Going Backward, Going Forward – Part II

Partial obsolete – see below

This post will be dedicated to 1-dimensional Q-Orders, studying the effects Axiom 6 Q-Topology has on them and preparing the ground for the n+1-dimensional case.
Axiom 6
  

We will study 4 cases: the Rational Numbers, the Real Numbers, the Cyclic Group Zn and the Circle Group, using for the latter geometric representations on the Unit-Circle when appropriate.

Here are the more formal definitions:
Lemma F 6

The proof for F 6.2-3 –we assume the normal order <, was already given –they are as full orders also partial orders-, for F 6.7-8 the reader, if in doubt, is kindly invited to go back at least to Going in Circles – Part III. The constraints -Reduced, Regular, Q-Connectivity- are obviously satisfied.  

Obsolete by the Great Simplification … yet still useful for heuristics .. and will be replaced soon.

Next we will have a look on the corresponding Line-Sets.
Lemma F 7

To show the intended above:
 
Please note that in all cases the red or blue marked sets designate or represent just one line, despite that apparently this line is not always connected.

Next we will look at l-closed points, and the open line-filter and open neighborhood-filter for each case.
Lemma F 8
As the definition for open L-intervals –hence the open neighborhood-filter- requires that only for l-closed points there is a surrounding L-interval, in case of Zn any interval is by definition open, as all points are open hence any neighbor-hood is open. Different in the other three cases.

Next a look on the topologies and whether they fit into Axiom 6. Nothing new, all rather very basics from elementary topology.
Lemma F 9 

No big surprises yet, as it is no surprise that Real Line and Circle Group both satisfy Axiom 6.7, mainly due to their Dedekind-Completeness.

This end our first round through 1-dimensional Q-Orders.

Going Backward, Going Forward – Part I

In the following 3 posts we will go backward and forward through the seven axiom-sets, on one side to get a better feeling for Q-Orders, on the other to relate Q-orders with classically known concepts. The final post will show that the Hawking-Topology is a Q-Order.

Let’s start in this post with some considerations about the Axioms 1 to 5 and their relation to partial orders.
Axiom 1   Q-Relation

 Lemma F.1   Q-Relation 
  
states that with the proper definitions, any partial order satisfies Axiom I.

Axiom 2 has similar effects for partial orders has it has for Q-Orders:
Axiom 2   Q-reduced

Lemma F.2   Q-Reduced
i.e. we don’t permit isolated points and points that could not be told apart by using the partial order are considered the same. Further on we will consider only reduced partial orders. Please note this condition is weaker than the standard distinguishing conditions.

Axiom 3.1 imposes an additional condition on partial orders, every sequence of three points a<b<c can be completed to have 4, yet Axiom 3.2 follows already from being a partial order.
Axiom 3   Q-regular

Lemma F.3   Q-Regular
  
Please observe that using brute force to prove F.3.2 requires to analyze (3*8)^4=331,776 combinations of binary conditions, which due to internal dependencies may be reduced to less then 6144, but still a substantial quantity, which again for a proof may be further reduced by applying internal symmetries. Instead of wasting three pages with either resulting valid combinations or detailed analysis of symmetries, we just put a small picture and invite the reader to do the latter her- or himself.
 
Suggestions for a 5 lines-proof are obviously welcomed. We let the brute force method being applied by a computer. With respect to F.3.1, we assume henceforth that all partial orders a regular.

Intentionally –it was designed that way- Axiom 4 turns out to be a property of reduced, regular partial orders.
Axiom 4   Q-Order

Lemma F.4  Q-Order
  
Again we will not reproduce the pages of a formal prove, but invite once more to have a look on some pictures.

We break the analysis  of Axiom 5 into two pieces, first the construction of the orientation-set, then the connectivity conditions.
Axiom 5   Q-Orientation

Lemma F 5.1  Q-Orientation

Actually we construct the two orientation-sets, then the F 5.1.3 is direct consequence of the construction, while F 5.1.4 is proven in few steps. This establishes the expected result F 5.1.5: partial orders have an orientation. But please note: the orientation is not the order of partial order itself, as orientation means always oriented cycles.

