Math Forum Discussions

fom

Posts: 1,968
Registered: 12/4/12

Re: Mistake
Posted: Oct 20, 2013 9:06 AM

Plain Text

On 10/20/2013 3:12 AM, William Elliot wrote:
> On Sun, 20 Oct 2013, fom wrote:
>> On 10/19/2013 10:46 PM, William Elliot wrote:
>
>>> Disregard the above post, it was posted prematurely. I will answer your
>>> last reply when I've completed my response to you.
>>
>> Are you getting anything worthwhile out of this? I downloaded the copy, but
>> it is quite a bit. What you are doing is rather kind on your part.
>
> It's huge and in need of reorganizing and rewriting.
>

So..., No Abel prizes quite yet.

Ok.

>> Also, I have a dedicated monograph to proximity spaces now. If ever you
>> have any further questions on the matter, I may be able to help somewhat
>> with a resource.
>
> I noticed you had a thread going on about proximity spaces which I didn't get
> into for lack of time and interest to wade through the formal logic overhead.

In summary, consistent theories are associated with a family
of proximities. The structure of the proximities in the
family reflect properties of both the language parameters
and the interpretation map. For example, in order a for
proximity in the family to be separated, the language can
have no constant or function symbols and the interpretation
map must be injective onto the variables.

These are proximities formed from the terms of the language
rather than from the presumed objects to which the terms
purportedly refer.

By the way, I just formulated an arithmetic based on
my set theoretic axioms and the fact that the natural
numbers enjoy a topology called the prime integer topology.

If you can mark posts in your newsreader, you should
do so for when you have time. The formal sentences are
long and ungainly, but the post includes some exposition.

It is not the usual arithmetic. Today I will begin
working on introducing a recursion relation for Pythagorean
triples to introduce canonical cardinality relations for
each natural number.

>
> Is that your monograph or is it something different?
>

When I said monograph, I did not mean my work.

It is a monograph by Naimpally and Warrack solely on
the subject of proximity spaces.

> They're interesting in that it may be a way to show uniform spaces are
> Tychonov. Is every proximity space a Tychonov space? Conversely?

Every separated proximity is Tychonov. Since, apparently,
most of the literature deals only with separated proximities,
the converse may appear true. But it cannot be. I have to
read more.

It seems as if there is a compactification called a Smirinov
compactification. The set of compatible proximities
for a Tychonov space is order isomorphic with the set of
Smirinov compactifications.

Every Tychonov space has a maximal proximity. It has a
minimal proximity if only if it is locally compact.