The Great Simplification - Part I (corrections)

As stated clearly in the presentation, this BLOG documents Work in Progress, not at all final results. Some decades ago, when I started to study seriously Mathematics, I always wondered: how the hell were those powerful initial axioms and definitions found, which then gave origin to such powerful theories? Most of my teachers (and most Text-Books) presented only the final results in the sequence Axiom, Axiom, Definition, Definition, Lemma, Lemma, Lemma .. and out of box jumps a wonderful theorem. Actually it took me some time to discover that the way the final results were presented had little, if any, to do with how they had been constructed. Similar I suspect most traditional papers as published by Journals are –may be due to the harsh space-limits of Journals and time-limits of potential readers- rather an intent to impress than to explain making understand how. (I do doubt however that it truly saves time for the really interested reader, because unless he or she knew already, they have to reconstruct a living body of knowledge by analyzing only its bar bones). 

So here goes a new round in my construction process: While working on a new post about Dedekind-Completeness and its extension to partial orders and Q-orders I got stuck in some little, tiny detail: from Net-Topology I know that there should be never adjacent open or adjacent close points. And Hawking-Topology requires that pieces of world-lines have to be continuous, hence connected, images of [0,1], which boils down to exactly the same requirement as for Net-Topology, yet it turned out to be impossible to deduce this simple property from the Axiom VI in its former form. Reluctant to dump not only already written pages but some 20 Lemmata or so and their proofs, I tried first –what I suspect many do- to patch the initial Axioms –in this case Axiom 6   -the results can be seen in The trapped Arrow of Time  Part II- but finally decided to start over again.

The reasons: Though very common, in my feeling for esthetics these patches damage any beauty of a true Axiom-System. Second the former versions used a concept –Lines- that has its own flaws already on conceptual level: it requires the Axiom of Choice twice for its definition, which makes it very cumbersome to use later on and its in a way unphysical, as a physical process can never correspond to a geometric world-line, again a consequence of the uncertainty-principle, both considerations mentioned already in The trapped Arrow of Time - Part I.

Yet as both Hawking and Petri use Lines very intensively, I gave them at least a try, but finally decided to drop the concept Line as something fundamental and tried to rewrite Axiom 6 and Axiom 7 without using it … and received as gratification a great simplification without –as to be shown- loosing essence in modeling, that is if there are lines, then they behave as before.

Here the new Axiom 6 corrected
Axiom 6 Q-Topology            
    
The Axiom starts with the definition of U-neighborhoods of a point using the Q-Relation itself. Each element of neighborhood U shall by member of some limiting pair (A 6.1.1), for each member of a pair there shall be a representative (A 6.1.2), and finally a point of the same interval closer to the point than some already included, shall be likewise included (A 6.1.3). For the notion closer see Going Backward, Going Forward - Part I and Going in Circles - Part II.

The picture below may be helpful:

A blue neighborhood { (0,0), {(0,0),(-1,1)}, {(0,0),(-1,-1)},)}, {(0,0),(1,-1)},)}, {(0,0),(1,1)} }, a red neighborhood { {(0,0),(0,1)}, (0,1), (0,0) }.
The color-families green and purple represent similar as before related pairs w.r.t. the point (0,0).  

Lemma S 1 shows that arbitrary unions and meets of two U –neighborhoods are U –neighborhoods.    corrected 
   
Axiom 6.2-6 Q-Topology    
   
As before, we introduce closed points as all those which can be distinguished from any other point by using the Q-Relation. An open set of the Q-topology includes for each closed point  a U-neighborhood. Lemma S 1 essentially proves already that O is a topology. We claim consistency of concepts: what is closed as Q-Relation shall be closed as Q-Topology. The complete cone of a point shall be open.

Here ends the presentation of the simplified Axiom 6.