The final lemma of this post transcribes the connectivity-condition for Q-orders into the language of partial orders.
Lemma F 5.2   Q-Connected

As preliminary result of our comparison Q-Orders versus Partial Orders, we obtain that under relatively weak constraints –Lemmas F.2 and F.3- plus the connectivity-condition –Lemma F.5.2- Partial Orders are models for the Axioms 1-5 for Q-Orders. This will ease the task to establish Hawking-Spaces –alas Causal Sets- as Models for Q-orders, because we can simply rely on the underlying partial-order and are almost done. By default these partial orders satisfy our weak constraints, while –advancing results- the connectivity-condition is one of their key-features.

Yet second this condition points already on a set of minimal constraints that a partial order must comply to become a candidate to be related by whatever structural-knowledge-preserving mechanism to the structure underlying GRT. It turns out that these have to take the form of Second-Order-Predicates, i.e. they can not be expressed as simple statements about relations among points.

Third it appears as if Q-Orders provide a proving-mechanics almost as strong as the transitivity respectively monotony from partial orders, essential not only for proofs but already for constructions like induction or convergence etc., yet without the disadvantage to have to believe that there is a universal beginning and a universal end for all and everything, a non-scientific hypothesis as it can’t be proven nor disproven. (See also Fotini Markopoulou (1)).

As a technical advantage, it will allow us to talk about systems with cyclic behavior –at least during some time-, not only –see Oscillator- a fundamental model in Physics but essential to introduce measurements or without cyclic clocks there is no time and without time there is no measurement at all, as there are no means to measure space as such.

This ends our first considerations about Q-Orders and Partial Orders.

1 Fotini Markopoulou, An insider's guide to quantum causal histories (1999), http://arxiv.org/abs/hep-th/9912137

The trapped Arrow of Time – Part III

Obsolete by the Great Simplification … yet still useful for heuristics

The central result of the cited articles from Stephen Hawking(1) and David Malament(2) is the proof that the path-topology, and only the path-topology, of space-time defines the time-like curves and viceversa, i.e. the time-like curves define uniquely the topology, where in turn the metric Tensor g may be reconstructed up to a conformal factor –in case of Lorentzian Manifolds- from the underlying topology.

This result is transcendental in our context, as hence time-like curves can be defined using only means of set-topology, that is without the heavy baggage of Pseudo-Riemann Manifolds etc. etc. and their implicit baggage of Real Analysis, Linear Algebra, Infinity anywhere etc. etc.

This stripped-down model of space-time can be extended without sacrificing its essential mathematical content to finite and countable models, something that can’t be done, at least no so easy, while –Einstein never said we had to- sticking to Lorentzian Manifolds supposedly as only feasible mathematical model underneath GRT.

… and we are almost there. Let’s see what still was missing in Part II:

Here we’ve got our already standard grid twice: once as-is, once flipped along the magenta axis of arrows, while the green arrows invert their direction. This transformation in blue coordinates corresponds to interchange the space- and the time-coordinate, a symmetry with profound physical interpretation. However –remember the coordinates by now have no meaning by themselves- the two grids are until now topologically identical: there is a 1-to-1 correspondence of boxes and connectors.

This means that only with this topology, the one defined until now, there is no way to preserve orientation or more general identify time-like curves only by means of topology. Actually, the picture already hints what to do: the time-arrow changed position (right side<>below), such that if we include the time arrow as additional connectors into the grid, we might be done.

We marked with a smiley two boxes, which before adding the additional connectors were topologically symmetric under the interchange of time and space and now are not. Noteworthy, the additional connectors were already present at some earlier stage of the development of net-theory and had their own name observables, as we will see –maybe- in a later post by no means a name by chance. And by then at least I knew already, that they are essential to define orientation respectively natural orders, i.e orders completely defined by their topology. So welcome back.

This is the content of
Axiom 7 Loops

Axiom 7.1 defines objects similar to the standard one-dimensional sphere, yet without relying on other concepts than our Axioms defined so far. The first line expresses that it should have just one dimension by requiring any 4 distinct points to be related. The second that it should comply with all Axioms defined so far, which as we’ve seen before among other orders all it points as on a circle. The third line requires double-connectivity, exactly what makes the difference between a Circle and the Real Line before one-point-compactification.

The unit-circle is one possible representation of S but likewise any other simple, closed curve i.e Jordan-Curve in the 2-plane or any homeomorphic image of the S, as from the point of view of Q they all are identical.

