Re: infinity

Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:

> *** T. Winter said:
>> In article <MPG.1d88f149318a1a7298a210@xxxxxxxxxxxxxxxxxxxxxxxxx> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> writes:
>> > *** T. Winter said:
>> ...
>> > > > For each L, there are 2^L strings of that length, and
>> > > > sum(x=0->L: 2^x) strings up to that length. If L is finite,
>> > > > then sum(x=0->L: 2^x) is a sum of a finite number of terms,
>> > > > each of which is finite, so there are a finite number of
>> > > > strings in a language with strings of finite length.
>> > >
>> > > The "so" part does not follow, it only follows if there would
>> > > be a maximal length.
>> >
>> > No, it follows that if the length is always finite, then this
>> > fact holds.
>>
>> No, it does not follow. *If* there is a maximal length L, then
>> there are finitely many strings with length less than or equal to
>> L. There is however no L such that all strings are in length less
>> than that L, so it does not follow.

> You miss the point.

Well, you would not see it if it poked you in the eye.

> For any finite L, the set of all strings less than or equal to L is
> finite.

Sure.

> To get an infinite set of all string less than or equal to L you
> need infinite L.

But the set of all strings is not a set of strings less than or equal
to some L.

> But, you have NO L which is infinite in your set of finite strings,
> so therefore you cannot have any infinite initial segment of this
> set.

The whole set of strings does not have the structure of an "initial
segment" since it has no end.

>> Also correct. This still does *not* show that in a set of finite strings
>> where there is no longest string (there are always longer ones available)
>> is also finite. So your assertion:

> Yes it does!!! You can only achieve an infinite set of such strings
> if you allow infinite strings, as shown.

Whining does not make it so.

> But, you do NOT allow such infinite strings, so you can NOT achieve
> an infinite set. It doesn't get much simpler than that. Maybe it is
> time to accuse me of quantifier dyslexia again, but that would only
> look stupid.

Well, it _looks_ quite stupid that you still don't get the difference
between "arbitrary large" and "infinite".

--
David Kastrup, Kriemhildstr. 15, 44793 Bochum
.

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