[FOM] Intuitionism, predicativism, and ill-defined domains
aa at tau.ac.il
Thu Nov 3 15:38:26 EST 2005
On Sun, Oct 30, 2005 at 07:44:45PM -1000, Robert Lindauer wrote:
> >I admit that I dont understand the essential difference between
> >"for any integer" and "for all integers". Maybe it is because
> >there is no difference of this sort in Hebrew. But it is OK for me
> >to replace in what I wrote above "for every integer n we come across"
> >by "for any integer n we come across" etc. The validity of excluded
> >middle would not be affected.
> 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
I doubt if this extra meaning you attach here to "all" has a real basis,
but this is not really important. As I said above, I dont mind using
"any" instead of "all" in case I dont mean to imply that the collection
involved is a completed object. Again: what is important
is that "for any x in ... either P(x) or not P(x)" is still necessarily
true whenever P(x) is a meaningful for any x in ..."
> Similarly, in the transfinite heirarchy of sets, the term "set of all
> sets" is disallowed because it might tend to lend credence to the idea
> that the class of sets is itself an object in need of scientific and
> mathematical examination.
I dont see the analogy. Calling something "set of all sets" implies
that it is a set. What is the connection of this with
your claim about "all"? But again: this discussion concerning
terminology is important only for someone who (for a misterious reason)
thinks thartr LEM applies for "all" but not for "any". So it is not
important for me.
> > I believe that the way I understand this is exactly the way
> >The Greeks understood infinity. Euclide never talks about an
> >infinite line as a whole: only about finite line segements
> >that can be extended indefinitely. Yet it was clear to him
> >that if we continue extending two finite line segments on both sides
> >then either they will meet at some point (of time and plane),
> >or else they will never meet.
> What about the third possibility - that they might meet at some time
> infinitely remote?
I dont understand this "possibility", and I am sure that neither did the
> >I am not quite sure what you mean by "indefinitely
> >extendible". If you have in mind something like the potential infinity
> >of the natural numbers or finite line segments then LEM applies.
> >If not then the only way I can understand "indefinitely extendible"
> >is "extendible to bigger and bigger domains", but then again, in each
> >of them classical logic is valid.
> Here you display adroit use of the distinction between "all" and
> "each". If one admits that LEM is valid for EACH of the indefinitely
> extended domains, then what's the difference between that and it being
> valid for ALL of them?
Of course I would not mind using "all" here. But since I was answering
Nik Weaver, I have consciously used the terminology he prefers (and thus
made sure that he understands the content of my claim. Whether he agrees
or not is of course another matter).
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