[FOM] Intuitionism, predicativism, and ill-defined domains

[Nik Weaver]

Arnon Avron:

> I admit that I dont understand the essential difference between
>"for any integer" and "for all integers".

Robert Lindauer:

> The term "all integers" would tend to lend credence to the idea of a
> specific object "all integers" which then would tend to be regarded as
> having properties, in particular, existence.  Whereas the alternate
> expression "any integer" only commits one to the assertion one is
> trying to make without thereby also committing one to the existence of
> transfinata.

The all vs. any distinction goes back to Russell.  See "Mathematical
logic as based on the theory of types", helpfully collected in
van Heijenoort's book "From Frege to Godel".  The discussion appears
in section II: "All and any", particularly page 158 of van H's book.
("In the case of such variables as propositions or properties, "any
value" is legitimate, though "all values" is not.  Thus we may say
`p is true or false, where p is any proposition', though we cannot
say `all propositions are true or false'.")

Russell's idea was that the "any" sense could be expressed by free
variables, whereas quantification required a concept of "all".  I
discuss this in section 2.3 of my Gamma_0 paper and reject Russell's
proposal in favor of the idea that "any" requires intuitionistic
logic, whereas "all" allows classical logic.  Of course, intuitionistic
logic was not available to Russell at the time he wrote this essay.

Nik Weaver
Math Dept.
Washington University
St. Louis, MO 63130 USA
nweaver at math.wustl.edu