Axiom 7.2 defines a subset of the set of mappings from S to Q, requiring that the mapping produces an image –a curve- with at least 4 elements –remember 4 points on a circle, that’s where we started- and is an continuous mapping in the respective topologies. In traditional settings –everything at least a Hausdorff-Space- one would continue -defining paths and curves- requiring an injective mapping, we ask only –in the second line- for some form of monotony, which actually preserves orientation and excludes overcrossings. Requiring an injective mapping would carry a Hausdorff-property over to Q, which means no finite and only quite weird countable models, against all our intention. The last part of Axiom 7.2 defines a class of subsets, those that are image of some closed path, it may be understood as a generalization of the concept of Jordan-Curves. Please note that J-Curves are always closed and that they may change direction while going through Q.

Axiom 7.3-4 introduces the concept of J-connected points of a set –they may be connect by a connected piece of a J-curve, completely in the set- and J-convex sets, i.e. sets where every two points may be J-connected.

Before continuing, let’s get back to our augmented model-grid, see how Jordan-Curves may look like and if we got now sufficient to tell time- and space-axis apart using only topological means.

First some Jordan-Curves (remember: connectors are noted by their adjacent boxes):
blue {(0,0), {(0,0),(-1,1)}, (-1,1), {(-1,1),(0,2)}, (0,2), {(0,2),(1,1)}, (1,1),{(1,1),(0,0)}}
red {(0,0), {(0,0),(-1,0)}, (-1,0), {(-1,0),(0,-1)}, (0,-1), {(0,-1),(0,0)}
We show two candidates for cones at (0,0) a black cone set, that would correspond to the time-axis and a light blue one, that might correspond to a space-axis. Finally we shadowed the area, the smallest where the asymmetry between time and space makes itself manifest.

We observe in this area: both halves of both cones are J-connected, i.e. there is a J-Curve inside that connects any two points. Going from half to half of a cone, every inner connecting J-Curve contains (0,0) and at least one additional point, that is not part of the respective cones. There is however a difference: any J-curve in one of the black cone-halves contains at least one third element (0,2) , (0,-2) that connects directly to the center point (0,0), while this element does not exist in the light blue one, a connection established precisely by the additional connectors we added.

Actually with these observations we’re done already, if we put them into a mathematical language in a way that extends to all our structures, avoiding pitfalls like for instances that already in the 2+1 Grid (2 space-, 1 time-coordinate), there are no longer halves of the space-cone.

Axiom 7.5-6 introduces some necessary technalities, first locally connected sets then, as not all our points are closed, what may be called a saturated open set, i.e. open sets that with a neighborhood of a point contain also the point itself, third the set of open points –if there are- and finally the closed hull of a point, meaningful if it’s a open point.

Now the core of the Axiom itself:
Axiom 7.7

Lets check against our model grid, if the Axiom 7.7 does indeed would we like that it does (and that way go through it line by line).

For the beginning, let’s just note that here connectors represent no problem, as they have just one box at entry, one box at exit, so most of the conditions are void.
Now boxes:
The first line of Axiom 7.7 says that there should be an open set that includes the hull of the box as a sort of limiting our scope to some open neighborhood. The gray shadowed area above may be such a neighborhood.
The second line asks that all open sets, which include the hull and are contained in our starting neighborhood, shall satisfy some conditions, i.e. once found, the axiom somehow propagates from outer to inner.
The third line asks that each of these contained open-sets should have a decomposition into 2 new open sets (sets with <- and –> on top, they will be our the local cones). Their join with the point-set {x} shall be the open-set on study, their meet contain only open points.
In our example,
{(0,0), {(0,0},(-1,1)}, (-1,1), {(-1,1),(0,2)}, (0,2), {(0,2),(1,1)}, (1,1), {(1,1),(0,0)},{(0,0),(0,2)}} is one of the candidates, it’s below dual the other.

Now the conditions, symmetric for both halves, the partition shall satisfy:
For every pair of distinct points y, z in the same halve and any neighborhood of the hull of the point x under study, there shall be always a J-Curve that runs within the saturated pre- (respectively post-) cone. It shall contain all three points x,y,z and at least one additional point in that neighborhood. This way we formalize the above idea of directly connected. A rapid look on the picture shows that this is indeed the case.

Be aware that the set of boxes {(0,0), (-1,1), (0,2), (1,1), (1,-1), (0,-2), (-1,-1)} is the smallest possible open neighborhood of the hull for x and the above introduced candidates for cones are the smallest possible saturated open sets to cover these boxes, then it’s easy to see that again our picture fits into the axiom: every closed J-Curve contains at least three boxes from the neighborhood of the hull .… and it’s likewise relatively easy to see why the space-cones do not fit.

