FOM: second-order logic is a myth

[Stephen G Simpson]

Me:
 > I actually favor the `Platonist' or realist view in many contexts.

Sazonov 25 Feb 1999 01:30:41:
 > In which contexts? Are there contexts in which you do not favor 
 > this view?

When I'm wearing my `naive working mathematician' hat, I usually take
the Platonist view, because at this point in history most
mathematicians think and talk as if actual infinities exist, the
powerset operation is well defined, etc.  When I'm wearing my
`f.o.m. professional' hat, I take a much more skeptical viewpoint.

Me:
 > the study of V_2 as a whole is fruitless and hopelessly
 > intractable.

Here I was speaking as both a working mathematician (`the study of V_2
is hopelessly intractable') and an f.o.m. professional (`the study of
V_2 as a whole is fruitless').

Sazonov:
 > Does not this (i.e. "fruitless and hopelessly intractable")
 > actually mean that the set of second-order validities V_2 is
 > meaningless (if considered outside ZFC or any other
 > [first-order] formal system which describes in its own way
 > the set of "all" subsets of arbitrary set)? 

No, that's not what I was trying to say, and I don't even agree with
that.  Some things are fruitful to study even if they are meaningless
outside ZFC or a similar framework.  For example, infinite cardinal
arithmetic has been fruitfully studied by Shelah, and he published a
thick book on it.

-- Steve