Re: Name the thesis: "Formal sentences capture informal ones"

tchow_at_lsa.umich.edu
Date: 01/30/05 

Date: 30 Jan 2005 21:30:16 GMT

In article <vcbmzurjyep.fsf@beta19.sm.ltu.se>,
Torkel Franzen <torkel@sm.luth.se> wrote:
>> (+) Con("PA") is an intension-preserving formalization of "PA is
>> consistent"
> (+) is not on the face of it an instance of (*), since it states not
>only that an itension-preserving formalization of "PA is consistent"
>is possible, but that a particular arithmetical formula is such a
>formalization.

Yes, that's what I meant by abusing the word "schema." What I probably want
is not either (+) or (*), but some schema (which I'm having difficulty
formulating) of which (+) is an instance.

>What is required of an intension-preserving
>formalization? In formalizing Con(PA) or the fundamental theorem of
>arithmetic in PA, we need to represent finite sequences of numbers
>as numbers. This can be done in many ways, but are they
>intension-preserving?

These are excellent questions, and part of the purpose of formulating the
thesis is so that these questions can be raised explicitly and debated.

I would say that *something like* (+) is implicitly accepted by most
people, and forms the basis for concluding that Goedel's 2nd theorem
effectively kills Hilbert's program as originally conceived. Detlefsen
has raised the possibility that this received wisdom is ill-founded,
and as I recall, his argument hinges on considering deviant consistency
predicates and, in effect, questioning whether standard consistency
predicates adequately capture the true meaning of consistency. Most
people don't buy this argument, and I think that it's because they
accept (+).

I think, then, that there is no doubt that (+), or something like it, is
firmly held by many people, despite the vagueness of the term "intension-
preserving" and the difficulty of formulating it more precisely. So as
long as you agree that the thesis I'm trying to articulate has many
instances that are widely accepted, I regard the questions you raise as
indicating the usefulness of articulating the thesis explicitly, not as
reasons to doubt the possibility of such articulation. 

-- 
Tim Chow       tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth.  ---Galileo, Dialogues Concerning Two New Sciences