The Lemma A.17

shows that, as intended, curves that connect points in different cones, run trough the tip x of the cones and contain at least one outside element.

We omit for the moment additional technalities, like that each cone shall be J-convex and their intersection shall consist only of open points, which however will be important in later contexts and posts.

We note without proof here –but there will be in a later post- already one fundamental result: the Q-Loops define Q and viceversa, actually not such a big surprise looking at homotopy-theory and it’s results.

This ends our presentation of Axiom 7.

1 S. W. Hawking A.R. King and P. J. McCarthy, A new topology for curved space-time which incorporates the causal, differential and conformal structures Journal of Mathematical Physics Vol. 17, No 2, February 1976
2 D. Malament, The class of continuous timelike curves determines the topology of spacetime Journal of Mathematical Physics, July 1977, Volume 18, Issue 7, pp. 1399-1404

The trapped Arrow of Time – Part II

Obsolete by the Great Simplification … yet still useful for heuristics

After the long introduction of Part I with so many caveats based on painful experiences, here the
Axiom 6 Topology 

As final goal we will construct Q-Path-Topologies, that is the category of topological spaces that correspond precisely to Q-Orders. The construction is done in two steps: first in this part we introduce a first topology for Q-orders, still too general in that it still does not encode completely the orientation by solely means of topology. This will be done in the next step by means of the Q-Path-Topology.

While the definition of Q-Path-Topology adds additional axiomatic constraints on q-orders, once done, we may just start with a Q-Path-Topologies, add a global constraint, that exclude weird paths, and obtain back our underlying Q-order.

As expressed earlier, the Hawking(1)-Topology for GRT shall be one model for Q-Path-Topology and the known Petri-Topology for some discrete Concurrency Structures another. Yet there is a large list of other models, like the Complex Plane, the Quaternion, alas the Minkowski-Space, all one way or the other related to our Leitmotif General Relativity Theory and Quantum Mechanics.

We start out with most simple set, which initiated our reasoning, Lines. Axiom 6.1 defines Lines as a level-2 set, where each two distinct elements of each of its level-1 member-sets –a single piece of a Line- satisfy two conditions:

1. They should be members of some q-set i.e. share some circle.
2. If there is a q-set that separates them, than at least one other element of this q-set should be also member of this Line.

We note that all one element-sets are trivially Lines and advise –why will be seen later- that this definition requires the Axiom of Choice.

To grasp the origin of the second condition, we have to take a step backward looking first at Q-convex sets.

Lemma A 12 

We note that the empty set and the whole universe are Q-convex sets. Due to the symmetric definition, the complement of a Q-convex set is Q-convex; a surprise may be for some: though obvious for Circle, not-so-obvious for a Line; yet –as one example with more detail in a later post- the Real Line and the Real Circle considered as Q-orders are equivalent, as they are in usual Topology after relatively harmless yet very useful 1-point compactification … and both have hence isomorphic Q-convex sets.

Lemma A 12 then states that with every separated pair {{a,b},{c,d}} a Q-convex set contains at least one of the sets in-between as defined in Q-Order Axiom Lemma 7 and sketched in the corresponding picture.

Finally, Q-convex does not require that any pair of 2 points inside are connected by an in-between set, only those that share some circle, not much of a surprise as our Lines have –among other- time-like curves of General Relativity as conceptual input: not any two points in GRT are time-like connected either; if, then in-between-sets between closed points correspond to the basis of the Alexandrov-II Topology as named by Stephen Hawking in the already mentioned article(1) and detailed –maybe- in some later post. (There is also another, the Alexandrov-I Topology, both named after the same Russian Mathematician Aleksandr Danilovich Aleksandrov, yet with completely different properties. We will need both, hence I and II).

Please note that Lemma 11 still does not need the Axiom of Choice, as the Lemma itself gives sufficient constructive means. The problem arises, when the second condition of the definition in Axiom 6.1 asks us to pick out single elements, where the choice of one element may exclude some others: as imaginable by looking on the pictures or based on the above reasoning, not all pairs of elements in the in-between set necessarily share some circle .. yet this is required by the first condition. It’s here where all the choice-trouble starts. Let’s cross fingers and believe that Lines do exist.

Axiom 6.2

constructs for each point a level-2 set of Lines, those on which the point is encircled by at least 2 other points, as we seen in the previous example. In traditional language one might call them the Line-Intervals that contain the point in question.

