Geometric Maths - Axioms and Definitions

Geometric Math introduces a novel approach to mathematical foundations, utilising the Universal set (I) and the Real number plane (Я²). The axioms establish equilibrium through the division of a dot, leading to foundational units (U) and the exploration of powers and dimensionality.

The Infinite Density Function reveals the reciprocal relationships between the Real and Communicative number planes. This revolutionary framework not only redefines numbers but also shows promising applications in areas such as compression algorithms, internet security, and advanced AI. Geometric Math has the potential to reshape mathematical thinking and address challenges in modern science.

Overview

Geometric Math pioneers a transformative approach to mathematics, redefining fundamental concepts like addition, subtraction, multiplication, and division. Rooted in the Universal set (I) and the Real number plane (ЯR), composed from real numbers between 0 and 1 and real numbers above 1 to infinity, referred to as reciprocal and whole number spaces. By dividing a dot which resides at the centre of infinity, creating points A and B. This initiates a dynamic interplay between the predecessor (P) and successor (S) of the whole numbers. Mapped into reciprocal space lays the groundwork for the Infinite Density Function, which can qualify the density of a bounded infinite set. Plus and minus operations gain new significance as they arise out of the division of zero, forming ±Я, into which the rich tapestry of interconnected numerical relationships is revealed. Geometric Math transcends traditional paradigms, offering a fresh perspective on foundational concepts and revealing profound implications for the future of mathematical theory and its real-world applications.

KEy Points

• Redefinition of Fundamental Operations: Geometric Math revolutionizes basic arithmetic operations, such as addition, subtraction, multiplication, and division, by grounding them in the Universal set (I) and the Reciprocal number space (Я). This transformative approach challenges conventional mathematical paradigms.
• Dynamic Interplay of Numbers: The division of a dot at the center of infinity into points A and B initiates a dynamic interaction between the precession and succession of numbers. This novel concept creates a rich and interconnected numerical framework, redefining the way numbers evolve and relate to one another.
• Infinite Density Function: Geometric Math introduces the Infinite Density Function, unveiling profound insights into numerical relationships. By exploring equilibrium, plus, minus, and division operations, it offers a fresh perspective on mathematical theory with far-reaching implications for real-world applications.

Abstract

This theoretical framework explores the foundational concepts of numbers within an infinite set, presenting a unique perspective on the nature of numbers, their relationships, and the inherent structure of mathematical space. The framework is centered around the notion of a fundamental set (I) that encompasses all conceivable entities regarded as numbers. At the heart of this set lies a central point (P), facilitating the operation of continuous division and the creation of unique polarities (P+ = +1 and P− = -1).

The framework elucidates the arithmetic manifestation of polarity and equality, laying the groundwork for the nature of addition and subtraction. Additionally, the formation of a unit measure (U) arises, which is a variable set comprised of 2 points on the number line with an infinite number space exhibited in between. In terms of ±Я the unit measure ±1 with 0 places at the center.

Through continuous division of reciprocal space (±Я), fractions are summed and the limit addressed as a finite value. For each  ±nЯ = the predecessor of ±Я is expressed as the where the infinite set diverges. For example, 2Я = 1, and 3Я = ½, and 4Я=⅓.

The convergent sum of reciprocal powers of Я² introduces the infinite density ratio, which defines the successor of Я. For example, 2Я² = ⅓ and so 2R² divided by 2R is equal to ⅓ ÷ 1 = ⅓ the successor of ½, found in 2Я. In the next example 3Я = ½ and 3Я² = ⅛. Therefore ⅛ ÷ ½ = ¼, which is the successor of Я. We can express this as a simple function (∞D(nЯ²:nЯ)) compares the convergence of 2R to 2R², contribute to the distinctive characteristics of the framework.

This framework proposes a novel perspective on the set of natural numbers (N), suggesting that each iteration in the reciprocal sequence (1Я, 2Я, 3Я, ...), and the notion of powers, (Я², 2Я², 3Я²..), formulates a methodology through which, the successor (S) and predecessor (P) of a unit can be establishes, leading to a unique understanding of infinite density ratios.

This exploration bridges traditional mathematical concepts with novel insights, challenging conventional notions and inviting further discourse on the nature of numbers and their interconnections within an infinite set.

