FOM: miniaturization/finitization

[Vladimir Sazonov]

Jan Mycielski wrote:

>         In JSL 51 (1986), 59 - 62, I gave a general method for turning the
> statement of the consistency of any theory T into a statement S(T) of
> finite combinatorics. Moreover S(T) reflects in a transparent way the
> original (model theoretic) intuition supporting T (as opposed to the
> purely syntactic meaning of consistency).

Stephen G Simpson replied:

> I propose that we try to get to the bottom of this issue right here on
> the FOM list, by simply comparing the corresponding statements side by
> side.
>
> First, let's look at finitary statements that imply Con(PA).

> The Paris-Harrington statement is:
...

> Now, what is your statement S(PA) exactly?  After you spell out S(PA)
> in complete detail here on the FOM list, we can judge whether it is as
> mathematically natural and appealing as P-H.

As I remember, this work of Jan Mycielski (JSL 51 (1986), 59 - 62),
simply demonstrates that any consistent first-order theory T is
in a sense "isomorphic" to a theory FIN(T) which is "finitary"
consistent: any finite number of axioms in FIN(T) has a finite
model. Thus, infinity which may be implicit in T (such as ZFC,
etc.) is "explained" via (possibly large) finite. I believe that
this is a very important result having a crucial philosophical
and methodological value.  It sheds a new light on what can be
done by mathematical formalization of finite and infinite and
explains infinity from a more proper, I would say, antiplatonistic
point of view. This FIN(T) probably gives not a *better*
understanding of (any) T, as Mycielski himself correctly writes,
but this gives definitely a *finitary* understanding.

As to comparison of this with Paris-Harrington, Friedman, etc.,
I think, in the first case, the main point is on *uniformity* of
translation T |-> FIN(T) and,  most important, on corresponding
finitary *meaning* of a *theory* T vs., in the second case,
consideration of a concrete (consistency or any other) *sentence*
as fixed, where we are rather interested only in finitization of
this alone (say, set theoretic) sentence.  In the second case a
naturalness of the resulting finitary sentence mainly from an
external point of view (such as combinatorial theory existing
independently of a given T), instead of a semantical view, plays
an essential role. These are somewhat orthogonal finitization
approaches having different goals.


Vladimir Sazonov