Re: infinity

David R Tribble said:
> Virgil said:
> >> Only for computer languages for which there is always some limit on
> >> both S and L. For non-computer languages, there is no inherent
> >> limit on either S or L, so no inherent limit on N. That TO limits
> >> himself does not mean that everyone is similarly limited.
> >>
> Tony Orlow (aeo6) wrote:
> >> If you limit your words to a finite length while using a finite
> >> alphabet, then you have limited the language as a whole to a finite
> >> maximum size.
> >>
> Virgil said:
> >> See, TO is doing it again.
> >>
> Tony Orlow (aeo6) wrote:
> > Actually I am exactly right, and you offer no alternative formula or
> > interpretation.
>
> 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, then any infinite set of normal digital
numbers must have infinitely long strings, as we do in the reals, to the right
of the digital point. For a whole number, the infinite digits must be to the
left, and this means the value is infinite, given how digital numbers work.
>
> 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.
>
> 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.
>
> >
> Perhaps a little device might help. Consider Tr(x), a function
> that takes the decimal fraction part of x and reverses its digits,
> so that the 10^-1 digit of x becomes the 10^0 digit of Tr(x), the
> 10^-2 digit becomes the 10^1 digit, etc. For example, Tr(1/4) is
> Tr(.25), which is 52. And Tr(6/25) = Tr(.24) = 42.
>
> Now perhaps you can tell me how digits Tr(1/3) = ...333 has?
Infinite. TR(1/3)=N/3-1/3
> Or maybe even Tr(pi) = ...3823979853562951413? If you know L,
> then perhaps can you tell me what the first (leftmost) digit of
> Tr(pi) is? Is Tr(pi) greater or less than, say, Tr(sqrt(2))?
I have no idea at this point, since I can't possibly know the Nth digit of Pi.
>
> We cantankerous Cantorians will point out that Tr(x) is undefined
> for all x that are not some multiple of 1/10^k for some finite k,
> i.e., Tr(x) is undefined for values of x with nonterminating
> decimal fractions, which produce infinite (divergent) values
> for Tr(x). That should not stop you, of course, from declaring
> that Tr(x) is well-defined for any x, since you posit the existence
> of infinite integers.
Eggggzactly!
>
>

--
Smiles,

Tony
.

Relevant Pages

  • Re: The Modified Halting Problem, Take ??? .
    ... What you write is not the same as saying all digits can be computed. ... With an infinite number the same process is used, ... We have infinitely many halting TMs, ... first you see 3 on the first tape, then you see 3.1 on the second tape, ...
    (sci.logic)
  • Re: The Modified Halting Problem, Take ??? .
    ... What you write is not the same as saying all digits can be computed. ... With an infinite number the same process is used, ... first you see 3 on the first tape, then you see 3.1 on the second tape, ... There are countably infinite Turing machines (meaning exactly TMs ...
    (sci.logic)
  • Re: abundance of irrationals!)
    ... >> aeo6 Tony Orlow wrote: ... >>> You claim an infinite set ... >> unlimited number of nonzero digits, do you think that if you ...
    (sci.math)
  • Re: infinity
    ... If p can take on infinite values, ... There is no reason to restrict your digits to finite positions. ... Every '9' digit corresponds to a member in set S ... finite strings ona finite alphabet, since the number of strings is finite for ...
    (sci.math)
  • Re: Well Ordering the Reals
    ... >> ending with an reversed infinite sequence of zeros, ... >> must ask where the intitial digits and terminal digits get connected to ... how do the initial or terminal digits connect up with them. ...
    (sci.math)
Loading