Superinfinity and Dimensions

[Aepos]

This is officially my longest post ever. It's probably my best, as well.

Speaking of imaginary numbers, I forgot that I wanted to post this up on YC for everyone to see. While bored in my classes the day before the day before yesterday, I ended up writing a 4-page dissertation on my analysis of the dimensions and infinity. It's a really fun topic in my opinion. I just wanted to put this out there for debate, so criticism is welcome.

Since I don't exactly have that paper with me right now, I'll go ahead a recreate this ideological brainchild of mine in my head... again.

As a quick note, if you look into set theory, a branch of mathematics, this is like addition is to calculus... very primitive. :D


My Analysis on the Concept of Superinfinity and the Dimensions:

I've always wondered about the concept of multiple occurrences of infinity; we're taught in school that it's a term used to express the idea of endlessness. We're taught "numbers are endless," which is true. Later down the road, we're taught "numbers can be divided into an infinite number of fractions," which is true as well.

Then I started to think. You would think that the "number line" is basically a line in the first dimension (from this point on, I'll refer to dimensions as Dn, where n is the dimension that is governed by a determinable number of certain axioms). But after much thought, I concluded that the number line was actually a 2-dimensional concept. Why? If you've never heard of the word 'infinitesimal', it's a word used to describe the idea that there are an infinite number of unique decimal values between ANY two numbers. It follow logically that on a number line, you can contain an infinite set of unique value within a finite range of values. Put into example, there are an infinite number of decimal places for any integer (e.g. for 1, there is 1.5, 1.75, 1.2, 1.003, 1.999999, ad infinitum). There are also an infinite number of integers (e.g. 15, 175, 12, 1003, 1999999, ad infinitum). The idea that and infinite set can be "contained" within another infinite set is what I like to call superinfinity. The fact that there are two degrees of superinfinity leads me to the idea that this number "line" is actually more like a number "plane."

[Edit: an hour after writing the above paragraph, I realized I'm going to have to think over that for a while, because I think I may have possibly disproved what I have stated above. So, if you're already getting skeptical, I'm not surprised. I'll post exactly why I might withdraw my above statement in a later post.]

In theory, a line is infinite, intersects two points that are both governed by D0 axioms and have multiple occurrences only because they exist in D1 and are therefore governed by axioms governing D1. Now,

There can be only a single line in D1, since D1 consists only of X (which we'll refer to as "length") and for there to be two lines, both lines must be equal, and therefore they are the same line.

[Note: In my original notes, I stated that a line in D1 was "either X or Y, but never both." I withdraw my statement, since after some consideration, to say that would mean it could also just be Z (depth), just be T (time), et al; while it may be true, it leads to serious ambiguity, which is what I'm trying to shy away from. I intend to be as clear and specific of everything as possible.]

[Another note: when I refer to a line, I'm referring to a line, not a segment or ray, which are both very different; a line is infinite. There are zero endpoints. A ray is simply self-contradictory in my eyes, since it describes both "finity," for lack of better words, and infinity. A segment is just a way to describe the measure of line between two collinear points; technically, a segment more of a convenient way of visually denoting distance than is anything else.]

You may have noticed that I have used the term 'superinfinity'. If you have never heard of this, it's most likely because you haven't. I have once again coined a term to describe something that I couldn't find a better term for. Basically, my definition of superinfinity is as follows:

Superinfinity is a way of describing inherent magnitudes of infinancy. The most basic way to show superinfinity is an exponential degree of infinity. Superinfinity does not occur naturally and is only an abstraction that describes powers of infinity from a perspective external of all dimensions. "superinfinity" and variants on the word are typically followed by a degree of superinfinancy (e.g. "superinfinity to the fourth degree", or "superinfinity sub-four"). It just happens that the degree of superinfinancy is equal to the dimension in which you externally describe it. In layman's terms, superinfinity is ∞^n where n is between 0 and 10, inclusively (I'm still debating how this works with D0, since x^0 is 1 (or if you can even say it's ∞^0 is 1 since it doesn't follow the same set of laws as numbers); but after thinking about it, D0 defines no infinities, and therefore ∞^0 is undefined, not 1).

All of that aside, I'd also like to introduce one of my ideas. You've all heard of parents, children, traits, and inheritance, regardless of whether biological or technological...

Well, I've formed a set of basic concepts of how they relate to dimensions and infinity. If you think about the zeroeth dimension, and then the first, you'll notice that two primary things change. The first is the number of traits. A child always inherits exactly 1 trait from its parent and defines exactly 1 new trait. The second, is the new trait that is introduced by a new dimension, the child dimension, that is not inherited from that dimension's parent dimension.

