Friday, April 30, 2010

Orders of Infinity

edited and abridged on 6 Feb 2018

A follow-up to my previous post Mass-Parity-Distance-invariant Universe where I described a Universe where negative mass explains dark energy...

Cantor's hierarchy of infinities

Let's look at its position in Cantor's hierarchy of infinities, as that seems more foundational than the other branches of mathematics, and move on up from there.

We know the zeroith order of infinity (a.k.a. finiteness) can define a logic system, by using two or more ordered finite values, i.e. false and true for boolean logic, more for other logic systems. The first order of infinity (a.k.a. aleph-0) can define an arithmetic system. The simplest is the natural numbers, created by starting at number 1 and applying induction. Extensions to this such as the integers and rational numbers are also aleph-0. Godel showed this arithmetic system cannot be both consistent and complete.

Things get more interesting at the second order of infinity. A next higher order of infinity is the power set of a lower one. The real numbers, introducing the square root of 2, extend the integers. We can prove they're at some higher order than the first order (integers, etc), but can't prove at what order they are. We therefore speculate they're at the second order (a.k.a. aleph-1), calling this the Continuum Hypothesis. Other number systems with the property of continuity (e.g. complex numbers, n-D manifolds) would then also be aleph-1, but complex numbers, which introduce the square root of -1, are "algebraically complete", not requiring any further extensions.

And what of aleph-2, the third order of infinity? The set of all curves (including fractal ones) is known to be at some order of infinity above aleph-1, hypothesized to be aleph-2. When looking at the curves, maybe it's best to consider the most complex curves first, such as 1D-lines with a (Hausdorff-) dimension of 2. The most famous of these are the Mandelbrot and Julia sets, which both happen to be defined on the complex plane. When we look at computer simulations of them, we see many concentric closed curves of various colors, reflecting the different integral-valued accuracies of calculation. If we could mathematically define a real-valued accuracy of calculation, would we still see concentric fractal curves? I suspect so, and that they would merge into a continuously-varying fractal structure with (Hausdorff-) dimension of 3, on the 2-dimensional complex plane. All of the Julia sets for the standard Mandelbrot look like they have this same concentricity property, though only some of them seem to have (Hausdorff-) dimension 2, the rest (on inspection) seeming to have dimension of less than 2. One is even a perfect circle, with only (Hausdorff-) dimension 1. And of course we could consider the nonstandard Mandelbrot views for all these Julia sets.

The Julibrot Set

Let's use these two fractals in a candidate mathematical model for our Universe. The Julibrot Set is defined as the topologically-4D union of all the Julia sets of the Mandelbrot set. If we consider real-valued accuracies of calculation for the entire Julibrot, we have a (Hausdorff-) dimensionality of somewhere between 4 and 6, embedded in the 4 topological dimensions. Now I suspect there could be a theorem concerning the (Hausdorff-) dimension in such a structure: I'll speculate it's 5 and see where that leads. If such a 2-complex-plane structure explains the 4D-spacetime of our Universe, then there's an extra non-expansive dimension supplied by considering the fractal curves at different positive-real-valued degrees of accuracy. This could explain the phenomenon of mass in our 4D-spacetime, along with certain rules of distribution within, which could be the law of Einsteinian gravity.

What would be the order of infinity for this Julibrot-based Universe? There could be a theorem saying this structure is necessary and sufficient to contain all curves, and is therefore at the third order of infinity, aleph-2. Our Universe could then be the minimumly-complexed structure that can exist at aleph-2. Furthermore, just as the complex numbers are algebraically complete at aleph-1-infinity, so also our 4D-spacetime with its inbuilt phenomenon of Mass, obeying rules both of gravitation and of quantum physics, could also be complete in some way at aleph-2-infinity.

The directed dimension and inbuilt polarity

Any recursively-applied polynomial equation seems to give the basic Mandelbrotly-edged shape on the computer, so presumably they all have high enough Hausdorff dimension to be a candidate model. But only simple Julibrot Sets are reflectionally or rotationally symmetric in 3 dimensions, but not in 4, giving a dimension that looks like Time.