Definitions

1. Universal Set (mathbbI): The set comprising all entities considered as numbers. Note that the set itself is considered to be a geometric construct, expanding to infinite dimension.
2. Center Point (mathbbP): A point situated at the center of the Universal Set I. Note that a point places in an infinite space will always be at the centre of that infinite space.
3. Division Operation: The operation that, for any point P in the Universal Set, results in the creation of two unique points. This defines the nature of 0÷2=+1 and -1.
4. Polarity and Arithmetic Manifestation: The introduction of the concept of polarity through the division operation. Points +P and -P  manifest positive (+) and negative (-) polarities, signifying arithmetic properties. With ±P we can construct the entire number line. However, in Geometric Maths, the process of division into 2 continues to establish the predecessor function which in this case is 2Я=1.
5. Equality: Mathematical equality established through equal distances within the Universal Set. The notion of equals is maintained as (+1) + (-1) = 0 and (-1) + (+1) = 0
6. Unit Measure Creation: The outcome of the division operation, creating a unit measure that represents a reference distance or object in space. The Unit measure can be considered as a variable set, which is formed from 2 points, with an infinite numbers space situated between them. In most cases a number is therefore represented from zero to the point of measurement, for example, the number 3 is measure from 0 to 3 with an infinite number space in between.
7. Asymptotic Approach to P: Continuous division defining the asymptotic approach to the center point P within the Universal Set. Division never exceeds the point of zero, it only approaches zero at different rates. However each division can be added up to form a bounded infinite set.
8. Continuous Division of Line overlineAB : Continuous division resulting in the subdivision of line overlineAB into infinitesimally smaller parts, defines the structure of all whole numbers (N) in reciprocal space (Я).
9. Reciprocal Number Space (Я): The infinite division of the line into whole number fractions nЯ establishes whole numbers (R) greater than 1, as x=1/x are equivalent. for every x there is always a 1/x, for all numbers greater than 1 including real numbers. Thus we can show that the carnality of Я is equivalent to R. We define this notion as ЯR. This symbol demotes the reciprocal nature of Я and R.
10. Eqaulity of +ЯR and -ЯR: Just as the unit measure +U1 can be divided to create +Я and +R, so the same can be applied to -Я and -R. Thus the structure of number space is divided into 4 equivalant portions. +Я, +R, -Я, and -R, each of which is equivalent.
11. Square Root and Reciprocal Powers: The square and root operations involving the extraction of two equal points, either multiplied or divided. As +Я and -Я are equivalent, +Я² will reflect in -Я², thus the ratio also maintains polarity. Thus in Geometric Maths, the concept of negative squares and powers is introduced.
12. Divergence of Reciprocal Powers and Roots: For any real number in ±Я, the application of positive and negative reciprocal powers, denoted as +Я^∞ and -Я^∞, respectively, approaches P, or more exactly, zero. Inversely, the root of any real number in ±R approaches ±P or ±1. Thus all roots and powers are contained within reciprocal space ±Я.
13. Infinite powers: Any real number can be raised to an infinite power, or root.  
14. Infinite Density Ratio: The infinite density function, denoted as ∞D:(nЯ²:nЯ), expresses the ratio of infinite densities between infinite sets. For example where the line is infinitely divided into 2,( 2Я) the results when added together converge to 1, whereas Я^2 converges to ⅓, which thus exhibits a density ratio of 1:3.
15. Infinite Density Predecessor Function The infinite density predecessor function, denoted as Pred(x), is found through continuous division of a line into parts, whereby, each smaller part is added together to converge towards the previous whole fraction, [katex]Pred(nЯ)=nЯ/nЯ-1[/katex]
16. Infinite Density Successor Function: The infinite density successor function, denoted as Succ(nЯ), is derived from the infinite density ratio between nЯ² divided by nR. This function establishes the Infinite Density Ratio that defines the successor of nЯ. Succ(n)= fracnЯ^2nR
17. Extending the unit measure: To the notation of the Infinite Density, we can add the subscript U to identify the unit measure so that: Я^X_u, where n is the divisor reiterated into infinity, Я is the convergent value when each part is added together, x is the power, which may also appear in the root position, i.e: and U is the unit measure. In most cases the unit measure is from 0 to 1, but when we extend the number line the infinite density of number decreases. Therefore when U=2, the 1st division of 2Я produces the result of 1, instead of 0.5, as it the case when U=1. Thus by tracking the rate of decent towards zero or by adding up each part to create Я, so the ratio or value of Я can be established. for example 2Я₁ = 1 whereas 2Я₂ = 1+1=2. Dividing 2Я₂ by 2Я₁, results in the ratio of ½.

Axioms

1. Universal Set (I): Let I be the set of all entities that can be considered numbers.
2. Center Point (P): There exists a point P at the center of I.
3. Division Operation: For any point P in I, the division operation results in the creation of two unique points mathbb+P and mathbb-P.
4. Polarity: The division introduces the concept of polarity. The points +P and -P represent positive (+) and negative (-) polarities, respectively.
5. Addition and Subtraction: Polarity gives rise to the concept of addition and division.
6. Equality: Equal distance produces mathematical equality (=).
7. Unit Measure (U): The division of zero into 2 results in a unit measure composed of 2 points with an infinite number space between, which zero at its centre.
8. Variable set: The Unit measure is a variable set whose start an end points can be altered. a whole number unit is thus considered as a unit measure between zero and the number (n).
9. Asymptotic Approach to zero, P: Continuous division defines the asymptotic approach to P within I.
10. Continuous Division of Line overlinemathbbAB: Continuous division results in the subdivision of overlineAB into infinitely smaller parts.
11. Reciprocal Number Space (Я): The infinite division of the line, defines whole numbers in reciprocal number space in both +Я and -Я. As 1/x is equivalent to x, for any real number above 1.
12. Roots and  Powers: the extraction of one of the two points. Additionally, the reciprocal powers exhibit behavior similar to both squaring and rooting, depending on the polarity.
13. Divergence of Reciprocal Powers: For any real number n in Я, repeated power operations converges towards zero, (Я^nrightarrow 0). Repeated root operations in Я converge on one (n√Я rightarrow 1). Higher root and powers diverge more quickly towards zero and one respectively.

Proofs

1: The point (P) sits at the centre of the infinite plane.

Theorem: For any point A placed on an infinite plane, Point A is at the center of the infinite plane.

Proof: Consider an infinite plane denoted as P, extending infinitely in all directions. Let A be an arbitrary point on plane P. To show that A is at the center of P, we will demonstrate symmetry.

For any vector v originating from point A, there exists an equal and opposite vector -v, such that the extension of plane P is infinite in both directions along v and -v. Since this holds true for any vector v, the plane is symmetric with respect to point A.

Therefore, by the principle of symmetry, we conclude that point A is at the center of the infinite plane P.

Q.E.D. (Quod Erat Demonstrandum): The theorem is proven.

2: Division into 2 equal parts.

Theorem: If a line segment AB is divided into two equal parts, say AC and CB, then AC is equal to CB, and AC is different from AB.

Proof: Consider a line segment AB, and let C be the point where AB is divided into two equal parts, AC and CB.

1. Equal Lengths: By the definition of equal division, AC and CB are of the same length. Mathematically, AC = CB.

2. Inequality: To show that AC is different from AB, assume the opposite. If AC = AB, then combining this with AC = CB implies CB = AB, violating the principle of division into two equal parts. Therefore, AC must be different from AB.

Q.E.D. (Quod Erat Demonstrandum): The theorem is proven. The division of a line into two equal parts results in two segments of equal length, and one of them is different from the original line segment.

3: Point is Different from a Line.

Theorem: A point is different from a line.

