FOM: my non-constructive proof
richman at fau.edu
Wed Jun 7 22:40:14 EDT 2000
martin at eipye.com wrote:
> Challenge to committed constructivists: should I have entertained the
> least doubt about the existence of such a Diopahntine set although I
> could not furnish an example? The constructive content of my proof if
> I cared to pay attention, would yield a pair of Diophanitne relations
> at least one of which does not have a Diopahntine complement. For an
> actual example, I had to wait twenty years.
Why don't you believe that your proof furnishes an "actual example"?
I'd like to clarify this situation by possibly simplifying it a bit.
Let P be a proposition (like "there is an odd perfect number") and
let's look for an integer n such that n = 1 if P is true and n = 0 if
P is false. Are you asking whether anyone should have the least doubt
about the existence of such an integer?
Certainly a constructivist would not claim, on general grounds, that
such an integer existed. An unwillingness to claim existence would
seem to be evidence of a doubt. On the other hand, the constructivist
might just like the finer distinctions that can be made when one
rejects the principle that such an integer exists for every
proposition. Like the distinction between your proof and the
construction of an "actual example".
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