I’m afraid, at this point the non-mathematicians will definitively stop reading this BLOG, unless we explain picturesque what we’ve done, advising before that the choice-trouble itself can not be visualized; nobody can see infinite many distinct points as distinct and infinitely close to each other -only imagine maybe- but here is the cause of the problem. Those mathematicians, who find picturesque explanations inappropriate –either you know or you don’t- may skip the pictures.

In the above grid, we will have a look on some of the intervals of the set of Intervals L(0,0) belonging to (0,0).
A blue interval for (0,0) : {(-1,1), {(-1,1),(0,0)}, (0,0), {(0,0),(1,-1)}, (1,-1)}
a red one {(0,1), {(0,1),(0,0)} ,(0,0), {(0,0),(0,-1)}, (0,-1)}
(Recall: connectors have no labels, they are identified by their adjacent points i.e. {(1,1),(0,0)} or {(0,-1),(0,-1)}

As indicated by the arrows, we have selected one out of the two possible directions, downwards. Obvious, the definition of an interval does not depend on the coordinate-system -blue or red- is a matter of convenience, nor on the direction chosen.

An interval includes all points between its endpoints on the same Line. Yet different to the usual, it’s not sufficient to give only the endpoints of an interval, as the following blue examples shows:
{ (0,2), {(0,2),(1,-1)}, (-1,1), {(-1,1),(0,0)}, (0,0), {(0,0),(1,-1)}, (1,-1) }
{ (0,2), {(0,2),(1,1)}, (1,1), {(1,1),(0,0)}, (0,0), {(0,0),(1,-1)}, (1,-1) }
Both intervals have the same endpoints – (0,2), ( 1,-1) - yet mean different paths.

Be aware of the above note about the Q-order equivalence of Real Line and Real Circle. For Q-orders not only the traditional Intervals but also their set-complements (!) on a line are Q-Intervals. This turns out to be no drawback, on the contrary will facilitate the construction of measures on Q-Spaces, as to be shown in a later post.

The Line-Intervals belonging to one point, hence form by set-inclusion a natural upper-complete partial order or upper-set allowing the construction of something similar to a set-filter belonging to that point, expressed more formally:
Lemma A 13 

We might have continued directly, coming up most probably with results similar to Keye Martin (2,3), yet decide a small detour, among other as their models do not include discrete model, essential within our framework. To understand, let’s have a look on the usual definition of what is a Topology and some of it’s most essential properties.

Lemma A 14 Topology 

The above states first the normal Axioms for a Topological Space plus two crucial properties: that all elements shall be distinguishable by means of their neighborhoods and that the space as such shall be connected, i.e. going by neighborhood in neighborhood in finitely many steps I can go from anywhere to everywhere. Then we recall the definition for Closed sets and that a Closed Set contains all its points of contact, which leads to the definition of open and closed points. In our context important properties are that in a distinguishing, connected topological space, no point can be both, open and closed, and that in a countable space of this kind not all points can be closed.

Axiom 6.3

retakes the results from Lemma A.13, and defines as l-closed those elements that can be separated from all others by a line or likewise, that are equal to their l-contact set.

Before continuing, let’s have a look on our Grid-Model, what might be open and what might be closed points. As one easily verifies, all connectors are closed, while the boxes are not, as their connectors x can not be separated by a line in Lx.

Now we continue with
Axiom 6.4-5 

We define the subset of open Line-intervals of a points, and a Level-3 set built with sets of open Line-intervals, where the sets of this Level-3-set Lx (Big Lambda in the formula) contains for every open Line-interval just one representative.

In our model, we use different color-families –the green and the margenta family- to illustrate chains of intervals (one contained in another with more tones).
So picturesque a set Lx in L is simply a set that contains all colors, and from each color family of intervals where if the larger interval is in Lx, so are Intervals it contains. The above finding is expressed more formally by
Lemma A 15
  

Please keep in mind that boxes are open by definition, there fore the blue set {(4,2),{(4,2),(3,1)},(3,1)} defines an open Line-Interval for the connector {(4,2),(3,1)} as the set {{(4,2),{(4,2),(3,1)},(3,1)}} is a member of L{(4,2),(3,1)}.

Axiom 6.5 finally uses Lx to construct the actual Filter Ux (a level-2 set). The sets U in Ux simply correspond to the union of the point-sets of sets of Line-Intervals or said differently but equivalent uX projects L-sets into Q. Please note, Line-Intervals and Ux are strictly locally defined, i.e. attached to some point.

