Re: Orlow cardinality question

David Kastrup said:
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
>
> > David Kastrup said:
> >> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
> >>
> >> > Actually, I have no problem with the naturals going on forever,
> >> > as long as they achieve infinite values, which I have no problem
> >> > with either.
> >>
> >> Unfortunately, the naturals have a problem achieving infinite
> >> values. They can't. While every possible limit gets exceeded by
> >> some naturals, there is no single natural that could exceed all
> >> limits.
> >
> > Then one can say the same about set size, which is incremented in
> > exactly the same steps as the maximal value.
>
> Rubbish. The set of natural numbers is defined by five axioms, not by
> somebody putting numbers into a bag. The size of the sets of naturals
> is not "incremented", it just is.
it's defined recursively with a base case and each element defined with respect
to its predecessor. It's incremented.
>
> > The set of all finite naturals is finite, indeterminate as its size
> > may be.
>
> Stomping your feet and sulking is not going to make it so.
No, the restriction of finiteness does that all on its own.
>
> >> > I really don't see what problems that introduces. If you want to
> >> > say the series of finite naturals has no end because the end is
> >> > not identifiable, then that is okay, but it is of the "boundless"
> >> > variety, and not infinite.
> >>
> >> "Infinite" as a set measure is defined as the ability to place a
> >> set into bijection with a proper subset. The successor relation
> >> does just that with the naturals. So the set is infinite.
>
> > That's one way to look at it. So? What does having infinite whole
> > numbers break in your world?
>
> The fifth Peano axiom. Any set containing 0 and including for each of
> its numbers its successors, contains _all_ elements of the naturals.
> Since the successor operation will give a finite number from a finite
> number, the induction axiom strictly rules out infinite numbers in the
> set of naturals.
Not after the infinite number of increments required to produce the infinite
set.
>
> Now if you want to, you can call the set size of the naturals a
> "number", or you can call it nonexistent. But what you _can't_ call
> it is a natural number. That name is already taken for the set of
> numbers defined by the five Peano axioms, and the size of this set
> can't be a member of this set itself, because it does not obey the
> axioms.
Yes it does. I have already shown how the size of the set IS the maximal
element. Same thing, a natural number, which is used to count things, like
natural numbers. Natural numbers are things used to count things, like natural
numbers, which are things used to count things, like natural numbers.........
>
> >> Whether it achieves this by being boundless or having infinite
> >> values or pink and orange elements is irrelevant. It happens that
> >> it does not contain any infinite element, but there are no bounds
> >> that would not be exceeded by some elements.
>
> > How very useless and unenlightening.
>
> It happens that mathematicians have found quite a few uses for this
> peculiar set. Now there _is_ a use for infinite "numbers" as well,
> for example to describe the size of the set of naturals more exactly
> as just "infinite". But whatever such numbers are, however useful or
> enlightening they might be: one thing they aren't by definition:
> members of the set of naturals.
The size of the set of naturals is in the set.
>
> The axioms for that set leave no place for such a number designating
> the size of the set _as_ _a_ _member_ of the set.
Poppy***!
>
>

--
Smiles,

Tony
.

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