Re: Orlow cardinality question

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.

>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.

Martin

.

Relevant Pages

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