Exempli gratia, take into account D0 and D1. D0 has a single trait, while its child, D1, has two traits. D0 is a point of indefinite (no) size and indefinite (no) position. It may seem very abstract at first, but it is required for all subsequent (child) dimensions to have definition. D1 is a line of indefinite magnitude, direction, size, position, etc., that is formed by an infinite number of points over X, which is the only defined axis in D1.

Now, let's switch to example mode. I'll now attempt to explain the concept of superinfinity via examples:

Dimension 0, a point:
[0 infinities (0 inherited, 0 defined), ∞^0]
--

Dimension 1, a line:
[1 infinity (0 inherited, 1 defined); ∞^1]
Imagine a line in the first dimension. That single line contains an infinite number of points. Since the line itself is infinite, the number of points that can be placed anywhere on that line are infinite.

Dimension 2, a plane:
[2 infinities (1 inherited, 1 defined); ∞^2]
Image a 2-dimensional circle on a plane. Then imagine a chord in that circle. The chord can be oriented and positioned in any way you would like to imagine it. I would normally like to think that a circle has an infinite number of chords, while when I take a deeper look, it seems more like a circle has a ∞^2 number of chords. This is why I believe this:

First, we need to get some terms straight. When I describe the orientation of the chord, I'm talking about the way it is "rotated", or direction of its line if you extend the chord. When I refer to its position, I'm talking just about the position of its point midpoint along the diameter perpendicular to it.

You can reposition any chord an infinite amount of times without reorienting it. But, when you do reorient the chord, you're setting your chord up for a whole new set of infinite repositions. This infinite set of infinite sets can be described as "∞ sets of ∞." In the same way you say "3 [times] 4" is "3 sets of 4," you can say "∞ [times] ∞" is "∞ sets of ∞." Simply put, this example expresses "∞(∞)," or more simply put, "∞^2."

Dimension 3, a space:
[3 infinities (2 inherited, 1 defined); ∞^3]
The simplest way to visualize ∞^3 geometrically, would to imagine a space, which is an infinite number of planes. Next, imagine a 3-dimensional sphere with a diameter parallel to the Z axis acting as a type of axis on which a 2-dimensional circle could spin in such a way that during a full 360 degree 3D rotation of the circle in the sphere, the circumference of the circle was always equal to the circumference of the sphere relative to the circle. Simplified, imagine the circle from the last example using the same exactly principles inside of a sphere in such a way that the sphere's diameter was equal to the circle's diameter (meaning that both diameters were the same line) and both center points were equal. If you were to rotate the 2D circle inside the 3D sphere around the only axis that is not already being used by the circle, the Z axis, you would have an infinite number of rotational positions of the circle. This means that the same principle introduced in the last example still applies, but we also have an infinite set of positions of the circle.

Therefore, we have infinite set of rotational positions of the circle that has an infinite set of reorientation of the chord that has an infinite set of repositions across the 2D circle. We now have 3 exponential infinite sets, or "∞(∞)(∞)" or "∞^3."

Dimension 4, space-time continuum:
[4 infinities (3 inherited, 1 defined); ∞^4]

If you are reading this, you are here! (Hardeeharhar!)

Surprisingly enough, this is the easiest dimension to exemplify. Think of time as yet another axis on an XYZ space. Hard, right? Exactly! Why is it so hard for us to perceive time (T) as a positional or coordinate axis? I mean, we've named X, Y, and Z! What's left? Well, since we're we're a 4-dimensional beings, why can't we perceive time as an axis on an XYZ space? Well, we'll get to that in just a moment! The simplest quick explanation I can give you is that we can't perceive a trait as an axis unless it has already been inherited at least once. Our dimension defined time, and it has not been inherited at least once. Therefore, we can perceive time indirectly by percieving the effects of T on XYZ. But being 4D beings, we can only say that the time axis is nonreal relative to our perspective. However, we know that time is ticking away because we can see the effects of time: organisms evolve, objects move, you're reading words... it's all due to time.

The best way to perceive an axis that you can't possibly comprehend is by detaching it from the graph. So while the graph of the 4th dimension is actually XYZT, we can only comprehend it as XYZ and T, T being an axis seperate from the XYZ graph.

On that note, we can percieve time as being a line. It is infinite. No endpoints. Thus, if the state of the sphere-and-circle from the last example can have any state over time, and we represent that state with a point on the T line, we can say that time hosts an infinite set of state points of the previous object. Therefore, "∞(∞^3)" or "∞^4."




[if you're reading this, I'm currently in the currently in the process of writing the rest of my leet ideology...]

