Re: infinity

Virgil said:
> In article <MPG.1d76ad0ae099937898a179@xxxxxxxxxxxxxxxxxxxxxxxxx>,
> Tony Orlow (aeo6) <aeo6@xxxxxxxxxxx> wrote:
>
> > David R Tribble said:
>
>
> > > Sure, given S symbols in L digits, you can represent exactly L^S
> > > unique values. But why does this apply to natural numbers?
> > S^L you mean. This applies to digital numbers, which for each base
> > consistute a symbolic language. If you are required to have infinite
> > strings in an infinite set of strings from a finite alphabet
>
> But one is not so required. One may quite nively have such strings.
> This allows an infinite set finite strings.
> It is only when there is some finite bound on the lengths of strings
> that the set of strings ust be finite.
if there is no finite bound on the lengths of the strings, then you have no
reason to claim they are all finite, see?
>
> That is, the set of strings does not have to be finite unless there is a
> longest allowable (finite) string. In which case, one should be able to
> specify its length.
And if the length is longer than any finite number you could name, what do you
call that? Unbounded means potentially infinite.
>
> SO if TO says that such sets of strings are finite, he must be able to
> give us that maximal length, or at least some finite upper bound on the
> lengths.
I am not the one claiming the lengths are finite. I am saying the lengths are
NOT finite if the set is infinite.
>
>
>
> > > Why is there some fixed (presumably finite) upper limit on the
> > > number of digits allowed for a natural number? Isthere some law of
> > > numbers that has previously gone unnoticed?
>
> > If there are any non-zero digits infinitely far to the left of the
> > digital point, then that represents an infinite value. If infinite
> > values are not allowed, then one cannot have non-zero digits
> > infinitely far to the left of the point.
>
> One doesn't have non-zero digits "infinitely" far to the left of the
> decimal point in standard natural numbers nor in standard real numbers,
> as far as that goes. Every real, except zero, has a first non-zero
> digit, it is always a finite number of places from the decimal point.
> > >
> > > It would help if you could specify what this limit L is. Something
> > > more concrete than 'N' or ceil(log(N)), whatever those are.
>
> > There is only a limit of finiteness on the length of the number if
> > you limit its value to finite values.
>
> Individual naturals are all finite, but the set of all of them has no
> finite upper bound. And the lack of such an upper bound is sufficient to
> prove that there is an injection from this set of naturals to a proper
> subset, which is the appropriate definition of being infinite for sets.
>

--
Smiles,

Tony
.

Relevant Pages

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