Subfinite Number

## Introduction

A **Subfinite** number is a hypothetical number that is said to be smaller than *nothing* without being the negative of *something*. Instead a subfinite number is neither *something* nor *nothing,* but *subthing*. The key feature is that no combination of somethings and nothings can ever arrive us at a *subthing*. Likewise no combination of subthings and subthings can arrive us at nothing or something. Thus, the subfinite realm and the ordinary number realm are completely disjoint. The existence of one can not be proven from the other. A subfinite number is also said to be closer to zero than zero is. Subfinite numbers tend to violate properties that are taken for granted for zero, and tend to behave strangely with non-zero non-subfinite numbers, specifically under addition.

The subfinite numbers should not be confused with the micro-zeroids. Subfinite numbers and micro-zeroids are not synonymous. All micro-zeroids are subfinite numbers, because by definition they are all "smaller than zero", however not all subfinite numbers are micro-zeroids, because micro-zeroids have the additional properties of being critical and simplex. That is, micro-zeroids are required to be "zero-like" and "smaller than zero", where as subfinite numbers only need to be smaller than zero. In some theories all subfinite numbers may be micro-zeroids, in which case one can ignore the distinction, but this is not the case in all zeroid theories.

## Definition

A **Subfinite Number** is a number which is both (1) strictly "smaller" than zero and (2) strictly closer to 0 than 0 is to itself

## Philosophical Implications

The above definition is not a formal or technical one. Instead it relies on an implicit conceptualization of a notion of "both relative size" and "relative position". It defines subfinite numbers as those that have a relative size smaller than zero, and a relative position closer to 0 than 0, assuming that this can be "interpreted" conceptually without actually defining either of these concepts. The **Assumption of subfinitude** then just asserts that this makes sense and that subfinite numbers exist. An Existential Axiom for subfinites appears to be required, because unlike the alephs of set theory, whose dependence only depends on the existence of infinite sets, the subfinite numbers don't have another mathematical object that they "measure" the size of. They are purely abstractified notion of quantity, taken to a place that doesn't have a referent in reality, either in terms of an actual amount one could point to, nor as a particular point in a positional space such as a number line. It is in some sense "invisible" or "non-existent". Nothing may be nothing, but it still refers to something we can point into in reality, say the absence of a substance or item type. This is why the existence of subfinites must simply be asserted since they aren't the "cardinality" (in a loose sense) of any other mathematical object.

Since there is no formal definition for what is meant by "relative size" and "relative position" as they are concepts that rely instead of physical and experiential notions, not mathematical ones, this can not serve as a "formal definition" of the subfinite. The above definition there is by its very nature informal.

In order to have a formal definition, we need to make conjectures about the strictly *mathematical properties* a subfinite number would have based on our conceptual model of the subfinite. This will be carried out in the next section.

## Prerequisite Knowledge

The most basic properties of subfinite numbers do not need to be merely asserted but can be derived some simpler postulates based on the idea that certain notions of size that operate with known sizes generalize to subfinite sizes, and certain notions of distance that operate with known distances generalize to subfinite distances.

What we will begin with, is something we will call **The Law of Small Numbers**. Rather than simply state the law, we will first show what this law is attempting to model and what motivates its creation.

One immediate problem we run into when talking about our modern number system and some notion of "smaller than zero" numbers, is that we already have a positional relation, the *less than relation*, that simply assumes that relative position or a total order is synonymous with relative size. This we must first disabuse the reader of.