Find a typo? Too bad!
- Brandon

[Aepos]

No, no, no, you have it ALL wrong, dumdum. A point in D0 cannot have a definite position, since it has no axes on which to position itself. It does not define any axes whatsoever. Without any axes, you CANNOT have a definite position. Matter of fact, you can't have a position at all. As abstract as it is, TECHNICALLY, there is no point at all, since we literally "define" a point in the fourth dimension. We cannot, in the most literal since, possibly comprehend anything external of our own dimension. Therefore we give D0 the abstract definition of being a "point", as we humans call it.

You're actually confusing dimensions there, which is what I JUST realized. When you try to plot a point in any dimension greater than the zeroeth, you have axes on which you can position the point. However, this is the only limitation broken by children of D0 on points. In D1, you can have a position, and subsequently in any dimensions beyond D1 which inherit D1's axis, X. You need only one axis to have position.

This actually leads into what I'm writing in my main post. It's very cool to think about: a point (D0) can only have a position when it is inherited by D1, since a point can only have position if it has a line (D2) on which to position itself. D1 (a line) can only have a position in D2 (a plane), since a line can only have a position if it has a plane on which to position itself. This pattern continues. You could call it a sort of dimensional postulate.

[Bipolarcat]

[Edit]Alright! I have found my error. I have been looking at this from a point of view subjective to the dimension in which I live, or rather the dimension I perceive. This brings up an interesting point (again, no pun intended): points, when perceived in D1-10, have definable position. Therefore, the traits perceivable in each dimension change as they are inherited by the next dimension. A point, when perceived as part of a line, has position in space, but when the two points that form a line, or rather direct a line, are looked at individually in the zeroth dimension, they no longer have position, because "position" is only defined by the dimensions above the object in question. An example: a point on an Cortesian two dimensional plane, say (2,3), when viewed in the second dimension, appears to have definite position, but when viewed in the third dimension, no longer can have definite position without extra traits being added to it. The new point is (2,3,∞), because the z axis has not been defined; when an extra trait is added, say a z position of 6, the point has definite position again, until it's viewed in the fourth dimension.

[Aepos]

No, Jason, if I have to take this argument to the phone, I will. The definition of a point that you are describing is the definition of a point when it is in a dimension that is greater than the dimension in which it is defined.

Of course a point could have a poisition on a line (D1), on a plane (D2), in a space (D3), over time (D4), et al.

But in D0, the dimension in which it is defined, it cannot have position, in much the same way, as I said earlier, a line cannot have position in the dimension in which IT is defined (D1). It must be inhereted by a child dimension before it can have any definition of position AT ALL.

@Bipolarcat: You're mother. Oh, and what are you doing this weekend?

[gorda001]

Woah - stuff to read.

Anyway, I'm sorry but somebody already got this idea - 150 years ago. It's called cardinality and countable/uncountable sets.

Basically, the real numbers are "more" infinite than the integers or the rationals.

We say that the natural numbers (0, 1, 2, 3, ...), the integers (..., -2, -1, 0, 1, 2, ...) and the rational numbers (1/2, 5/3, 2, ...) are all the same size (the same cardinality). This is called Aleph-null and it's written like this:


The irrational numbers (√2, √5, etc), the real numbers and complex numbers all have the same cardinality too. But this is different to the cardinality of the integers and rational numbers. It's impossible to map one natural number to every real number. The cardinality of these numbers (the "continuum") is 2^Aleph-null and is written like this:

[Aepos]

... ****it.

Well at least I think like a mid-19th-century-er. It's a nice thing to come up with on the spot, though, at least. :D

Any idea on what level this kind of thinking is tought?

[gorda001]

Certainly degree level - but that's what wikipedia is for eh? :D

[Aepos]

Coolio. Umm, +rep much?

:D

Kidding.
- Brandon

[hot_cakes]

The irrational numbers (√2, √5, etc), the real numbers and complex numbers all have the same cardinality too. But this is different to the cardinality of the integers and rational numbers. It's impossible to map one natural number to every real number.

You mean the other way around, of course :)

I find the result that Q is countable to be quite surprising. I forget the proof now, but it was quite straight forward.

Edd

[gorda001]

I find the result that Q is countable to be quite surprising. I forget the proof now, but it was quite straight forward.

Yeah it's a great proof. Lay the fractions out in a 2D grid:


Then you can go in a kind of zig-zag. 1/1 -> 2/1 -> 1/2 -> 1/3 -> 2/2 -> 3/1 ->4/1 -> 3/2 -> 2/3 -> 1/4 -> etc

Simplifying the fractions and removing the duplicates you get:
1, 2, 1/2, 1/3, 3, 4, 3/2, 2/3, 1/4, ...

Rinse and repeat ad infinitum for countable rationals!

EDIT: Wikipedia has a nice image:
http://en.wikipedia.org/wiki/File:Diagonal_argument.svg