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Topic: The Distinguishability argument of the Reals.
Replies: 2   Last Post: Jan 3, 2013 5:07 PM

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Jesse F. Hughes

Posts: 9,776
Registered: 12/6/04
Re: The Distinguishability argument of the Reals.
Posted: Jan 3, 2013 8:18 AM
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Zuhair <zaljohar@gmail.com> writes:

> On Jan 3, 3:23 pm, gus gassmann <g...@nospam.com> wrote:
>> On 03/01/2013 5:31 AM, Zuhair wrote:
>>

>> > Call it what may you, what is there is:
>> > (1) ALL reals are distinguishable on finite basis

>>
>>  > (2) Distinguishability on finite basis is COUNTABLE.
>>
>> What does this mean? If you have two _different_ reals r1 and r2, then
>> you can establish this fact in finite time. The set of reals that are
>> describable by finite strings over a finite character set is countable.
>> However, not all reals have that property.

>
> I already have written the definition of that in another post, and
> this post comes in continuation to that post, to reiterate:
>
> r1 is distinguished from r2 on finite basis <->
> Exist n. n is a natural number & Exist d_n, k_n: d_n is the n_th
> digit
> of r1 & k_n is the n_th digit of r2 & d_n =/= k_n)


So, what does statement (2) mean and what sort of argument can you give
in its favor?

It seems like you mean something like:

Let S be any set such that every pair of distinct elements r, s in S
are distinguishable on finite basis. Then S is countable.

I don't see why that should be true at all. It seems plainly obvious to
me, for instance, that the set of paths in the full binary tree are an
example of such a set S, but that there are uncountably many of them.

I know you said that this "argument" of yours may not be capable of
being expressed as a formal proof, but you have to give me *some* reason
to think that (2) is at least plausible.

--
"The Hammer is not force. It is absolute power. The Hammer is from Idea Space.
That's the real world. Here is the magical realm.
You are creatures in that realm, who do not quite understand.
But it doesn't matter. There is a story to be told..." James S. Harris, poet.



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