Arcanum

## Introduction

An **Arcanum** is a specialized Algebraic Structure tailor made to deal in Zeroid Theories. Most Algebraic structures assume universal identity elements and orderly structures, which don't allow for Zeroids to exist, since all identity elements and annihilators are absolute.

An Arcanum provides a simplified framework that can be used for a variety of Zeroid Theories, not just one. Additional structure can be added to an arcanum to provide more structure than the arcanum provides itself. The Arcanum mainly codifies the relationships between the numbers of the physical world and the numbers of the abstract subfinite world.

An Arcanum assumes that there are two fundamental operations, addition and multiplication, and that any two elements combined with them must form yet another element that must be part of the Arcanum. It also assumes the existence of numbers whose properties violate ones deemed to be absolute, especially those in connection with zero. The Arcanum then imposes certain assumptions onto those operations and elements.

The need for an Arcanum arose from the need to have some way of organizing the principles that govern the operations of Zeroid Theory, and one that formalizes those operations without regards to their *interpretation*, thus simplifying their description. The Assumptions were chosen to specifically model the conjectured or asserted behavior of the subfinite.

The Assumptions are broken into different groupings. We'll begin with the overall definition and work our way back to the assumptions.

## Definition

An **Arcanum** is a class equipped with addition and multiplication which contains two disjoint subclasses which form a **Minor Arcana** and a **Major Arcana** with respect to the addition and multiplication operators such that (1) all exofinite numbers in the Arcanum exist within the Minor Arcana and all subfinite numbers in the Arcanum exist in the Major Arcanum (2) the Arcanum as a whole obeys all the basic, minor, major, and arcane assumptions.

A **Minor Arcana** is a non-empty class equipped with addition and multiplication which only contains **exofinite numbers** and exofinite numbers generated from those exofinite numbers and for which the basic and minor assumptions of an arcanum hold.

A **Major Arcana** is a non-empty class equipped with addition and multiplication which only contains **subfinite numbers** and subfinite numbers generated from those subfinite numbers and for which the basic and major assumptions hold.

## Basic Assumptions

**Basic Assumption I.** Addition is a binary function which is closed and commutative.

**Basic Assumption II.** Multiplication is a binary function which is closed.

**Basic Assumption III.** Left and Right Multiplication distribute over addition.

## Advanced Assumptions

### Minor Assumptions

**Advanced Assumption I.** There exists an element, called the Honorary Zeroid, also called more simply Zero, and denoted by 0.

**Advanced Assumption II.** There exist elements, called Exofinite numbers, such that adding the Honorary Zeroid to them is equivalent to performing no action on them.

**Advanced Assumption III.** The Honorary Zeroid is an Exofinite Number.

**Advanced Assumption IV.** The sum of two exofinite numbers is exofinite.

**Advanced Assumption V.** The product of two exofinite numbers is exofinite.

### Major Assumptions

**Advanced Assumption VI.** There exists numbers, called Subfinite Numbers, which are not part of the exofinite numbers.

**Advanced Assumption VII.** The sum of two subfinite numbers is subfinite.

**Advanced Assumption VIII.** The Product of two subfinite numbers is subfinite.

### Arcane Assumptions

**Advanced Assumption IX.** The Sum of an exofinite number and a subfinite number is never equal to either.

**Advanced Assumption X.** The sum of an subfinite number and an exofinite number is exofinite if and only if the exofinite number is not the honorary zeroid.

**Advanced Assumption XI.** The product of an exofinite number and a subfinite number is always a subfinite number.

## Extensions

An Arcanum may be extended by defining subtraction as the inverse of addition, defining two divisions which are the left and right inverses of multiplication, and defining an exponentiation based on left multiplication, along with an order of operations. An Arcanum extended in this way is known as an **Arcane Arithmetic**. If in addition order is imposed on the elements, besides mere equality or inequality, it becomes an **Ordered Arcanum**. If it has both extended operations and has ordered laws, then it's an **Ordered Arcane Arithmetic**.

For a simpler and more well behaved framework for Zeroid Theories, especially one that is ideal for first learning about them, see Sanctuary.