We may consider the (non-zero) real numbers as composed of two parts, its "sign", either positive or negative, and its "size", usually referred to as its magnitude in this context. The sign indicates in what direction one must head from zero to reach the number, either right or left, and the magnitude tells us how far one must go in that direction. This generalizes to other types of numbers. For example in the complex plane, the magnitude of a (non-zero) complex number is still a positive real number, but now it's "sign" is replaced with any straight-line 2-dimensional direction from the "origin" (zero). We can actually go further, as long as the number system as a normative notion of what is meant by the notion of magnitude. That is, one that is (1) represented only by positive real numbers by non-zero elements (2) whose value depends on the euclidean distance in n-space from the origin and (3) whose magnitudes obey the law of magnitudes that says the magnitude of a product is the product of the magnitudes of it's elements (4) that this property is commutative and associative. It turns out that these three conditions due hold and generalize beyond complex numbers. These 3 properties are also true of Hamilton's Quaternions, and Grave's and Cayley's Octonions. This is a consequence of the fact that the Real Numbers, Complex Numbers, Quaternions, and Octonions, are the only four *normed division algebras* that can exist (by The Hurwitz Theorem). Sedenions due not preserve the notion of distance in this way. Specifically, while one can define the "magnitude" of a sedenion as it's 16-dimensional distance from sedenionic-zero, this will be a positive real number and will represent a euclidean distance, when we multiply two sedenions the magnitude of their product is not necessarily the product of the magnitudes. Specifically there exist pairs of sedenions, which have non-zero magnitudes, yet whose product is sedenionic-zero, meaning that while the magnitude of their product is zero, the product of their magnitudes is not.

With that in mind, assume that we only work in number systems in which there is a normative notion of "distance" and "magnitude" that can assign a real value to any non-zero element of the class, and that the magnitude of the product is always equal to the product of the magnitudes. We will also assume for moment, and for simplicity, that this does not involve any infinite nor infinitesimal quantities, though it should be understood that generalizations of these exactly these sorts *are* considered in Zeroid Theories.

We may not that such systems have a certain set of numbers, we may call **Normoids**. When we multiply any number in the system by a normoid of the system the product has the same magnitude as the original number. This is because Normoids always have a magnitude of 1. Numbers whose magnitude is greater than any normoid we may call the **Large Numbers**, and numbers whose magnitude is less than any normoid we may call the **Small Numbers**. In the real number line two points, -1 and +1 are the normoids. Between them lie all the "small numbers", and outside this boundary lies all the "large numbers". In the complex number system there is the unit circle which contains all the normoids. Those complex numbers within the circle are the small complex numbers, and those outside the circle are the large complex numbers. We can continue. In the Quaternions the normoids are the quaternions lying on the unit glome of the 4 dimensional quaternion-space. Those points bounded by the glome are small quaternions, and those outside the glome are the large quaternions. Lastly the normoids of the octonions are the points on the unit 7-sphere (a 7-dimensional hypersurface embedded in a euclidean 8 dimensional space) are the normoids. Points bounded by the the unit 7-sphere are the small octonions, and points lying outside it are the large octonions.

Note that in all of these cases, the 0 of that space is by definition a "small number". Also in all of these systems we may note that the normoids do not effect the size of the product, the large numbers of that set make the product "larger", and the small numbers make the product "smaller".

Let |z| denote that magnitude of a number z (in any normed division algebra). Let left multiplication be treated as a function applied to the right term. Let N be a normoid, L be a large number and s be a small number. We may codify the above stated generalizations as follows:

From these statements we can easily derive the **Law of Large Numbers** and the **Law of Small Numbers**. The Law of Large Numbers (not to be confused with the Law of Large Numbers in statistics) states that the product of two large numbers is large. This follows from the fact that left multiplication by a large number makes the product larger than the right element, which is already large. Likewise the Law of Small Numbers states that the product of two small numbers is small. This follows from the fact that left multiplication by a small number makes the product smaller than the right element which is already small. Therefore, if we were to take our algebra and split it into two subsets, the large numbers and small numbers, both sets would be closed under multiplication.

*"The Product of Two small numbers is no larger than the smaller of the two"*

From this we derive the First and Second Law of the Subfinite in the next section.

## Subfinite Magnitudes

A hypothetical number which is *smaller than zero*, or *closer to zero than zero*, must be a small number itself and "exist" (or fail to exist in this case) within the "orbit" of the normoids.

Assume The Law of Bonded Smallness applies. A small number attempts to "shrink" the number it is operating on. Zero is no different. Let \_ be an arbitrary subfinite number (which may or may not be critical).

Now consider the product 0\*\_

because 0 is a small number it attempts to result in a number *smaller than* \_. That is it attempts to "shrink" \_. But since \_ is already smaller than 0, the product must be shrink to something smaller or the same size as \_, thus we have that the product of 0 and a subfinite number is subfinite! If the subfinite number, \_, is critical, then the product will be \_, because it can be made no smaller by a larger number (just like zero). If on the other hand it is non-critical, it may be scaled by 0 and will result in a yet smaller subfinite number.

