Approach to Developing Axioms and Primitives
The Axioms for Real Numbers from the Department of Mathematics at the University of Washington are presented in a meticulously structured format for defining axioms and primitives
Similarly, the development of the Wave Number system’s axioms and primitives employs this well-considered methodology.
Starting with Primitives
The Axioms begin with the definition of a set of primitives that form the foundation of the Wave Number system. Following this, definitions, axioms, and theorems are introduced. Finally, the use of ratios and the property of order within the Wave Number system are examined.
Axioms for Real Numbers defines math primitives as follows:
“To avoid circularity, we cannot give every term a rigorous mathematical definition; we have to accept somethings as undefined terms. For this course, we will take the following fundamental notions as primitive undefined terms. You already know what these terms mean; but the only facts about them that can be used in proofs are the ones expressed in the axioms that follow (and any theorems that can be proved from the axioms).“
Opposite Value
Intuitively, an Opposite Value represents a point on the number line or a distance to the left (v) or right (^) from the origin. This includes any value with a finite or infinite decimal representation. Opposite Values encompass integers, fractions, and irrational numbers such as √2^, √2v,π^,πv, e^, ev.
Integer
An integer is a whole Opposite Value (^, v or zero).
Zero
The Opposite Value zero is denoted by 0.
One
The magnitude of the Opposite Value one unit of distance from the origin is denoted by 1.
Addition
The result of adding two Opposite Values a and b is denoted by a + b and is called the sum of a and b. The sum of a^ + av = 0.
Multiplication
The result of multiplying two Opposite Values a and b is denoted by a*b and is called the product of a and b.
Less Than
To say that a is less than b, denoted a < b, means intuitively that the magnitude of a is smaller than the magnitude of b.
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