Proof: A point is a geometric entity with no dimension, whereas a line is a one-dimensional object. Therefore, a point and a line are distinct entities.

Q.E.D.

4: Non-polarity of zero

Theorem: A point at the centre of an infinite space is non-polar.

Proof:

1. Assume there exists a point �P at the center of an infinite space represented by the line ��AB.
2. Since �P is at the center, there is no reference point within the infinite space to establish polarity.
3. Polarity requires a comparison with other points, but within an infinite space, there are no distinct reference points for comparison.
4. Without external points of comparison, �P lacks directional bias or inherent positive/negative attributes.
5. Thus, �P is non-polar within the infinite space, embodying the neutral reference characteristic of zero.

Q.E.D.

5: Polarity of all other numbers

Theorem: All numbers, except zero, exhibit polarity.

1. Consider any non-zero number �n in the infinite space.
2. If �n is positive, it means that �n is in one direction away from the center point.
3. If �n is negative, it means that �n is in the opposite direction away from the center point.
4. The polarity of �n is established by its directional placement relative to the center point.
5. Thus, any non-zero number �n exhibits polarity based on its position within the infinite space.

Q.E.D.

6: Division of a point

Theorem: If a point is divided into two points, then it creates two distinct points.

Proof: Let P be a point. When we divide P into two points, let's call them Q and R. Since the division results in two distinct entities, Q and R are unique points. Therefore, if a point is divided into two, it creates two distinct points.

Q.E.D.

7: Division of Zero

Theorem: Dividing zero into two equal parts results in a positive one and a negative one.

Proof:

1. Initial Division: Consider dividing zero into two equal parts, denoted as P_1 and P_2. frac02= P_1 and P_1
2. Equality and Polarity Creation: Due to the equal division, P_1 and P_2 are equal in magnitude but opposite in polarity. This introduces the concept of polarity. P_1 = P_2
3. Polarity Manifestation: Recognize that P_1 can be assigned the value of positive one (+1) and P_2 can be assigned the value of negative one (-1). P_1 = +1 and P_2 = -1
4. Resulting Equation: Substituting these values back into the initial division equation yields the desired result. 0÷2= +1 & −1.

Q.E.D.

8: Polarity is a prerequisite for addition and subtraction.

Theorem: Addition and subtraction require at least the existence of plus one and minus one, signifying polarity.

Proof:

1. Start with the concept of addition. For any number �n, adding zero doesn't alter the value, but adding 11 increments it positively, and adding −1−1 decrements it negatively.
2. Now, consider subtraction. Subtracting zero doesn't change the value, but subtracting 11 decreases it, and subtracting −1−1 increases it.
3. The presence of both 11 and −1−1 allows for positive and negative changes in value, introducing the concept of polarity.
4. Therefore, addition and subtraction operations inherently involve the existence of at least plus one and minus one, signifying polarity.

Q.E.D

9: Division precedes addition and subtraction

Theorem: Division, as demonstrated by the division of zero into two points with polarity, precedes the concepts of addition and subtraction.

Proof:

1. Division of Zero: As established in the previous proof, dividing zero into two equal parts (P_1 and P_2) results in the creation of polarity, with P_1 representing positive one +1 and P_2representing negative one (-1). frac02=P_1 and P_1.
2. Polarity Introduction: The act of division introduces the concept of polarity, signifying the presence of both positive and negative values.
3. Foundation for Addition and Subtraction: Since division is shown to create polarity, it forms the foundational concept that leads to the introduction of addition and subtraction. In the established example,P_1 and P_2 can be combined (P_1 + P_2), representing the addition of positive and negative values, and subtracted (P_1 - P_2), representing the subtraction of such values.
4. Conclusion: Therefore, division, as exemplified by the division of zero, serves as a precursor to the concepts of addition and subtraction.

Q.E.D.

10: Two Points Create a Line

Theorem: Given two distinct points, they determine a unique line.

Proof: Let A and B be two distinct points. Draw a line connecting A and B. This line is uniquely determined by the two points.

Q.E.D.

11: Line Can Be Divided Infinitely

Theorem: A line segment can be divided into an infinite number of smaller parts.

Proof:

1. Assumption: Consider a line segment AB, the space between two points.

2. Arbitrary Division: Arbitrarily choose any two distinct points on the line segment, say P and Q, where P is closer to A and Q is closer to B.

3. Infinite Subdivision: No matter how close P and Q are chosen, the line segment PQ also contains an infinite number of points.This process can be repeated infinitely within any smaller segment.

4. Iteration to Infinity: For any subsegment chosen within the original line segment, the process of selecting two points, subdividing, and finding infinite points can be iterated ad infinitum.

5. Conclusion: Therefore, irrespective of the initial choice of points, any line segment AB contains an infinite number of points within its bounds.

QED (Quod Erat Demonstrandum): Thus, it has been demonstrated that any line segment, regardless of its length, contains an infinite number of points.

12: Infinite Divisibility of a Line AND NUMBER REPRESENTATION

Theorem: Infinite Divisibility of a Line Corresponds to All Possible Numbers

Proof:

1. Assumption: Suppose there exists a point, (a number), on the line that cannot be represented by the infinite division of the line.

2. Infinite Divisibility: Given the infinite divisibility of the line, any point on the line can be expressed as a result of the infinite division into smaller parts.

3. Contradiction: The assumption of a point not being representable contradicts the nature of infinite divisibility, as every conceivable point is a result of this process.

4. Completeness of Division: The infinite division of the line covers every possible point, leaving no gaps or unrepresented numbers.

5. Conclusion: The impossibility of a point being excluded from representation confirms that the infinite division of the line corresponds to the entirety of possible numbers.

QED: Thus, the proof affirms that any point on the line, representing any number, is inclusively covered by the infinite divisibility of the line, demonstrating the completeness of representation through division.

 

13: Cardinality Between Positive and Negative Real Numbers

Theorem: Unity of Cardinality Between Positive (+ЯR) and Negative Real Numbers (-ЯR).

