Exofinite Number

## Introduction

In Zeroid Theory, the Exofinite Numbers are a way to collectively refer to the finite, infinite, infinitesimal, and anything else which is not subfinite, in a single category. From the perspective of Zeroids, these distinctions aren't as important as the fact that all these numbers are *external to zero*. That is, they all exist at a distance of themselves from 0. The key thing is that we can not reach the zeroids by means of operating with these numbers, and in this sense they are "closed". The Exofinite numbers as a category contains everything that is considered to be a "normal number". That is, anything outside of the numbers that are proposed only in zeroid theories.

Informally an Exofinite Number is a number which is no smaller than zero. That is, an exofinite number is larger than or the same size as zero, although in some early formulations zero was excluded from the exofinite.

Because Exofinite numbers are both larger than zero and further away from 0 than 0, they are held to have certain properties that always hold.

When an object possesses these properties in Zeroid Theory it is assumed to be exofinite and therefore larger than 0 (obviously 0 is equal to and the same size as itself). For this reason, when making numbers smaller than 0 it was realized that they would have to violate these properties, or else 0 would always be smaller than them.

The most common properties are:

**I. An Exofinite Number plus Zero returns the exofinite number:**

x+0 = x

**II. An Exofinite Number minus itself returns 0:**

x - x = 0

**III. An exofinite number times Zero returns Zero:**

x\*0 = 0

**IV. An Exofinite number divided by Zero is Undefined:**

x/0 = Undefined

The first subfinite number "discovered" violates all four of these, all on account of being smaller than and/or closer to zero. The Number \_ is said to have these four properties:

I. The sum of Underscore and Zero does NOT equal underscore (and in fact the sum is larger):

\_+0 != \_

II. Subtracting underscore from itself does not result in 0 but rather underscore. It has a self-distance which itself is a subfinite number:

\_ - \_ = \_

III. The Product of Underscore and Zero is NOT equal to Zero, but rather underscore (that is underscore annihilates zero):

\_\*0 = \_

IV. Underscore divided by 0 is defined and equals underscore:

\_/0 = \_

Note that, every real number is exofinite by these 4 criteria. Every complex number is exofinite. Every finite number is exofinite.

You might think that infinitesimals are too small to be exofinite, but we also find that, despite how small infinitesimals are, subtracting them from themselves is still 0, which implies 0 is smaller than any of them.

What about mutliplying infinities by 0? These depends a bit on the model, however in Cantorian set theory there is a definition for multiplying by Zero called the Cartesian product in which every cardinal number times 0 is 0. More importantly, even in theories, like calculus, where infinity \* 0 is indeterminate, even here no one would argue that infinity + 0 = infinity can can't equal anything else. 0 does not have any effect on the infinitesimal or finite, let alone the infinite.

For the most part these 4 properties can be assumed of all these objects that are held to be larger than Zero.

Another important property that Exofinite Numbers possess is that they are not equal to their own negation. That is:

x != -x

This too is violated by zeroids and subfinite numbers. The only Exofinite Number equal to its own negative is Zero.

Although there is no largest Exofinite Number, Zero is said to be the smallest possible Exofinite Number.