We note that Ux inherits Lx from the filter properties.
Lemma A 16

Axiom 6.6 introduces the Topology.

The Open sets of this Topology O are those sets that for each l-closed point contain some set of its corresponding open filter-sets Ux. We leave the verification that Axiom 6.6 defines a topology as previously described in Lemma A 14 to the interested reader, yet using Lemma A 16 it’s not so difficult either.

However there are counter-examples in already in the case of countable base-set Q, showing that neither distinguishing nor connected follow from previous Q-Axioms and this definition of O-Topology.
Axiom 6.7-9

We claim connectivity in a stronger form: all Lines shall be topologically connected, a necessary condition if we would like to have Lines as continuous images of paths (as continuous mappings of [0,1]), essential for the Hawking-Topology.  The connectivity in Q was already established in Axiom 5. 

Similar we claim that the open filter-sets Ux form a base for the neighborhood-filter Ox and hence for the whole topology O.

Axiom 6.9 encodes a center piece of GRT: the cone formed by open time-like curves through a point is open (4). Whence automatically satisfied in countable and finite cases, it was much harder to find as essential axiom for the other Q-structures. We will introduce a model to discuss this in the next post.

Finally, just for the sake of consistency in our definitions:
Lemma A 18

that is l-closed w.r.t. lines and close w.r.t. the topology have the same meaning.

This ends the presentation of Axiom 6.

1 S. W. Hawking A.R. King and P. J. McCarthy, A new topology for curved space-time which incorporates the causal, differential and conformal structures Journal of Mathematical Physics Vol. 17, No 2, February 1976
2 K. Martin and P. Panangaden. Spacetime topology from causality arXiv:gr-qc/0407093v1
3 K. Martin and P. Panangaden. A domain of spacetime intervals in general relativity arXiv:gr-qc/0407094v1
4 Stephen Hawking, Roger Penrose, The Nature of Space and Time, Princeton University Press, 1995, ISBN 0-691-05084-8

The trapped Arrow of Time – Part I

Obsolete by the Great Simplification … yet still useful for heuristics

The Axioms 1-5 allow to speak about q-sets –our hammer- and points –our nails-, yet we don’t have a space where to put eventually constructed buildings –physical processes-.

The wrong way to get a space would be simply assume it, as Einstein showed convincingly a century ago refuting thereby Immanuel Kant, who another century before had declared space & time as logical aprioria beyond material experience.

There is a second caveat already raised by Einstein –see The Challenge-: though a specific space & time may be extremely practical for a description and hence a necessary heuristic tool, the Physics described should not depend on the specifics of the used space & time: Any space & time should do, as long it produces the same pattern of coincidence.

However –advancing results to be proven in later posts- some additional notes seem necessary:

• Einstein writes about point-events, where only point is the mathematical essence while event is already an interpretation, which apparently turned into a misleading trap for many, many, when they tried to identify the point-events of Einstein with quantum-leap-events from Quantum-Mechanics as introduced by Niels Bohr and Werner Heisenberg. At least I myself was caught in this trap until very recently, though Wolfgang Pauli –as I found out only after getting out of the trap- already noted in 1953 that this identification is false: due to the uncertainty principle QM-events can never ever correspond to closed GRT-points. This is a basic fact of physical life, by no means subject to, less result of impossibilities of observation, as what not is, can not be observed either, notwithstanding bad results from bad observations. It’s more over a structural property and not a consequence of erratic behavior of nature: we hold with Einstein God does not play dice.
• The principle of General Covariance, even more its expression as diffeomorphism covariance, is for Einstein a sequitur of the idea of background independence, yet he never stated anywhere that the realm of Differential Manifolds would be for him the only domain for admissible Coordinate-Systems. As long as we conserve the pattern of coincidence other Coordinate-Systems may do as well (and should do). We are about to show –even though we still have a long road ahead- that Q-Orders might be another, more general candidate.

There is another old saying The reinvention of the wheel results easier with a model at hand, so before continuing lets introduce a a model.

The above highly regular structure has two types of “points”, boxes and connectors. To illustrate Coordinate-Systems just as labels, we have labeled the boxes twice, once in blue, once in red, connectors in this simple model apparently need no own labels, as they can be uniquely identified by their limiting boxes. The arrow on the right indicates the possible directions.

