🚀 go-pugleaf

In an earlier post, Paul Budnik writes:

   For example the real numbers definable in any consistent formal system
   are countable.

and in his last post, he explains what he meant:

   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):

    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.

One has to be a bit cautious drawing philosophical conclusions from the
Loewenheim-Skolem theorems.  (Or rather, one doesn't have to be, but angels
fear to tread.)  Let me rush in anyway: say L is a countable first-order
language rich enough to express "all of physics", as currently understood,
and in particular includes enough of the theory of R (real numbers) for
traditional QM, QED, GR, etc.  (L could be the language of ZFC, for
example.)  Philosopher X objects to L, claiming that uncountable infinities
do not really exist, and so demands that physics be reformulated on another
basis.  Philosopher Y retorts that a mere distaste for the uncountable is
no grounds for rejecting L, since we can make the same assertions and just
interpret them as applying to an elementary countable submodel.

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.