****Update:
Wed Nov 05, 2025 2:44 pm
If we really look at the number line it is fundamentally the recursion of 0. There are no axioms to this system, it is premised upon the distinction of 0 thus has zero axioms.
Symbolic definitions for formalism:
“R” is the recursive sequence.
“r” is the isomorphism of the recursive sequence as number(s) for further recursive sequence. One sequence can result in several isomorphic numbers simultaneously.
The distinction of 0 as 0 is 1 number: R(0,0)r[1]
The distinction of 1 as 1 is 2 numbers: R(1,1)r[2]
the distinction of 1 as 1 as 1 is 3 numbers: R(1,1,1)r[3]
so on and so forth.
Negative numbers are the spaces between each recursive number, by degree of isomorphism, where the space is the absence of complete unity as one and zero. A negative space can be seen on a number line where the number 3 has 1 space between it and 2, 2 spaces between it and one and 3 spaces between it and 0. The absence of the negative space would effectively result in 3 being one of those numbers, thus with each number there is a relative negative space (as a negative number).
Given each negative number is a recursion of 0, the negative number is an absence that occurs between numbers and as such observes a relative void space where 0 occurs as a negative recursion (given each number is a recursive sequence). Negative recursion is recursion between recursive sequences that allow distinction of the sequences themselves by degree of contrast.
Negative recursion is isomorpnic to positive recursion. Given numbers are recursive sequences of zero positive and negative recursion are synonymous to positive and negative numbers. Negative recursion is a negative number, a negative space by default. For example if 1 is (0,0) then -1 is -(0,0).
In these respects where the standard number line extends in two directions from zero, the number line is now effectively 1 dimensional as overlayed positive and negative recursive sequences. So where 1 occurs on the number line there is no negative number as only the distinction as 1 exists, where 2 occurs there is a -1 because of the linear space between 2 and 1, at 3 there is -2 and -1 as there is a linear space between 3 and 2 and 3 and 1.
The distinction of negative sequences occurs by their isomorphic positive sequences: -1 and -2 have 1 between them, -3 and -2 has 1 between them, -3 and -1 have 2 between them. Negative recursion and positive recursion, hence negative number and positive number, are isomorphic to eachother by contrast induced distinction.
Negative recursion is simultaneously both a meta recursion and isomorphic recursion. Meta in the respect that it is recursion within recursion, isomorpnic in that as a meta-recursion it is a variation in appearance of recursion but of the same foundations.
A recursive sequence is repetition of a distinction, the foundational distinction is 0 as 1 distinction, but recursion of zero does zero become distinct.
1 leading to 2 leaves a space of -1: R(1,1)r[2,-1]
1 leading to 3 leaves a space of -2: R(1,1,1)r[3,-2]
so on and so forth.
Fractions are the ratios of numerical recursive spaces within themselves, these spaces are effectively recursive 0. Given a fraction is effectively a fractal on the number line, what a fraction is are fractal emergence of recursive sequences: a recursive sequence of zero folded upon itself through isomorphic variations of it. In these respects a fraction is equivalent to a mathematical “super positioned sequence”; over-layed sequences as a new sequence. A fraction is a process of division that is complete in itself as a finite expression, ie. 1/3 as 1/3 or 2/7 as 2/7.
In these respects an irrational number is a process of recursion that is non-finite outside its isomorphic expression as a fractional number. By these degrees, irrational numbers are recursive processes that are unfixed, they are unbounded recursion. While notions such as x/y may symbolize such states in a finite means, a number such as .126456454…334455432… still observes recursion by degree of each number in the sequence itself. In these respects the second notion observe multiple degrees of recursive sequences happening simultaneously as each number itself. An irrational number, on a number line is a fixed point regardless, where a fraction such as 2/7 cannot only be observe as a single point but spatial as both 2 and 7 simultaneously as a visual line space. In these respect the number line expresses an irrational number as two over layed recursive sequences as two over layed numbers as spaces.
The space of 1 and the space of 2, on the number line, observes the space of 2 as a fractal of one and the space of 1 as a fraction of two.
The space of 2 and the space of 3, on the number line, observes the space of 3 as a fractal of 2 and the space of 2 as a fraction of 3.
Now the number line contains within it the six degrees of arithmetic, addition/subtraction/multiplication/division/exponents/roots by degree of recursion.
The recursion of 1 as 2 is addition, same with -1 as -2: R(1,1)r[2]
Short hand example:
3+7=10 as R(3,7)r[10]
-7-3=-10 as R(-3,-7)r[-10]
The recursion of this act of addition is multiplication, where “R” stands for recursion the nested R is due to addition nesting: R((1,1)R(1,1,1))r[6] or R((2)R(3))r[6]
Shorthand example:
2×25=50 as R((2)R(25))r50
The recursion of multiplication is exponentially: where “R” stands for recursion and the number is the degree of nested multiplication:
3*3=9 as R3(3)r[9]
Subtraction is the addition of a negative space and a positive space: R((-1,)(1,1))r[1] or R((-1,2)r[1]
division is the recursion of the addition of negative spaces in a positive space, where “R” stands for recursion the nested R is due to addition nesting and the "-’ addition is to showing nested negatives as degrees of subtraction:
R((1,1,1,1,1,1)-R(1,1,1))r[2] or. R((6)-R(3))r[2]
To divide a negative number is for the negative number to occur recursively as a negative space, this is negative recursion regardless as what divides is negatve recursion within negative recursion itself. Dividing by a negative number effectively is self-embedded negative recursion.
Fractions are fundamentally that process of division, thus to observe a fraction is to observe negative recursion in the isomorphic form of the symbolic nature of the fraction itself.