Some q-set using red labels {{(0,-4),(0,-2)},{(0,-3) ,(0,-1)}}
The same using blue labels {{(-4,-4),(-2,-2)},{(-3,-3),(-1,-1)}}
A connector in red: {{(0,-4),(0,-1)},{{(0,-3) ,(0,-2)},(0,-5)}}
A connector in blue: {{(-4,-4),(-1,-1)},{{(-3,-3),(-2,-2)},(0,0)}}
Parts of a red line {{(m,2n),(m,2(n+1))},{(m,2n+1) ,(m,2(n+1)+1)}}
Parts of a blue wave {{(m,2n),(m,2(n+1))},{(m+1,2n+1),(m+1,2(n+1)+1)}}
… much more on this in a later post.

The shown grid may be considered as just a window to larger grid, that extends above and below, left and right, yet looking only on the window there is no way to tell whether the grid will extend to eternity respectively infinity or will bend somewhere, somehow returning into itself.

Final observation: take the grid as such, i.e. it’s located nowhere nor occupies anything else. Obviously its representation may need pixel on a screen, bits and bytes in some memory, ink and paper elsewhere, but all these representations, including those that use labels as their means of representation, are only shadows as Plato would have said of the idea grid.

Advancing in content, the picture may be considered as a window on a 2-dimensional discrete Minkowski-Space, once labeled as usual with a time- and a space-coordinate, once with mixed 2 space-time coordinates as first introduced by Kurt Gödel. In the latter form it’s also known as Petri-Grid, as it is a cornerstone in Carl Adam Petri’s General Net-Theory.

Please note that all this is until now pure interpretation of two, among many other, possible label-systems, yet as the labels considered as numbers may reflect in their arithmetic knowledge about the underlying structure, the above chosen labels seem to be useful, each for a different purpose.

Yet the picture has also another interpretation as the structure of an one-dimensional discrete Random Walk, which in turn is the discrete equivalent for the Diffusion-Equation.

Now this equation is the real counterpart to the one-dimensional imaginary Schrödinger-Equation, which models the 1 dimensional harmonic oscillator, whose solution-structure at minimal energy can be represented by Génesis (more about Génesis and Q-Orders).

Génesis is not only the smallest possible Q-Order, but also corresponds to the smallest possible folding of the above defined blue waves.

Hence if our Hauptvermutung is correct and all falls neatly more or less in its place, then the empty discrete 1-dimensional Minkowski-space is not empty at all, but rather constituted not filled by zero-point energy waves, just as it should be … but this nice result is still many, many posts away.

There is a third caveat, this time from Logic. Except Axiom 5, all preceding Axioms stayed within First Order Logic, i.e. referred to individual elements, not to properties of non-constructive sets (Obviously the domains (Q,Q) themselves are many times non-constructive). Though we used already Circle as a heuristic means, talking formally about the Geometry of Space & Time will require to talk formally about sets and their properties, relations among sets and their respective properties and finally functions and sets of functions and their properties.

And here we are in some very fundamental troubles for 2 reasons:

1. The risk of circular nonsense-definitions ala Bertrand Russell or the famous lying Cretans.
2. The almost theological decision to accept or not to accept the Axiom of Choice (or variants, like Zorn’s Lemma and others)

Well, with respect to the first we will stick to some typographic and definition-discipline to avoid –hopefully- trivial definition pitfalls, using different typos for different levels of classes:

1. Small letters for elements, CAPITAL letters for sets, two small letters (or more) for relations among elements.
   Bold letters refer to sets defined before, as we use one of the standard conventions to name Natural Numbers, Integer, Real, Complex and finally Quaternion (or Hamilton Numbers).
   Sometimes we use a subscript to indicate that the definition is to be understood as local i.e. as attached to some specified element(s).
2. Fraktur letters for Sets of Sets and their elements, hence in their definition (:=, :<=>) on the right side appear either sets of level 1 as elements or the elements are taken out of an already defined set of this level.
3. Greek Letters for Sets of Functions/Relations and their elements
4. Though we use recursive and implicit definition as a powerful tool, only primitive recursion is used (with a single, harmless exception sometimes: the implicit definition of equivalence classes).

There will be –maybe- a later post explaining in more detail the whole symbolic language, notation and logical mechanics used. Yet I fear that a 5 pages “must read first” primer about language and notation had been counterproductive. Therefore I hope that the actual snippets together with their verbal transliteration and picturesque illustration are sufficient to capture the intended meanings.

With respect to the second, we accept the Axiom of Choice, yet will try to avoid its usage whenever possible with reasonable effort or else raise the red flag.

This ends the long introduction for Axiom 6.

