FOM: Re: Arbitrary Objects

[Gordon Fisher]

Jesper Carlstrom wrote:

> Can an "arbitrary" object be thought of as an element in some free
> structure over the one considered? For instance, saying "let x and y be
> arbitrary integers" seems like considering Z[x,y], i.e. the free
> commutative ring over {x,y}.
>
> Jesper Carlstroem
> Stockholm University

I am coming late to this discussion, and to this list.  I am
wondering what the substance of this debate is.  On the
face of it, I don't see any substantial difference between
"Let x and y be arbitrary integers" and "Let Z be (or "denote")
the set (or group, or ring) of all integers, and take any x and
y in Z" or more briefly "let x and y be any (or "any two") integers".
(All of these, it seems, allow that x and y may be the same two
integers.)

Is the problem that no one is psychologically capable of selecting
just _any_ two integers, but is liable, say and in general ("in general",
another mysterious phrase?) to pick a number from 1 to 10, and
not to pick, say, -(10^7)* 0.314157 ?  Or perhaps that there is
a smaller probability of the latter choice than the former?

Is there some finitist or intuitionist problem here, involving some
view that since, in view of the (countable) infinity of integers, there
are some integers that never will be selected by anyone, one can't
honestly  speak of selecting just any two integers?

What about "let x and y be any two real numbers"?  Is the axiom
of choice of any relevance in this discussion?  How about picking
any two numbers from each of two subsets of the set of real
numbers?

Gordon Fisher     gfisher at shentel.net