Re: INFINITY Revisited



> This thought experiment seems like a method prone to generate
> misunderstanding.
>
> What is the proper mathematical term for a decimalic digit "dj" for the
> jth place to the right of the decimal point in a number represented in
> decimalic form? For example: 0.333...dj. Knowing proper terms would
> be useful when trying to precisely specify and discuss issues on these
> matters. I hope the language I use in this post will not be
> misunderstood.
>

Its normally written as the jth digit d_j but you are fine with dj.

> The sequence of rational numbers 0.3, 0.33, 0.333, ... converges to a
> real limit point (as dj the number of decimalic digits approaches
> infinity) of 0.333...; where 0.333... is the "repeating" decimalic
> representation of the rational number 1/3.
>

Yes

> Does the list of natuarl numbers corresponding to the infinite set of
> natural numbers enumerate all of the digits dj where j = 1 -> oo of
> 1/3 = 0.333... ?
>

No. 1/3 doesn't appear in the listas you have written it. It appears in
others, such as the standard mapping of p/q to Natural numbers.


> 0, 1, 2, 3, ...
> 0 . 3 3 3 ...
>
> 0,1,2,3, ..., n, ...
> 0.3 3 3 ..., dj, ...
>
> Can the set of natural numbers be put in a one to one correspondence
> with all of the digits of any specific decimalic number (including any
> nonterminating rational or irrational number)?
>

Yes, trivially.

1 -> 3
2 -> 1
3 -> 4
4 -> 1
5 -> 5

If you want a mapping between N and approximations to pi, pick
1 -> 3
2 -> 3.1
3 -> 3.14

If you want pi on the list,

1 -> pi
2 -> 3
3 -> 3.1
4 -> 3.14


> Can there ever be any decimalic digits dj of a real number that such a
> one to one correspondence (between the digits of the real number and
> the infinite set of natural numbers) miss? Cantors diagonal proof
> seems to depend upon this.
>

Sort of. We need to know that if you count up from 1 you eventually get to
all numbers. This is provable from the axioms of arithmetic.


> Does such a rule serve as an alternative definition of a real number
> eliminating numbers such as 0.000...1 ( the termination of the sequence
> of numbers 0.1, 0.01, 0.001, ... that apparently lacks a real limit
> point) from consideration as real numbers?
>
> <snip>

Yes, the rule says that the limit point of 0.1, 0.01, 0.001 is 0, and tough
titty if that is not part of the sequence. You can construct things that
look like 0.000..1, "infintismals", but they don't help with Cantor.


>
> Don Whitehurst
>


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