Going in Circles – Part III

There is an old saying When the only tool at hand is a hammer, the world appears to be a bunch of nails. Our hammer are the sets {{a,b},{c,d}}, the nails are the points, Axiom 5 defines then how the world appears to us.

Axiom 5 ORI Orientation

Axiom 5.3 is still quite easy to understand: it claims that the relation Q is connected by q-sets. Whatever cut into two pieces, there will be always 2 q-sets that differ only in 1 element to connect the pieces. This claim generalizes Q-Order Axiom 4 ORD as a careful inspection shows: the implications of Axiom 4 establish exactly this type of connection between their left and right sides.

Lemma A 9 shows that any two q-sets can be connected in a finite number of steps.

Lemma A 10 shows how this connectivity extends to Q: any two points are connected through a finite number of q-sets.

Hence our world is, as intended, connected only by hammers and nails. In a later post about Q-Topology we will see the equivalence to path-connected.

Now the harder parts of Axiom 5, which are intrinsically related to one of the most complicated problems in Algebraic Topology: the problems whether a space has a systematic orientation. In our case there is also a close, independent and direct relation to Group Theory. Both relations will be touched, maybe, in some later posts.

It should be mentioned that there have been related, relevant investigation also in the realm of Petri-Nets, e.g. the Orientation of Concurrency Structures by Olaf Kummer and Mark-Oliver Stehr(1), the Construction of globally Cyclic Orders by Stehr(2) and the proposal of cycloids as basic building blocks by Carl Adam Petri. Again these subjects will be touched, maybe, in a later post.

For our current purpose of motivation only, some picturesque considerations and some simple supporting lemma appear sufficient.

We know already that the points on any circle may be visited one after one in exactly two fashions or orientations: clockwise or counter-clockwise. And –as the following picture shows once more- interchanging (a,b) to (b,a) in {{a,b},{c,d}} reverses the direction.

Axiom 5.1 constructs first a set, a double-cover, that for each {{a,b},{c,d}} contains exactly two sets of sequences to run the circle, either clock- or counter-clockwise. We do not define which is which (same column = same orientation).

The idea of double-covers in Topology is to split the double-cover later into 2 subsets such that each original element has exactly 1 orientation, applying consistency rules to guarantee a orientation defined as uniform for all elements. From our experience with 5 points, we know it should be sufficient to have a consistency rule just for 5 points and then extend this rule. Let’s see.

Lemma A 2 tells us which are the q-sets for this configuration, which enables us to construct manually the double-cover and the desired single-covers .. to find the rule.
Lemma A 11
 

At this stage we know that to be consistently clockwise oriented all quintuples of 5 points must fit into the above definitions and that then the rule expressed as lemma holds.

Axiom 5.2 converts these findings for 5 points into a claim. We claim: there shall exist a split of the double-cover into two single-covers, such that the found rule holds for all quintuples in each of the single-covers. We define then that these Q allow a consistent orientation. Due to Axiom 5.3, there are exactly 2 possible global orientations. Therefore it’s sufficient to define the orientation for any {{a,b}},{{c,d}} to have it defined it for all Q or: there is only one direction of the arrow of time -even though it might be actually cyclic- and its orientation extends from anywhere to everywhere in finitely many steps.

Though somewhat surprising, it’s almost mathematical routine to convert a property found in a specific case into an axiomatic claim: all new model shall have this same property. And it’s relatively risk-less w.r.t.o inconsistencies as the specific case serves as a model in the sense of formal logic consistency.

However and again this is an axiomatic claim, not a fact: the world should be as we think it were, something that can not be proven, only disproven by a found counter-experiment. Similar as in the preceding cases of global claims, we will raise a red flag whenever Axiom 5.2 is actually used. (Question for the interested reader: why there can’t be a counter-experiment to axiom 5.3?)

With these considerations, we end the presentation of the first 5 Axiom-Sets for Q-Orders.

(1) Olaf Kummer and Mark-Oliver StehrPetri's Axioms of Concurrency - A Selection of Recent Results , Proceedings of the 18th International Conference on Application and Theory of Petri Nets, Toulouse, June 23-27, 1997, Lecture Notes in Computer Science 1248, © Springer-Verlag , 1997
(2) Mark-Oliver Stehr Cyclic Orders: A Foundation for Concurrent Synchronization Schemes. University Paris 7, Group Preuves, Programmes et Systemes, December 16, 2003. Part I: Thinking in Cycles