FOM: corresponding?

[Randall Holmes]

I didn't notice the following aspect of Kanovei's note.

He said:

Apparently, the only mathematical point here is that 
some properties of mathematical structures S, 
not expressible in the *corresponding* 1st-order 
language, 
become expressible if quantification 
over P(S) is allowed. 

I reply:

No, not expressible in _any_ first-order language.  The property of
being a model of true arithmetic is not expressible at all in any
first-order theory, however strong.  (In the following sense: any
first-order theory which describes a certain structure S in its domain
in terms compatible with S being a model of true arithmetic has models
in which S is in fact not a model of true arithmetic).

The issue here is entirely one of what one can define or express;
one gains nothing in terms of proof machinery which cannot be gained
by working in a stronger first-order theory.  The point which is
being belabored is that the reference of mathematical language is
an important foundational issue.

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes