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.
>> 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.