Visual Proof:

1. Setup: Divide a page into two halves: left side for negative numbers, right side for positive numbers.

2. Representation: Place negative numbers on the left, positive numbers on the right, with zero as the center.

3. Folding Test: Fold the page along the center line (zero), observing that each positive number aligns with its negative counterpart.

QED: The visual proof demonstrates the unity of cardinality between positive and negative real numbers by folding the page and revealing a one-to-one correspondence, with zero as the point of unity, demonstration, establishing the equivalence of numbers in both spaces.

14: Cardinality between Reciprocal and Whole Number Space

Theorem: Unity of Cardinality Between reciprocal (±Я) and Negative Real (±R) number spaces.

Visual Proof:

1. Setup: Consider a page representing real numbers, divided into two equal spaces, with the first space denoted as Я (reciprocal space) and the second space denoted as R (whole number space).
2. Observation:
   Acknowledge that each point in R corresponds to an equivalent point in Я, establishing the 1:1 correspondence between reciprocal space and whole number space. i.e x=1/x.
3. Folding Demonstration:
   Illustrate the folding of the number page, emphasizing that the spaces Я and R align perfectly, demonstrating the equivalence of numbers in both spaces.
4. Mutual Representation:
   Highlight that every real number in Я has a reciprocal representation in R, and vice versa, showcasing a mutual representation between the two spaces.

QED: The visual proof demonstrates the unity of cardinality between reciprocal space (Я) and whole number space (R) through the folding demonstration, establishing the equivalence of numbers in both spaces.

15: Quadratic nature of numbers

Theorem: Any calculation which results in a value in +R, -R, +Я, or -Я  Will have an equivalent answer in the other three quadrants. 

Visual Proof:

1. Foldability Property: If a page representing the unified number space structure can be folded in such a way that each quadrant aligns perfectly, preserving its proportions, then the spaces are equivalent.

2. Calculation Equivalence: For any calculation performed in one quadrant, there exists an equivalent calculation in each of the other three quadrants, ensuring a one-to-one correspondence.

Conclusion (QED):

• The foldability property guarantees the structural equivalence of the quadrants, and the calculation equivalence ensures that any result obtained in one quadrant has a corresponding equivalent result in the other three, affirming the unified nature of the number space.

16: Convergence of reciprocal multiplication towards Zero.

Theorem: If a number  in reciprocal space Я is continuously multiplied by another number Яs in reciprocal space, Я×Я_1×Я_2×Я_3…, the result approaches zero.

Proof:

1. Let �r be any positive number in reciprocal space (�<1r<1), and let �s be another positive number in reciprocal space (�<1s<1).

   1. Recursive Multiplication: Consider the sequence r, r_1, r_2, r_3where each term is obtained by multiplying the previous term by �s.

   2. Divergence Calculation: Express the sequence as a recursive formula: ��=�×��an=r×sn, where �n represents the number of iterations.

   3. Limit Analysis: Evaluate the limit as �n approaches infinity: lim⁡�→∞�×��=0limn→∞r×sn=0

      The limit tends towards zero as �n grows indefinitely.

   (QED): This mathematical proof shows that the continuous multiplication of a number in reciprocal space by another number in reciprocal space leads to a sequence that converges towards zero.

17: Convergence of reciprocal division towards one.

Theorem: If a number �r in reciprocal space is continuously divided by another number �s=>r in reciprocal space ( �/�/�/�/…r/s/s/s/… ), the result approaches one.

Proof:

Let �r be any positive number in reciprocal space (�<1r <1), and let �s be another positive number greater than r in reciprocal space (�<1s = <1 =>r).

1. Recursive Division: Consider the sequence dfracrs,dfracrs_1,dfracr s_2,dfracrs_3... , where each term is obtained by dividing the previous term by �s.
2. Convergence Calculation: Express the sequence as a recursive formula: a_n=dfracRs^n=1, where �n represents the number of iterations.
3. Limit Analysis: Evaluate the limit as �n approaches infinity: lim _nrightarrow infty a_n=dfracRs^n=1

   The limit converges towards one as �n grows indefinitely.

(QED): Continuous division of a number in reciprocal space by another number in reciprocal space leads to a sequence that converges towards one, providing a mathematical proof of the convergence towards one in reciprocal space.

18: Convergence of number Division greater than 1 towards one.

Theorem: If a  number greater than 1 = �R is continuously divided by a value s = >1 <=R, (for example dfracRs,dfracRs_1,dfracR s_2,dfracRs_3 ), the result approaches one.

Proof:

Let �r be any positive whole number, and let �s be another positive whole number.

1. Recursive Division: Consider the sequence dfracrs,dfracrs_1,dfracr s_2,dfracrs_3... , where each term is obtained by dividing the previous term by �s.

2. Convergence Calculation: Express the sequence as a recursive formula: a_n=dfracRs^n, where �n represents the number of iterations.

3. Limit Analysis: Evaluate the limit as �n approaches infinity: lim _nrightarrow infty a_n=dfracRs^n=1

4. The limit converges towards one as �n grows indefinitely.

(QED): Continuous division of a number (R) greater than 1 by another number (s) great than 1 but equal or greater to R leads to a sequence that converges towards one, providing a mathematical proof of the convergence towards one in whole number space.

19: Divergence of Whole number multiplication towards infinity.

Theorem: If two whole numbers �a and �b are continuously multiplied (�⋅�⋅�⋅�⋅…a⋅b⋅b⋅b⋅…), the result diverges towards infinity.

Proof:

Let �a and �b be any positive whole numbers.

1. Recursive Multiplication: Consider the sequence �,�⋅�,�⋅�2,�⋅�3,…a,a⋅b,a⋅b2,a⋅b3,…, where each term is obtained by multiplying the previous term by �b.

2. Divergence Calculation: Express the sequence as a recursive formula: ��=�⋅��an=a⋅bn, where �n represents the number of iterations.

