🚀 go-pugleaf

In article <COLUMBUS.92Nov16105941@strident.think.com>, columbus@strident.think.com (Michael Weiss) writes:
> In an earlier post, Paul Budnik writes:
>    This well known result is called the Lowenheim and Skolem theorem. The
>    idea of the proof is that a formal system is a computer program for enumerating
>    theorems. The names of all real numbers created by such a program are
>    obviously countable. Of course the mapping of these real numbers to names is
>    not definable within the the formal system and thus these reals cannot be
>    shown to be countable within the system.
> 
> This is not the correct statement of the Loewenheim-Skolem theorem.  Here is
> one formulation (the so-called "downward" Loewenheim-Skolem theorem):

I did not say that the above was a statement of the theorem, but that it was
the idea of the proof. I suspect that the `downward' L-S theorem is a 
generalization of what I was referring to.

>     Any model of a countable first-order theory has a countable elementary
>     submodel.  (By "countable theory", we mean one whose language is
>     countable.  "Elementary" means, roughly, that any assertion true of the
>     model is true also of the submodel.)
> 
> Budnik's statement about there being only a countable infinity of definable
> real numbers is correct, if by "definable" we mean "definable by a formula
> in a countable theory".  This fact is essentially trivial, since there are
> only a countable number of formulas in such a theory.

But all formal system have only a countable number of formulas. You and
many mathematicians may believe that there exists a Platonic heaven in
which all reals reside and that some formal systems include sentences
referring to each of these reals. That is a philosophical conjecture that
is open to question.

> ... 
> In short: there may be good reasons for radically changing the foundations
> of physics (and there may not-- pace, defenders of the status quo(ntum)!),
> but the Loewenheim-Skolem theorem is not one of them.

I do not think anyone is going to accept or reject a physical theory based
on L-S. I do think it provides a valuable perspective in thinking about these
issues. A formal system is a computer program for enumerating theorems.
Whether it is anything beyond that and whether the infinite totalities
it supposedly refers to exists is an open question. There is no physical
evidence for such objects. It is hard to understand what such physical
evidence might be.

Follow ups are directed to `sci.logic'.

Paul Budnik