Re: infinity

In article <MPG.1d8a10695b27bd3f98a228@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:

> William Hughes said:
> >
> > Tony Orlow (aeo6) wrote:
> > > William Hughes said:
> > > >
> > > > aeo6 Tony Orlow wrote:
> > > > > William Hughes said:

> > > > > Yes, I saw it. It's basically the "largest finite" argument.
> > > >
> > > > No it isn't. The difference is that you claim that:
> > > >
> > > > a: the largest finite natural does not exist
> > > >
> > > > b: the sum of all finite naturals does exist but cannot
> > > > be named.
> > > >
> > > > -William Hughes
> > > >
> > > >
> > > No, I never claimed the sum of all finite naturals "exists".
> >
> > No I suppose you didn't. You have claimed that:
> >
> > a: the sum of a finite number of finite integers is finite.
> >
> > b: there are only a finite number of finite naturals.
> >
> > I concluded from this that you were claiming that the sum of all
> > finite naturals exists. On the other hand I cannot see how you can
> > believe a and b and not believe that the sum of all finite naturals
> > exists.

> Taking the sum of all finite naturals depends on identifying the last
> of them and completing the sum,

Proving that a sum exists does not require actually finding it.

Lemma: the sum of any two FNN's (finite natural numbers)
exists and is a FNN
Proof: any standard test on foundations.

(1) The "sum" of a set of one FNN is that FNN. (2) Given for any
FNN, n, that the sum of a set of n FNN's exists, it
follows from the lemma that the sum of a set of n+1 FNN's
exists.

Thus, by mathematical induction on the set of all FNN's, for EVERY
FNN n, the sum of a set of n FNN's exists.

Corrolary: if the set of all FNN's is finite,
then the sum of all FNN's exists.

> but we all know it is impossible to identify any largest finite.

The only viable reason why it can be asserted to be impossible to find
a largest finite is if no such thing exists.

> The contradiction that is being pointed out now as proof that the set
> of finite numbers is infinite derives from the contradiction inherent
> in supposing any largest finite number.

WRONG! Such a contradiction depends only on the assertion that there
are only finitely many FNN's.

If a set of two or more FNN's is finite then the sum of all its members
exists and is larger than any member of that set.

Thus no finite set of FNN's can contain all FNN's.

> So, in this sense, neither the largest finite nor the sum of all
> finites exist, even though we can say that conceptually they are both
> finite numbers.

TO can say what he pleases, but that does not make any of his claims
true, particularly when his claims lead to immediate contradictions.
>

> >
> > >I proved that any
> > > set of strings up to a given finite length is finite, and cannot
> > > be infinite unless that length is allowed to be infinite. If all
> > > strings in your set are finite, then the condition required for
> > > the infinite set is not met, and as unbounded as your set may be,
> > > it is not infinite.

And others have proved that the set of all finite strings cannot be
finite. For example:

For any finite set of finite strings, any concatenation of all of them
is a finite string.

If the original set contained more than one string of positive length,
that concatenation, being longer than any member of that set, cannot be
a member of that set.

Thus no finite set of finite strings can contain ALL finite strings.

So much for TO's delusions!

> So, the question is why I believe that the set of finite naturals is
> a finite set. I think the contradiction I pointed out above, between
> the infinity of aleph_0 as a set size and the required finiteness of
> it as the largest finite number in the set, points out pretty clearly
> that you have a problem here. If the elements are necessarily finite,
> then the set cannot possibly have an infinite size, because the size
> is always equal to an element in the set.

That is nothing like the problem of the direct proof above that no
finite set of FNN's can contain all FNN's.

> I hope I have explained my thinking to your satisfaction

Vain hope that TO can explain such self-contradictory thinking to
anyone's satisfaction.
.

Relevant Pages

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