formalizations [thus even including
Goedel's constructible hierarchy] is informative.  They turn out
to be infinitely feeble compared with the original mathematical
systems they attempted to objective.  Indeed, these attempts
to secure  mathematics from paradox by invoking constructibility,
[again, referring clearly to Goedel's constructable hierarchy]
or formalizability,  end up by losing most of it"

To make clear he is also including something that goes
well beyond Hilbert and totally decidable systems, he then
once again says 

"In other words, a "constructible universe" is at best only 
an infinitesimal fragment of "mathematical reality"".

That can only be a reference, in this context, to the phrase
"constructible universe" (and its subsequent abuse) used 
a few paragraphs previously in reference to Goedel's 
constructible hierarchy.