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.