Roots is the recursion of division, where “R” stands for recursion the degree of negative recursion is implied by "-’ :
2✓9=3 as -R2(9)r[3]
3✓27=3 as -R3(27)r[3]
Shorthand example:
50/2=25 as R((50)-R(2))r[25]
7/3=2 1/3 as R((7)-R(3))r[7/3]
The six modes of arithmetic are based upon addition as recursion where subtraction, division and roots are negative recursive sequences within positive recursive sequences.
A negative recursive sequence is the absence between positive recursive sequences. Number is a recursive sequence; evidenced by the number line number is recursive space. Arithmetic is fundamentally recursive addition. By degree of recursive space, all number is recursive 0 and the line is a recursive 0d point. Math is rooted in recursive “void” (0/0d point) that is distinct as 1.
Quantity is dependent upon form as quantity is dependent upon form, form is fundamentally spatial, the number line is numerical space.
Recursion terminates as the distinction of the recursive sequence as a number itself. The isomorpnkc expression of a sequence as a number allows potentially infinite recursion to terminate as isomorphic finite number. Each recursive sequence is simultaneously a set of numbers, thus a sequence is a set of numbers.
Recursion occurs recursively through isomorphism. Negative and Positive recursion observe the embedding of recursive sequences within recursive sequences isomorphically. This can be observed in positive and negative numbers, as the number lines, as well as fractions being not only self-enfolding recursive sequences but effectively the isomorphic expression of sequences between each other as a given relation.
Numerical identity is the recursion of the distinction of 0 as 1 distinction. Identity is distinction.
The composition of a number recursive distinction.
All numbers, as rooted in recursive zero, are effectively different degrees of isomorphisms from each other thus associativity is the recognition of a universal holographic state.
Proof in this meta-system is expression of distinctions as distinctions, these distinctions are the processes of recursion thus the operator “R” is not so much an operator but the embedding process as a distinction:
- Addition: R(n,n) and R(-n,-n)
- Subtraction: R(n,-n) and R(-n,n)
- Multiplication: R(nR(n)) and R(nR(-n)) and R(-nR(n)) and R(-nR(-n))
a. +++”R(R())” is Recursion of Recursion
- Division: -R(nR(n)) and -R(nR(-n)) and -R(-nR(n)) and -R(-nR(-n))
a. +++”-R(R())” is Negative Recursion of Recursion
- Exponents: Rn(n) and R-n(n) and Rn(-n) and R-n(-n)
- Roots: -Rn(n) and -R-n(n) and -Rn(-n) and -R-n(-n)
The nature of variables within Algebraic theory translates that all variables are recursive sequences that are superimposed with trans-finite or infinite other sequences until a variable is chosen. The algebraic nature of recursion by degree of the foundations of arithmetic operations being recursive sequences where said foundations are necessary for algebra to occur.
Any formalization of such a calculus would effectively fall within the function of the calculus by degree of the standard formalism being an isomorphic variation of it. All mathematical systems built upon axioms are built upon assumption thus negating, in and by degree, a fully rational expression. This system has zero-axioms as distinction is not an axiom given to assume distinction is to make the distinction of assumption. The distinction of 0 as 1 distinction observes an isomorphic foundation that is further expression by recursion.
“R” is embedded within the sequence itself, “r” is the inversion of the sequence by degree of isomorphic symbolism. “R” and “r” are not operators in the traditional sense but rather embedded distinctions.
The closure is always evident by degree of the sequence always being an expression of a distinct 0, that which it contains. 0 contains itself as a distinction by degree of its folding by recursion.
Given each number is a recursive sequence of numbers, each number within each sequence is a recursive sequence as a form of meta recursion. 1 as a distinction of (0,0) observes a recursive sequence of (.1,.1,.1,.1,.1,.1,.1,.1,.1,.1) as 1 itself where .1 as a fraction of 1 is an unfolding of 1 within itself through zero. .1 observes this same nature as (.01,.01,.01,…) and the recursion of recursion occurs infinitely.
To visualize this one can look at a line segment composed of further line segments, with each line segment following the same course.
In these respects each number is an infinite set that is finite by degree of isomorphic symbolism that grounds it by degree of a distinction. So observe “n” is to observe a holographic state of distinction, bounded by the distinction of 0, where “n” effectively is a process of distinction where the observation of a sequence is a distinction of one sequence among infinite. A number is a superimposed state of numbers thus effectively a number is equivalent to a variable in a manner that is more fundamental than what a variable is in standard algebra.
A number is a recursive sequence within a recursive sequence as a recursive sequence. In these respects “n” is a set and the recursion of “n” is a recursion of sets. Standard arithmetic, in this system, is fundamentally involved with the recursion of sets as a new set.
A sequence is always complete given its beginning and ending are founded on the recursion of 0, by recursion of 0 a sequence always contains itself thus regardless of the degree of progression the beginning and end are always the same.
All is provable within the system by degree of its nature of distinction of 0 as foundational. The system begins with the distinction of 0 and any complex expression of the system is contained as itself by degree of the expression being a distinction of 0. There are no rules beyond the system as recursive distinction is self-generating and woven throughout all formalisms.
All mathematical systems contained within this system are complete by degree of the system having no axioms beyond it while the system provides the foundations for such mathematical systems by degree of the number, by which they exist, being recursive sequences of 0. Given a mathematical system must have an unprovable assertion beyond it that cannot be proven, this system contains its proof as its structural emergence as self-referencing distinctions of 0 at all levels. In these respects math’s are complete by this system.