Note that, if for any subfinite number, s, the product 0s is smaller than s, we find that multiplication is non-associative. Because

0(0s) is smaller than 0s

yet (0\*0)s = 0s

We can avoid this by assuming subfinites obey a scalar law and have a bilinear product. That is, the product of non-subfinite numbers, x and y, with subfinites \_ and .. have the following properties:

x(y\*\_) = (xy)\*\_

(x\*\_)(y\*..) = (xy)(\_..)

We should also note that for non-zero small numbers, left-multiplying them by a subfinite number should always result in a smaller or same size subfinite number. If we multiply a subfinite number by a normoid we may assume that the result is a subfinite number of the same size. It also follows from the Law of Bounded Smallness, that the product of two subfinite numbers must be subfinite. This only leaves what happens when we multiply a subfinite number by a large number.

For this we observe that any Large number must be larger than 0, therefore it must be a "something". Can a something times a *subthing* yield a something? Let's say there was such a thing, call it x. Let .. be an arbitrary subfinite number, not necessarily critical or simplex.

Then we have x.. = y

where y is something. By definition then y/.. = x but also y/x = ..

However since y and x are both something, and we stated in the beginning any combination of somethings and nothing can never produce a *subthing*, we have a contradiction. This also suggests 0/.. is undefined, since there is no thing, nor subthing which one could multiply .. by to get 0. We can say 0 is incommensurate with ..

## Subfinite Distances

What happens if we had a subfinite number to an non-subfinite number? Here we can a notion of distance, ie. the idea of being a subfinite distance away from an ordinary number. Such numbers are sometimes called mixfinite numbers, and they have to be handled with care in any arcane theory.

Our prototypical example will be 1+\_. If we take the magnitude view, we may assume that since 1+0=1, this is because 0 has a *negligible effect* on 1, and since \_ is subfinite and smaller than 0, therefore it must also be *neglible* and therefore 1+\_=1. This is a little like how infinitesimals are ignored in non-standard analysis. However \_ is not even infinitesimal. Instead we note that 1+0=1 because 1 has a **self-distance** of 0. That is, 1+0=1 actually because 1-1=0. Since \_ is closer to 0 than 0 is to itself, 1+\_ is said to be closer to 1 than it is to itself. We now are a distance towards 1 that is smaller than 0 in size, which effectively brings us closer to 1 than 1. 1+\_ presents certain technical problems. On a most basic level, even though 1+\_ clearly appears to be larger than 0, it's size does not appear comparable with other numbers. Is for example 1+\_ larger than 1? No because that would suggest we *added something* *positive* to make 1 larger. Also it's worth noting that 1+\_ = 1+(-\_) = 1-\_. So it can be thought as being subtracted as well. In this form we don't think to think of it as an \_ distance from 1, but rather as the total distance from \_ to 1. This distance can not be longer than the distance from 0 to 1 nor shorter, since \_ does not even exist on the number line. Some weird properties of 1+\_ include its produces with other non-subfinite numbers:

(1+\_)\*2 = 1\*2 + \_\*2 = 2+\_

(1+\_)\*0 = 1\*0+\_\*0 = 0+\_

(1+\_)\*(-1) = -1+\_

(3+\_)+(2+\_) = 5+\_

(1+\_)+0 = 1+\_

(2+\_)(3+\_) = (2+\_)3+(2+\_)\_ = 2\*3+\_\*3+2\*\_+\_\*\_ = 6+\_+\_+\_ = 6+\_

We get essentially a perfect copy of the exofinite numbers, with all the members closer to the respective counter-parts then the counter-parts are to themselves. If we create a minor arcana from these they are closed among themselves since (x+\_)+(y+\_) = (x+y)+\_ and (x+\_)(y+\_) = (xy)+\_, just as the exofinite numbers. If we combine an exofinite number with these "mixfinites" we just get the mixfinites, suggesting they are "stronger" in some sense. They have an absolute priority.