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

[Arnon Avron]

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 
> transfinata.

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
greeks.

> >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).

Arnon Avron