3. Limit Analysis: Evaluate the limit as �n approaches infinity: lim⁡�→∞(�⋅��)=∞limn→∞(a⋅bn)=∞

   The limit diverges towards infinity as �n grows indefinitely.

QED: Continuous multiplication of two whole numbers leads to a sequence that diverges towards infinity, providing a mathematical proof of the divergence towards infinity in whole number space.

20: Nonexistence of Unbounded Infinity

Theorem: Unbounded infinity does not exist within the framework of reciprocal real (Я) and real (R) number spaces.

Proof:

1. Reciprocal Space (0 to 1): In reciprocal space, as numbers approach 0, the reciprocal process (1/�1/x) diverges towards infinity. However, the process is confined within the bounds of the unit interval (0 to 1).

2. Whole Number Space (1 to Infinity): Multiplication whole number space, increases towards infinity, which produces a reciprocal that converges at zero. The process appears unbounded in the positive direction, yet is revealed as being bounded through the examination of reciprocal number space.

3. Mirror Reflection: The reciprocal and multiplication processes are mirror reflections across the boundary between number above and below one. The divergence towards infinity in reciprocal space corresponds to the convergence towards 0 in whole number space.

QED: The reciprocal space is confined to the unit interval, whose behaviour is mirrored in the multiplication of numbers in whole number space. The existence of reciprocal and whole number spaces, with their mirror reflection properties, implies that unbounded infinity does not exist within this mathematical framework. 

20: Unity of Cardinality whilst Changing Infinite Density

Theorem: Cardinality can remain consistent while infinite density changes.

Geometric Proof:

1. Consider two concentric circles, A (inner circle) and B (outer circle), with each point on their circumferences representing real numbers.
2. Draw a straight line from the center that intersects both circles, creating a continuous path connecting points on A to corresponding points on B.
3. As we move along this straight line, points on A (inner circle) align with points on B (outer circle), ensuring a constant cardinality.
4. While cardinality remains consistent, the density of infinity changes due to the larger circumference of the outer circle, making the points on B more spread out.

QED: The geometric example of two concentric circles and a connecting straight line demonstrates the unity of cardinality while showcasing the change in infinite density. As the line connects points from the inner circle to the outer circle, the constant correspondence maintains cardinality, while the larger circumference of the outer circle alters the density of infinity.

21: Quantifying Infinite Density

Theorem: The density ratio (infty D_r) can be expressed as infty D_r=dfrac1Delta r.

Definitions:

1. r_a : Radius of the inner circle (Circle A).
2. r_a : Radius of the outer circle (Circle B).
3. Delta r : Difference in radii Delta r=r_b-r_a.
4. Density is the concentration of points or values within a given space.

Geometric Proof:

1. Consider N_a as the number of points on the circumference of Circle A.
2. Correspondingly, N_b is the number of points on the circumference of Circle B.
3. Since there is a one-to-one correspondence, N_a=N_b 
4. Density is inversely proportional to the difference in radius: infty D_r=dfrac1Delta r

Visual Representation:

• Graphical representation can visually validate the impact of Delta r on density.
• As Delta r increases, (infty D_r) decreases, indicating a lower concentration of points.

QED: The mathematical proof establishes that the density ratio (infty D_r)  precisely quantifies the change in infinite density based on the difference in radii.

22: Consistency of Infinite Density while changing Cardinality

Theorem: Consistency of infinite density can be maintained whilst the cardinality of an infinite set changes.

Proof:

1. Example: 2R Division: Consider the line continuously divided into 2 (2Я), resulting in the series: ½,¼,⅛...
2. Summation of Parts: Summing up the consecutive parts: ½ + ¼ + ⅛... = 1
3. Implication:The sum of parts equals the unit measure from 0 to 1.
4. Density Consistency: Infinite density remains consistent yet the cardniality between whole number fractions of 2Я differs.

QED: The proof demonstrates that with consistent infinite density, cardinality can change.

23: Consistency of Cardinality whilst changing infinite density.

Theorem: Consistency of cardinality can be maintained whilst changing infinite density.

Proof:

1. Basic Definition: nЯ_u=dfrac1n-1 for nin mathbbN
2. Density Expansion with U:nЯ_nrightarrow Convergence Value as Urightarrow ∞
3. Generalized Pattern: nЯ_u=dfrac1n-1,dfrac2n-1,dfrac3n-1 … as Urightarrow infty .
4. Density Expansion Rate: lim _urightarrow infty nЯ_urightarrow Convergence Value + dfrac1n-1 for each increment of U.
5. Convergence Patterns:
   • 2Я_U: dfrac11,dfrac21,dfrac31 ...(Convergence Value = 1)
   • 3Я_U: dfrac12,dfrac22,dfrac32 ...(Convergence Value = ½)
   • 4Я_U: dfrac13,dfrac23,dfrac33 ...(Convergence Value = ⅓)

QED: The proof demonstrates that even with consistent infinite density, the cardinality changes.

24: Unity Between Infinite Density at F=∞ and Infinite powers at F=1

Theorem: nЯ where effort (F) ranges from 1 to ∞ (F=∞) exhibits the same infinite density as nЯ^x where effort = 1 (F=1) and x ranges from 1 to infinity (x=∞).

Proof:

1. Equation 1 ( nЯ^x=1_F=∞ = (n-1): The convergent series of 2Я=dfrac12+dfrac14+dfrac18, which sums to 1. Thus, the convergence of 2Я leads to a cardinality and infinite density equal to 1.
2. Equation 2 (nЯ^x=∞_F=1): The powers of 2Я where F=1, for all X ranging from 1 to ∞, converges to 1. This convergence is indicative of the same cardinality and infinite density as observed in Equation 1.
3. Changing the value for ndoes not change the equilibrium of these equations.

QED: The proof demonstrates that Therefore, both equations nЯ^x=1_F=∞ and nЯ^x=∞_F=1 exhibit the same cardinality and infinite density, demonstrating their equivalence.

25: Unity of Cardinality for infinite powers of Я

Theorem: Unity of Cardinality and Progression in nЯ^k.

