Re: infinity

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.

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.

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.



> > 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.
.

Relevant Pages

  • Re: infinity
    ... The natural number, n, in the set of naturals, N, is finite iff the set ... If there is no finite upper bound to a set of finite strings, ... >> then the set has to be infinite. ...
    (sci.math)
  • Re: infinity
    ... >>> Sure, given S symbols in L digits, you can represent exactly L^S ... If you are required to have infinite ... >> strings in an infinite set of strings from a finite alphabet ... if there is no finite bound on the lengths of the strings, ...
    (sci.math)
  • Re: infinity
    ... > Cantor-infinite, and so the set of strings will also be ... Sure, Cantor-infinite, but not actually infinite. ... the Cantor-infinite set of finite naturals. ...
    (sci.math)
  • Re: Logarithm of transfinite numbers
    ... uncountably infinite set of strings? ... representation of naturals conundrum, which is where I came in. ... A language is a set of strings constructed from a specific alphabet ...
    (sci.math)
  • Re: abundance of irrationals!)
    ... As we enumerate the naturals starting at one, ... infinite number of times to achieve an infinite set, ... Your "procedure" for incrementing a set is an unending one: ... We define the naturals by strings of digits. ...
    (sci.math)