Re: infinity

Randy Poe said:
>
> aeo6 Tony Orlow wrote:
> > Daryl McCullough said:
> > > Tony Orlow wrote:
> > > >
> > > >Daryl McCullough said:
> > >
> > > >> >> [1] For all s a string, exists L a length s.t. L > len(s)
> > > >> >>
> > > >> >> [2] Exists L a length s.t. for all s a string, L > len(s)
> > >
> > > >> So are you saying that [2] means the same thing as [1], even though
> > > >> the quantifiers are reversed?
> > >
> > > >No, I am saying that [1] implies [2].
> > >
> > > But [1] is true even if we restrict L and s to be *finite*:
> > >
> > > [1'] For all s a finite string, exists L a finite length
> > > s.t. L > len(s)
> > >
> > > In contrast, [2] is certainly *false* if we restrict s and L to be
> > > finite:
> > >
> > > [2'] Exists L a finite length s.t. for all s a finite
> > > string, L > len(s)
> > >
> > > --
> > > Daryl McCullough
> > > Ithaca, NY
> > >
> > >
> > True. I'll concede this point, that there is no largest finite length,
>
> OK.
>
> > but if
> > all lengths are finite, then none are infinite, and the language cannot be
> > infinite.
>
> Empty words, no reasoning. Let's look at the actual
> expression you try to write.
Empty words? All lengths are finite, and for any finite length the language up
to that length is finite, so the language is finite.
>
> > The only way to get sum(x=1->k: S^x) to be infinite for finite k
>
> Forget "finite k". Just tell me what the expression is
> that includes ALL FINITE STRINGS. If you put k=10 in
> that expression, that leaves out the strings of length 11.
> If you put k = googolplex in there, you leave out the
> strings of length googlplex+1.
>
> What expression can you write down that does not leave out
> ANY FINITE STRINGS?
The statement applies to ALL finite strings. If ALL strings are finite, then
there are NO infinite lengths, and the sum can NEVER be infinite.
>
> Is there a value of k you can put in there that covers
> all the finite strings?
All n in N.
>
> Is there an expression of the form sum(x=1->k: S^x) that
> doesn't leave any strings out? For what value of k?
All finite values.
>
> - Randy
>
>

--
Smiles,

Tony
.

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