Date: Mar 31, 2013 9:51 PM
Author: fom
Subject: Re: Matheology § 224

On 3/31/2013 8:32 PM, Virgil wrote:
> In article <1qKdnWnmuLpMQMXMnZ2dnUVZ_qqdnZ2d@giganews.com>,
> fom <fomJUNK@nyms.net> wrote:
>

>> On 3/31/2013 1:44 PM, Virgil wrote:
>>> In article
>>> <3d98da78-e43c-4550-812c-6436200744ec@vh9g2000pbb.googlegroups.com>,
>>> "Ross A. Finlayson" <ross.finlayson@gmail.com> wrote:
>>>

>>>>>> In a Complete Infinite Binary Tree, every binary rational path has only
>>>>>> finitely many left-child nodes or only finitely many right-child nodes,
>>>>>> whereas every other path has infinitely many of each.

>>>>>
>>>>> That is nonsense. 0.0101010101... has infinitely many of both sorts.
>>>>>
>>>>> Regards, WM

>>>>
>>>>
>>>> Well, you see Virgil has introduced a term in context the "binary
>>>> rational path"

>>>
>>> The standard definition of a binary rational is a rational whose
>>> denominator is a power of 2.
>>>
>>> In binary place value notation, they are the infinite strings starting
>>> at the binary point, then having onlybinary digits of 0 or 1, which end
>>> with either a string of infinitely many 0's or infinitely many 1's.
>>>
>>> Thus in a Complete Infinite Binary Tree they correspond to infinite
>>> paths with either only finitely many 1's or only finitely many 0's.
>>>

>>
>>
>> I will not disagree with your statement concerning "binary
>> rational path", but I did do a search and did not come up
>> with anything useful. That does not mean much since there
>> are far more pages with "binary" and "rational" used in a
>> context different from yours.
>>
>> I did, however, find dyadic rationals.
>>
>> http://en.wikipedia.org/wiki/Dyadic_rational
>>
>> These "standard terms" are sometimes a pain.
>>
>> What is important is that they are a dense subset
>> in spite of not being the entire class of rational
>> numbers.

>
> If "binary rational" is only my own phraseology, it is still at least
> unambiguously defined above.
>


Heck. I don't care. Ross brought it up. He
brought it up with me before -- and was correct to
do so. Your statement was perfectly understandable,
but I had made a not-so-subtle point about WM making up
terms.

What I have learned through the years is that almost
any sizable mathematical presentation is built with
a handful of idiosyncratic definitions among the
system of definitions upon which it is built. There
is probably some finite combinatorics document that
uses the exact phrase you used.

(As an example: do a google search on "almost
universality" and most of the hits are newsgroup
post for which I am responsible. The phrase is
from Jech. I had thought it would be a "standard
term". Well, it seems to have not made it to the
internet.)

What is important is that the eventually constant
sequences of the complete infinite binary tree have
the same structural characteristics with respect to
density as they do in the Baire space.