Proof:

For any positive whole number n, let Я=dfrac1n-1. The seriesnЯ demonstrates unity in cardinality and a consistent progression, defined as:

1. The initial term: nЯ^1=Rtimes n+left( n-1right)
2. Subsequent terms: nЯ^k=1=Яtimes nЯ^k+( n-1), where k ranges from 1 to infinity

QED:This generalisation affirms the equivalence between the continuous division of a line into n  parts and the powers of nR, showcasing unity in both cardinality and progression.

26: Infinite density of whole number fraction in Я

Theorem: The Aleph 0.5 sequence represents the sum of fractional parts in a stepwise manner.

• Starting with k=1, each step adds dfrac12 to the sum.
• The process continues up to n where the sequence accumulates fractional contributions, which maintain a half step correspondence to n.

Generalisation:

1. Expression for Aleph 0.5 Sequence:S_n=dfrac12+dfrac12+dfrac12 ldots+dfrac12
2. General Form of the Sum: S_n=sum ^n_k=1dfrac12

Claim: 1n=1m , 2n=1.5m , 3n=2m… Proof:

1. Expression for 1_n: 1=dfrac11
2. Expression for 2_n: 2=dfrac22 + dfrac12
3. Expression for 3_n: 3=dfrac33 + dfrac23+dfrac13
4. General Form for k_n:
5. k_n=dfrackk + dfrack-1k + ... + dfrackk.
6. Simplification of k_n:
   • k_n = /dfrack+(k-1)+...+1k
   • k_n=dfrackleft(k+1right)dfrac2k
   • k_n=dfrack+12
7. Comparison with m:

• Let m represent the steps in the sequence (1,2,3,…).
• Substitute k=m into the simplified expression:

Conclusion: Thus, for every step n in the sequence, k_ncorresponds to m_m, where mrepresents the step in the sequence.

27: Unity of Cardinality for 2d planes

Conjecture: Consider the conjecture that folding the 2D plane along both the x-axis and y-axis simultaneously establishes a one-to-one correspondence between points on the plane and the intersection points formed by folding.

Explanation of the Concept:

1. Coordinate Correspondence:

   • Let �(�,�)P(x,y) represent a point on the 2D plane, where �x corresponds to the x-axis, and �y corresponds to the y-axis.

2. Folding Operation:

   • Conceptually fold the plane along both the x-axis and y-axis simultaneously, denoted as �(�)F(P).

3. Intersection Point:

   • Define the intersection point after folding as �(�,�)I(x,y), where �x and �y correspond to the unique points on the x-axis and y-axis after folding.

Formal Proof:

1. Folding Operation Formalization:

   • Formally, the folding operation is defined as �(�)=�(�,�)F(P)=I(x,y).

2. One-to-One Correspondence:

   • The folding operation �F establishes a one-to-one correspondence between points on the 2D plane and the intersection points formed by folding.

3. Generalization to Infinite Plane:

   • Extend the folding concept to cover the entire infinite plane, allowing coordinates �x and �y to take on any real values.

4. Mathematical Proof of Correspondence:

   • For any point �(�,�)P(x,y) in the infinite plane, �(�)F(P) uniquely corresponds to the intersection point �(�,�)I(x,y) formed by folding along the axes.
   • This follows from the folding operation preserving the relative positions of points.

QED The conjecture has been demonstrated to be true through the explanation of the concept and the formal proof. Folding the 2D plane along both axes indeed establishes a one-to-one correspondence between points on the plane and the intersection points formed by folding.

27: Measure of infinite density through the hypotenuse of a right angled triangle.

Conjecture: For any Cartesian coordinate system with a right-angled triangle formed by the x-axis and y-axis, there exists a direct relationship between the one-to-one correspondence of points on the xy-plane and the measurement of infinite density through the hypotenuse. The hypotenuse undergoes changes in length as the density of points on the xy-plane is altered through scale reductions, with the limit of the hypotenuse approaching a measurement of infinite density.

Proof:

1. Coordinate System and One-to-One Correspondence:

   • Consider a Cartesian coordinate system with the x-axis and y-axis forming a right-angled triangle.
   • Assume a one-to-one correspondence where each point (x, y) on the xy-plane uniquely corresponds to a point on the x-axis and y-axis.

2. Hypotenuse Measurement:

   • LetHrepresent the hypotenuse of the right-angled triangle formed by the x-axis and y-axis.

3. Initial Density:

   • In the initial configuration, where the one-to-one correspondence covers the entire xy-plane, the hypotenuse is H_0=sqrt2 as per the Pythagorean theorem.

4. Reducing Scale on the Y-axis:

   • Introduce a scale reduction on the y-axis, e.g., reducing it to 0.5 while keeping the x-axis unchanged.
   • The new hypotenuse H_0=0.5 is measured using the Pythagorean theorem: H_0= sqrt1^2+0.5^2=sqrt1.25
5. Density Shift and Hypotenuse Change:
   • • As the density shifts due to the y-axis scale reduction, the hypotenuse changes accordingly.
     • The change in hypotenuse reflects the altered density, illustrating a direct relationship between density and hypotenuse measurement.

6. Generalization to Infinite Density:

   • Extend this concept to an infinite density scenario by allowing the scale reduction to approach zero.
   • The limit of the hypotenuse �limitHlimit​ as the scale reduction approaches zero represents the measurement of infinite density.

QED: The formal proof demonstrates that infinite density can be measured by the hypotenuse of a right-angled triangle in the context of a one-to-one correspondence on the xy-axis. The relationship is showcased through the Pythagorean theorem and the alterations in the hypotenuse as the density shifts.

28: Change in Density and Cardinality in the Folded Plane

Conjecture: In a Cartesian coordinate system, folding a square along its diagonal, with a 1-to-1 correspondence between points on the folded plane and the x-axis and y-axis, reveals a direct relationship between 1-to-1 correspondence, cardinality, and the changing surface area density.

