Re: infinity

In article <MPG.1d98ee1655c0bd6998a2fb@xxxxxxxxxxxxxxxxxxxxxxxxx>,
Tony Orlow <aeo6@xxxxxxxxxxx> wrote:

> The problem with the finite naturals is the restriction of finiteness
> on the elements, which causes the set to be finite.

Only in TOmatics. In standard mathematics, the infinite sequence
f(n) = n has infinitely many finite terms diverging upwards.

> Except that, for every element with an infinite number of successors,
> there are an infinite number of elements with an infinite number of
> predecessors.

Another only in TOmatics property. In standard set theory, for the
sequence f(n) = n, each of infinitely many terms has finitely many
predecessors and infinitely many successors, and the set of calues is
unbounded above.


> Just be satisfied that it is the smallest positive
> number which is not zero. That's as good as it gets.

Half of any positive number is smaller (closer to zero) than the number
itself, so TO's "good as it gets" isn't good enough!

TO is off dreaming in TOland again without a clue about how things
really work.
> >
> > In any case, it is trivial to prove that the number of finite
> > numbers is larger than any finite number. Pick any old finite
> > number x. The set {1, 2, 3, ... x, x+1} has x+1 elements, which is
> > greater than x. Therefore the number of elements in the set of
> > finite number is larger than any finite number.
> That is not a proof of any such thing.

Sure it is! A finite set has a finite number of members. Let N be that
number, then there is a set with n+1 numbers.

Thus there is a set with more than any finite number of numbers
whatsoever.


> This is a proof that there is
> no largest finite.

>From which one can show the set of finite naturals is unbounded.

Form which one can show the set of finite naturals is Dedkind infinite.

Which, outside of TOmatics, shows that the set is every kind of infinite,
.