It is the inherent nature of all things that they are a compilation of two different and distinct things. It is axiomatic that these two things are space and value. The value of any given thing being what it is, while the space is what it occupies.

It is true that, abstract or otherwise numbers are a thing, therefore they must also contain a compilation of space and value. It is an axiomatic truth that space is the labeling of quantities of dimensions. It is an axiomatic truth that value is the labeling of quantities of existence, other than dimensions.

It is an axiomatic truth that space and value exist in one of two forms. So that any given quantity of space or value is first labeled as defined or undefined. It is reasonable to say that any given number, that has had both its quantities of space and value labeled as undefined, requires no further question as to its nature. If however a given number, has had both its quantities of space and value labeled as defined, it is then necessary to further define the given quantities. That is to say what is the nature of the space and value's that are defined.

There are four axiomatic steps in the further defining of a defined quantity of space and value. First it is that, after a given quantity of space and value is labeled as defined, a symbol is given to identify the amount of quantities given. Second it is that the given amounts of defined space and value are labeled as finite or infinite. Third it is that the given amounts of defined space and value, that are finite or infinite, are labeled as fractional or whole. Fourth it is that the given amounts of defined space and value, that are finite or infinite, fractional or whole, are labeled as positive or negative.

It is the case that all forms of the defining of quantities of space and value, are from the perspective of our humanity. This then shows that there is a collection of only four kinds of numbers. That is there are numbers that possess an undefined space and an undefined value. Otherwise represented as a ( Uv + Us ). Such a number not requiring further defining. There are numbers that possess a defined value and a defined space. Otherwise represented as a ( Dv + Ds ). Such a number requiring further defining. There are numbers that possess a defined value and an undefined space. Otherwise represented as a ( Dv + Us ). There are numbers that possess an undefined value and a defined space. Otherwise represented as a ( Uv + Ds ).

It is reasonable to say that natural numbers have both their quantities of space and value labeled as defined. That is that a natural number is a ( Dv + Ds ). It is then through the process of further defining, that a natural number such as 2 is labeled as having ( 2Dv + 2Ds ). The symbol 2 then is the symbol identifying the amounts of quantities contained. It is then that the given quantities are labeled as finite. Otherwise represented as a ( 2DvF + 2DsF ). It is then that the given quantities are labeled as whole. Otherwise represented as a ( 2DvFW + 2DsFW ). It is then that a positive is assigned to the compilation of space and value, and it is so on for any natural number.

It is also the case that fractions are labeled as a ( Dv + Ds ). That is any given fraction has both its quantities of space and value labeled as defined. So that such a number as .2 is labeled as ( 2DvFF + 2DsFF ). Then a positive is assigned to the compilation of space and value. Additionally, a fractional symbol may replace the decimal symbol.

It is also the case that infinite numbers are labeled as a ( Dv + Ds ). So that such a number as 2infinite is defined as a ( 2DvIW + 2DsIW ). As well as fractional infinites, such as .2infinite. Which is labeled as ( 2DvIF + 2DsIF ). Then a positive is assigned to both compilations of space and value, and it is so on for any infinite or fractionally infinite number.

Remaining are numbers that are a ( Uv + Ds ) and numbers that are a ( Dv + Us ). Such numbers do not necessarily require further defining. As an undefined quantity of space or value composites the given number. So then such numbers can only be limitedly defined relative to the given defined quantity. If then a number possess a defined value and an undefined space, the sum is then relative to the defined value. So that such a number as ( Dv + Us ) is then a 1 relative. Otherwise represented as a 1r.

If then a number possess an undefined value and a defined space, the sum is then relative to the defined space. So that such a number as a ( Uv + Ds ) is then a zero. As no quantity of value is defined, and as one quantity of space is defined. The space of zero is clearly defined on any number line. The equation ( 1 + (-1) = 0 ) proves this in that, if zero did not occupy a defined space on the number line, then the equation would equal ( -1 ), and not zero.

It is the case in multiplication and division, that neither number given is an actual number. Not in the fashion that each symbol contains both space and value. It is that one symbol is representing a value, and that one symbol is representing a space. It is the case that in multiplication the labeling of the given symbols as space or value in a specific order is not necessary. The sum yielded is always the same.

It is the case that in division the labeling of the given symbols as space or value in a specific order changes the sum that is yielded. So that as an axiom the first given symbol is labeled as value, while the second given symbol is labeled as space.

It is then that in multiplication the given value is placed additionally into the given spaces. Then all values are added in all spaces. It is then that in division the given value is placed divisionally into all given spaces. Then all values are subtracted except one.

So that in the equation ( 2 x 0 = X ), there is a given defined value of ( 2DvFW ), that is placed additionally into the given defined space of ( Ds ). Then all values are added in all spaces. This process then yields the number 2.

Whereas the equation ( 0 x 2 = X ), there is a given undefined value of ( Uv ), that is placed additionally into the defined space of ( 2DsFW ). Then all values are added in all spaces. This process then yields the number zero.

So then in the equation ( 2 / 0 = X ), there is a defined value of ( 2DvFW ), that is placed divisionally into the defined space of ( Ds ). Then all values are subtracted except one. This process then yields the number 2.

Whereas the equation ( 0 / 2 = X ), there is an undefined value of ( Uv ), that is placed divisionally into the defined space of ( 2DsFW ). Then all values are subtracted except one. This process then yields the number zero.

As an addition to all current field axioms these ideas are expressed as stated.

" For every A in S there exists a Z1 and Z2, constituting A, such that any A in operation of multiplication or division is only representing Z1 or Z2 in any given equation. Such that Z1 for all A's other than zero equal A. Such that Z2 for all A's other than zero equal A. Such that Z1 for zero equals zero. Such that Z2 for zero equals 1. "

It is possible that further defining of the given defined value of a relative number, and the given defined space of a zero, is applicable and necessary. It is possible to either leave the same, or adapt exponents and logarithms. Naturally further axioms will be needed for adaption. Such as exponents of zero existing as a space representation of zero (z2). Logarithms of zero existing as a value representation of zero (z1).

It is possible to here-in re-address the idea of the continuum theory. If the definitions for numbers and their groups, are adapted as stated, and with further exploration into the defining of ( Dv + Us ) relative numbers, ( Uv + Ds ) zero numbers, ( Uv + Us ) undefined numbers, and their placement onto the number line. The idea here being to show all numbers originating from and returning to ( Uv + Us ) on any given number line.