Orders of Infinity

You had to know it was coming…another calculus post!

If you have absolutely no interest in calculus, then I don’t recommend reading this, it will probably just be frustrating.  I don’t really expect anyone will follow this, but I did my best in wording it; it is a difficult concept.  I might rather call it an anti-consept because it is not an established idea, but an idea of ideas of ideas of …

Here it is:

Allow me to begin with a definition: an “arithmetic dimension,” n, is an algorithm of numeric manipulation that is defined according to a series of summations or subtractions of the input from or to itself repeated to the n-th level of recursion.

Therefore, the first arithmetic dimension is addition/subtraction, the second is multiplication/division, and the third is exponentiation/root.  These are the only commonly used and defined dimensions, but there are in fact infinite arithmetic dimensions.  Consider it this way: addition is pre-defined, multiplication is the addition of the multiplicand to itself repeated the number of times indicated by the multiplier, exponentiation is the multiplication of the base by itself repeated the number of times indicated by the power.  Therefore, it is clear that the fourth arithmetic dimension is the raising of the input to the power of to itself repeated the number of times indicated by the “secondary input.”  Thus it is recursively defined: the n-th arithmetic dimension is the application of the (n-1)-th dimension using the input as both the input and the secondary input, repeated the number of times indicated by the secondary input.

I bother presenting this definition before we begin because I am not aware of its existence elsewhere.  Therefore, I will invent a notation for it: let the n-th arithmetic dimensional operation applied to an input and a secondary input be expressed as

a n: b

and read as “a dim n b.

I am interested, at present, in only the application of the addition-based side of each of the arithmetic dimensions, so this notation and reading will assume a positive based definition and only acquire a negative one if 0≤b<1 as is inherently true from the nature of arithmetic.  The concept I wish to use this for at present (though I’ve already found it has many applications beyond this concept) is that of the orders of infinity.

It is often said that there are infinite subsets of infinity.  This statement, while true, only looks at the negative-based arithmetic dimensions.  That is, we can subtract, divide, or root infinity by any finite number and get an output of infinity.  What is looked at less often is the opposite, the positive-based arithmetic dimensional operations applied infinity.  Infinity can be added to, divided by, or raised to the power of any finite number, and once again, the end result is infinity.

Of course, the calculus literate know that while all this is true–that is, while any finite operation applied to infinity outputs infinity–the qualities of the infinity outputted by these different operations vary.  That is, while infinity squared still equals infinity to the first power, the ratio between infinity squared and infinity to the first power is equal to infinity, but the ratio of infinity to the first power and infinity to the first power is equal to 1.  Therefore, when we apply any positive operation of an arithmetic dimension higher than 1 to infinity, we get a higher order of infinity that can be appreciated via other arithmetic operations.  (Notice this statement excludes the first arithmetic dimension because infinity + x, where x is finite, is still the same order of infinity.)

This is a pretty big deal considering the following:  In a very important sense, all the orders of infinity are not equal to each other–they are in fact infinitely different, but that, I will admit, is irrelevant.  The real issue in considering them equal is that it would disrupt a pattern when we start using infinite secondary inputs.  That is, ∞ -1: ∞ (also written as ∞ – ∞) is equal, not to infinity, but to zero, but ∞ 1: ∞ would be said to equal ∞.

But now consider something like ∞ ∞: ∞.  You shouldn’t be reading this sentence yet–you should still be considering.

…Ok you can go on reading now.

That above mentioned quantity is the highest arithmetic dimensional highest order of infinity.  However, the raising of such a concept introduces a second set of dimensions: We have thus far defined an arithmetic dimension relative to the use of arithmetic operations, but we might also now consider an “arithmetic dimension” its own operation which can be used, in a similar respect, to define an arithmetic dimension dimension.  I know, the terminology is silly, but it’s the most natural wording that arises.

A second “arithmetic dimension n dimensional” operation with an input of and a secondary input of 2 could be written out long hand as follows:

(a n: a) n: a

and a secondary input of 3:

((a n: a) n: a) n: a

After defining this, we could give it some sort of notation (perhaps a c: n: b), and then define the arithmetic dimension dimension dimension.  We could keep going about this to infinity, plugging the algorithm into itself, with each additional dimension requiring an additional input (though we might just default to assigning this input a value of ∞).  And then, if we really had so much time on our hands, we could begin constructing a series of that series, calling the arithmetic dimension the first in the series, the arithmetic dimension dimension the second in the series, and so on to infinity.  Then we could begin to construct a series of that series, and a series of the series of that series, and so on.  In short, there is no limit to the fractal of orders of infinity.

All that probably seemed pointless, but its not; my point is this: if one travels to a high enough order on any of these dimensions, the first order of infinity in that dimension is considered equal to zero (the trivial case is to compare ∞^1 to ∞^∞).  That concept taken and applied to the infinite sets of infinite series of recursion is a powerful thing.  It is even recursive in itself, because we could use this model of infinity to evaluate the infinite system of dimensions that we have used to arrive at the model.

Spooky.  I know.

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