Re: abundance of irrationals!)

Randy Poe said:
>
> aeo6 Tony Orlow wrote:
> > Randy Poe said:
> > >
> > > aeo6 Tony Orlow wrote:
> > > > So, it would
> > > > seem the set is infinite, and it seems the value grows by 1 at
> each
> > > step, so it
> > > > would become infinite after an infinite number of steps. Show me
> > > where it says
> > > > otherwise.
> > >
> > > What does the word "after" mean in connection with a
> > > process which never ends?
> > >
> > >
> > It means somehwere in the infinite realm. Are you now arguing that
> there is no
> > such thing as infinity?
>
> No, I argue that there is nothing that happens "after" an
> infinite process. I argue that the question of what happens
> "after an infinite number of steps" is not of interest
> in a system in which everything is generated by finite
> iterations.
>
> > What is the purpose of your question?
>
> To point out that your belief in infinite-sized elements
> rests on an ability to carry out an infinite process to
> the end, and that there is no end.
>
> - Randy
>
>
And what do bijections between infinite sets rest on? How about inductive
proof? Infinite sums? They all rest on the notion that if we define a process
recursively then it is essentially infinite, but that doesn't mean we can't
figure out what happens at infinity. You have been listening to Zeno too much.
You want to claim that the set of natural numbers is infinite because it goes
on forever, but you want to claim that the members are all finite because you
can't, really, count that high. You say There is no largest finite number; is
there a largest finite set, and if not, then how do you know this extremely
large set of all finite numbers isn't just an extremely large finite size? The
fact is, if you restrist the naturals to finite values, then the set MUST be
finite. Finite values -> finite digits per value -> finite number of values
that can be represented. Please explain which of those two implications you
disagree with.
--
Smiles,

Tony
.

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