By looking at the Julibrot's constituent Mandelbrot and Julia sets on the computer, including their colored accuracy levels, we see that the Mandelbrot-real dimension is the assymmetric dimension, i.e. the Time dimension. As a bonus, the Mandelbrot-imaginary dimension is the only one that's reflectionally symmetric, therefore the one along which positive and negative matter flew apart, i.e. the Axial Space dimension. The Julia sets at each point below the standard Mandelbrot Time axis (where y=0 on the plane) are similar to the corresponding one above, except for being reflected through their Julia-real axis. This gives the appearance of the Mass rotating in opposite directions in each half of the structure, perhaps suggesting the opposite handedness of gravitons and neutrinos in each half of the mass distribution in our Universe. But in each half of the Mandelbrot Mass distribution, when we look toward the Time axis, the Mass appears to rotate in the same direction, perhaps suggesting why space appears to have an inbuilt polarity.

Higher orders of infinity?

So we've seen a possible model of our physical Universe by regarding Mathematics not as something that simply describes the Universe, but as something which the Universe is at some order of infinity.

If our Universe is what exists at the third order of infinity, then what might constitute the next higher order of infinity? Using Cantor's theorem, by considering the power set of our Universe, we might say it's the set of everything that could have happened, could yet happen, and would yet have happened in our Universe, except for the number of dimensions. But how would we place our own specific Universe of actual happenings in this picture, with its unique quantum reductions into actualities? One person might say the power set is at a higher order of infinity than our own specific Universe of actual happenings, because of Cantor's theorem. But someone else might say the Universe of happenings is at a higher order, being the real, intentioned actualization, while the possibilities are at a lower order, being just the canvas for the actualization, so to speak. It sounds like an argument between atheists and theists, and perhaps never able to be proven either way.

So possibly our own consciousnesses can't really comprehend very high up the ladders of infinity. At the first order, we can't logically define a consistent and complete system. At the next order, we must hypothesize its actuality as The Continuum in our own Universe. And at another order or two higher, where our own consciousnesses dwell, we can't logically prove which of the two orders is the higher and which is the lower. If we could look higher up the hierarchy of infinities to the infinite level, then the hierarchy itself would be able to be counted by an induction-based counting scheme, and so become self-referenced, and even itself an inconsistent and incomplete arithmetic system. In fact, because different instances of our human consciousnesses can't agree which of the third and fourth orders of infinity is higher than the other, we can't even make the known physical representations of the orders of infinity into a propositional logic system, not knowing which to call True.

Moving each order of infinity up the ladder seems to require utilizing some new well-known mathematical concept. Moving from finiteness to aleph-0 requires Induction, and from aleph-0 to aleph-1 requires Continuity. If an MPD-invariant Universe is at aleph-2, then moving there requires Probability. This would be the lowest order of infinity which contains the entities of Time and Mass, as distinct from Space. And such entities, Time and Mass, are required for the mathematics of computational complexity, Time as a resource and Mass for building Turing machines. Perhaps the next order of infinity, aleph-3, requires the concept of Computation, which could explain the phenomena of consciousness. Such Computation is also required for calculating fractal curves in aleph-2 spacetime.

Computational complexity

Perhaps computational complexity will play an important role in a theory of the Universe. There's now many known complexity classes. If we can slot the theory of computational complexity directly onto a mathematical structure which defines a space-distinct Time, which behaves according to the laws of our Universe, then many of these complexity classes, such as PSPACE and P(TIME), may meld into one when we factor in the effects of Special and General Relativity, such as time dilation and space contraction. PSPACE problems require more computation "power" than P(TIME) problems, but if space can become time due to high acceleration or a nearby strong gravitational field, perhaps ultimately they're really the same complexity class.

Similarly, the distinction between the P(TIME) and NP(TIME) complexity classes may not exist at Planck scales because of the quantum nondeterminism. Perhaps the electric field generated by human brain neural structure makes use of such quantum nondeterminism to produce the effect of consciousness. Perhaps both large-scale (relativistic) and small-scale (quantum) effects together reduce the many complexity classes down to a mere few. They seem to fall into four broad groups: logarithmic, polynomial, exponential, and recursive.

Do these complexity groups each match up somehow to the various orders of infinity I've described? Do aleph-0-infinite structures like the integers relate somehow to logarithmic-space computation? Do aleph-1-infinite structures like the complex numbers relate to polynomial-resource computation? Is our Universe an aleph-2-infinite structure? Is it an MPD-invariant "canvas" for our Universe of actual happenings, related somehow to exponential-resource computation? Are the quantum reductions that form the actual happenings in our Universe, including our own consciousnesses, an aleph-3-infinite structure? Related somehow to recursively-enumerable computation? And what could possibly lie beyond that?

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