Proof:

1. Original Surface Area Density:
   • Define D_square as the original surface area density: dfraca_squarecardinality
2. Surface Area of Folded Triangle:
   • Apply the Pythagorean theorem to calculate the surface area of the folded right-angled triangle: A_folded = dfrac12 × base × height
3. 1-to-1 Correspondence:
   • Establish that the points on the folded plane exhibit a 1-to-1 correspondence with points on the x-axis and y-axis.
4. Y-Axis Scaling to 0.5:
   • Scale the y-axis to 0.5 while keeping the x-axis unchanged.
5. Comparative Measurement:
   • Measure the changing density and cardinality by considering the quarter square formed by scaling the y-axis to 0.5.
   • Calculate the surface area of the quarter square using the Pythagorean theorem: A_0.5 = dfrac12 × base × height .
6. Comparative Density and Cardinality:
   • Compare the density and cardinality ratios for the quarter square (R_0.5).
7. Conclusions:

• • Highlight the direct influence of 1-to-1 correspondence in the folded plane on changing surface area density and cardinality with y-axis scaling to 0.5.

QED: The proof supports the conjecture, demonstrating the inherent connection between 1-to-1 correspondence, cardinality, and the changing surface area density in a folded plane with y-axis scaling to 0.5

29: infinite dimensional tetrahedral numbers

Conjecture:

For every non-negative integer n, there exists a multidimensional tetrahedral numberT_n^(m+1) such that:

T_n^left( m+1right) = dfracn^left(overlinem+1 right) left( m+1right) !

where n^left( m+1right) is the m+1-th dimensional triangular number.

Proof:

1. Base Case (m = 1):
   • For m=1, the formula becomes T _n^2=dfracn^22!
   • This represents regular triangular numbers, which are sums of consecutive natural numbers.
2. Inductive Step:
   • Assume the formula holds for k i.e T_n^left( k+1right) =dfracn^left( k+1right) (k+1)!
   • Now, consider m=k+1.
   • The formula becomes T_n^left( k+2right) =dfracn^left( k+2right) (k+2)!
3. Expression for n^k+2.
   • Use the assumed formula for T_n^left( k+1right) to express n^k+2 in terms of n^k+1: n ^left( k+2right) =sum ^n_i-1n_i^left( k+1right)
4. Substitute into T_n^left( k+2right) : Substitute the expression for n^k+2into the formula for T_n^k+2.
   • dfracsum ^n_i=1n_i^left( k+1right) left( k+2right) !
   • This represents the sum of the n^k+1-th triangular numbers, which is a n^k+2 dimensional tetrahedral number.
5. Conclusion:
   • By induction, the formula holds for all non-negative integers m
   • Thus, from whole numbers, a multidimensional tetrahedral number space is generated.

QED

30: infinite density of all k-space triangular numbers

Conjecture: The limit of the summation of all triangular numbers in all k-dimensions  (where n^k becomes n^n) as k approaches infinity is equal to dfracn^ne

Proof:

1. Begin with the rising power formula for triangular numbers in k-dimension:
   T_n^overline(m+1) = dfracn^overline( m+1) ( m+1)!
2. As m approaches infinity, the rising power becomes an entire set of natural numbers, denoted as:
   k :T_n^k=dfracn^left( k+1right) left( k+1right) !
3. Let k approach infinity, leading to the substitution of m for n:
   n:T_n^n=dfracn^nn!.
4. Note the mathematical constant e is defined as sum ^infty _n=0dfrac1n!=1+dfrac11!+dfrac12!+dfrac13!ldots
5. The limit of this series, as k approaches infinity, converges to dfracn^ne.

QED

31: generalised Infinite density of k-dimensional reciprocal triangular numbers.

Conjecture: For every positive integer n within the domain of k -dimensional triangular numbers defined as T_k^n=dfracn^kk! a formal relationship emerges between n^2 and the Sophomore's Dream constant S. Specifically, n^2 converges to S and n! converges to the mathematical constant e. This convergence is expressed as lim _nrightarrow infty =dfracn^nn!=T infty ≈ 0.4745227492 providing a rigorous connection between k-imensional triangular numbers, the Sophomore's Dream constant, and the mathematical constant e.

Proof:

1. Expression for k Dimensional Triangular Numbers:
   • The general formula for k Dimensional triangular numbers is T_n^k=dfracn^kk!.
2. Construction of e:
   • The mathematical constant e is defined as sum ^infty _n=0dfrac1n!=1+dfrac11!+dfrac12!+dfrac13!ldots
3. Sophomore's Dream Constant S:
   • The Sophomore's Dream constant S is associated with the series sum ^infty _n=1dfrac1n^napprox 1.291285997.
4. Expression for T_k^n useing e and S
   • When k=n as K moves to infinity, the expression for T_n^n becomes T_n^n=dfracn^nn!=dfracSe.
   • This reveals that n^n converges to the Sophomore's Dream constant S, and n! is equivalent to the mathematical constant e.
5. Convergence Relationship:
   • The convergence relationship is rigorously expressed as lim _nrightarrow infty dfracn^nn!=Tinfty

QED: Thus, the conjecture stands affirmed through the exploration of k-dimensional triangular numbers, the construction of e, and the convergence of n^n to S. The convergence ratio is captured in the constant Tinfty approx 0.4745227492, and its reciprocal value dfrac1Tinfty approx 2.105038736.

32: generalisation of limits for k-dimensional reciprocal triangular numbers.

