Re: Orlow cardinality question

Martin Shobe said:
> On Tue, 31 May 2005 12:52:34 -0400, Tony Orlow (aeo6)
> <aeo6@xxxxxxxxxxx> wrote:
>
> >David Kastrup said:
> >> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
> >>
> >> > imaginatorium@xxxxxxxxxxxxx said:
> >>
> >> >> Well, you've been shown this before, over and over again. The
> >> >> pofnats do not have an endpoint at the right. There is therefore no
> >> >> contradiction whatsoever in the fact that all of the elements of
> >> >> the pofnats, not being the rightmost point, are finite. All you
> >> >> need to do is to think about this carefully and seriously, with
> >> >> your finite-based preconceptions turned off.
> >> >
> >> > This is the lion's share of the problem you folks have with seeing
> >> > what I am saying.
> >>
> >> We are not having any problem in that area. You come in loud and
> >> clear, and you are not getting it, loud and clear.
> >>
> >> > This focus on the "largest finite", nonexistent leaf nodes and
> >> > endpoints, is totally futile when talking about the finite naturals.
> >> >
> >> > You claim to have a set of finite integers, represented as finite
> >> > length strings constructed from a set of 10 digits, which is also
> >> > finite. If your maximum string length L is finite,
> >>
> >> The "maximum string length L" does not exist, and so can't be finite.
> >> Every single existing string length is finite, and none of them is
> >> "the maximum". If you allow some infinity measure as a descriptive
> >> shortcut, then the _supremum_ (not! the maximum, which does not exist)
> >> of the string length is infinite.
> >>
> >> > and your symbol set S is finite, then the number of strings given by
> >> > S^L is also finite. Any finite number raised to a finite power
> >> > yields a finite result. So, how can you claim a set defined this way
> >> > is infinite? You must have infinite length strings if you want an
> >> > infinite set of strings from a finite set of symbols.
> >>
> >> Uh, no. You string length is unlimited over the set, meaning that
> >> there can't be a fixed maximum for the complete set, but every single
> >> one of those lengths is finite.
> >Why?
>
> There can't be a fixed maximum for the complete set becuase if there
> was, we could append another member of S to the end of such a string
> and have a finite string with a greater length.
>
> Every single one of those lengths is finite because the set in
> question is the set of all finite strings.
And what is the upper bound on the length of those strings? If it's finite, you
should be able to count to it. How many times do I have to count to get to the
end?
>
> >The string length is unlimited. If you have an infinite number of numbers,
> >why can't they have infinite values?
>
> Because you would be talking about a different set. You can certainly
> talk about sets with infinite strings. But there are no infinite
> strings in the set of all finite strings.
Yes, only a finite amount of finite strings in your set, unless of course
you're working with an infinite set of symbols.
>
> Martin
>
>

--
Smiles,

Tony
.

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