Conjecture: The convergent values of k-dimensional triangular numbers follow the pattern dfrackk-1 i.e.
sum ^infty _n=0dfrac1beginpmatrix n+k-1 \ k-1 endpmatrix

Proof:

1. Base Case (1D Triangular Numbers):
   sum ^infty _n=0 dfrac1beginpmatrix n+1-1 \ 1-1 endpmatrix=sum ^infty _n=0dfrac1beginpmatrix n \ 0 endpmatrix=1
   The base case holds true.
2. Inductive Step: Assume that the conjecture holds for k-1 dimensional triangular numbers, i.e:
   sum ^infty _n=0 dfrac1beginpmatrix n+k-2 \ k-2 endpmatrix=dfrack-1k-2.
3. Show for k-dimensional Triangular Numbers:
   
   sum ^infty _n=0 dfrac1beginpmatrix n+k-1 \ k-1 endpmatrix=sum ^infty _n=0 dfrac1beginpmatrix 0+k+1 \ k-1 endpmatrix + dfrac1beginpmatrix 1+k+1 \ k-1 endpmatrix + dfrac1beginpmatrix 2+k+1 \ k-1 endpmatrix+
   • ...Using the Pascal's identity:

     beginpmatrix n \ k-1 endpmatrix+beginpmatrix n \ k endpmatrix+beginpmatrix n+1 \ k endpmatrix =dfrac1beginpmatrix k-1 \ k-1 endpmatrix+dfrac1beginpmatrix k \ k-1 endpmatrix+dfrac1beginpmatrix k+1 \ k-1 endpmatrix +...

   • Combine the terms with a common denominator:

     = dfrac1beginpmatrix k-1 \ k-1 endpmatrixleft( 1+dfrac1k+dfrac1k+1+ldots right)

   • Simplify the series inside the parentheses using the formula for the sum of reciprocals of consecutive integers:
     = [katex]dfrac1beginpmatrix k-1 \ k-1 endpmatrixcdot dfrackk-1[/katex]
   • Simplify further:
     = dfrackk-1
     Thus, by induction, the conjecture holds true for all kgeq 1.

QED: The pattern is a valid representation of the convergent values ofp
k-dimensional triangular numbers.

33: Change in geometric formation changes density

Conjecture: A change in the geometric arrangement of numbers results in a corresponding change in their density.

Mathematical Proof:

1. Consider a geometric arrangement where two triangles form a square. Each individual triangular series converges at 2, as represented by the equation dfracnn+1.
2. When the two triangles are joined to form a square, the overall series converges at 
3. dfracπ²6, as indicated by the Basel problem.
4. Split dfracπ²6 into two equal parts, giving dfracpi sqrt6times dfracpi sqrt6.
5. Divide each side by 2 for each triangle, obtaining dfracpi 2sqrt6times dfracpi 2sqrt6.
6. Multiply the result, obtaining dfracpi 24
7. Divide dfracπ²6 by dfracpi 24, resulting in 4, which is the original density of the 2 triangles.
8. Thus the ratio difference bewtween a triangle and half square is, 2:dfracpi sqrt6

QED: The geometric proof demonstrates that altering the geometric arrangement of numbers leads to a change in density. Specifically, the transformation from two triangles to a square, and the subsequent comparison with the Basel problem, reveals a ratio of 4. The convergence of the triangular series at 2 and the Basel problem at dfracπ²6, further supports this conjecture.

Additional concepts

Infinite Powers

1. Logical Formulation: If each mathbbN is already an infinite set mathbb∞^∞(( N=infty )), then ( n_1 times n_2 times n_3 ) is ( infty^ infty ), where we substitute 1 for ( N ).
2. Formal Statement: The application of the Infinite Power Conjecture challenges traditional notions of uncountable cardinality, suggesting a paradoxical form of measurability in the real number continuum.

Measurability of Cardinality through Infinite Density

1. Equivalence to mathbb∞^∞ Given the established equivalence of mathbbR to mathbb∞^∞, this equivalence extends to mathbbW (whole number space), indicating that mathbbW is also characterized by mathbb∞^∞.
2. Infinite Density in mathbbW The property of infinite density observed in mathbbR is inherent in mathbbW as well. Any measurement between two points in mathbbR results in an infinite number of points due to the infinite density. Since mathbbR and mathbbW are equivalent, mathbbW inherits the property of infinite density.
3. Measurable Cardinality: Despite the uncountable nature of the cardinality of mathbbR and mathbbW, the presence of infinite density implies a form of measurability. The unit measure mathbbU establishes the density of mathbbR, and as mathbbR=W, all real numbers greater than 1 in mathbbW exhibit a 1-1 bijection with all reals in mathbbR.

Consider the setsmathbbR and mathbbW defined as follows:

• mathbbR represents the reciprocal number space, consisting of numbers greater than zero but less than 1.
• mathbbWrepresents the whole number space, consisting of numbers greater than 1 up to infinity.

1. Construct a line segment mathbbAB representing the entire number line from 0 to infinity.
   Divide the line into two segments:
   1. mathbbR
   2. mathbbW.
2. Fold the line along the point mathbbP at its center, ensuring a one-to-one correspondence between the elements of mathbb{R and mathbbW.
3. Due to the law of equilibrium defined by mathbbA to mathbbB with mathbbP at the center, the lengths of the line segments in mathbbR[/katex and [katex]mathbbW match.
4. The equivalence in lengths ensures a bijection between the elements of mathbbR and mathbbW, establishing that the cardinality of the real numbers in mathbbR
5. mathbbR is equivalent to the cardinality of the real numbers in mathbbW .

This conclusion is grounded in the geometric properties of the line, emphasizing that the infinite density in
mathbbR does not alter the cardinality when compared to the whole number space mathbbW.

In summary, the application of infinite density challenges traditional notions of uncountable cardinality, suggesting a paradoxical form of measurability in the real number continuum.

Cantor's Diagonal Proof and Base (∞) Framework:

Cantor's Diagonal Proof involves assuming that all real numbers in a list can be enumerated and then constructing a new number not on the list through a diagonalization process. This leads to the assertion that the set of real numbers is uncountable.

In the base (∞) framework, where each real number has a unique symbol, Cantor's diagonalization paradox doesn't hold. In this context, each real number is already uniquely represented, and there is no need for a diagonal process to discover an unlisted number. Thus, the argument for the uncountability of real numbers, based on Cantor's diagonalization, is not applicable in the base (∞) framework.

Formal Statement:

Let mathbbC denote Cantor's Diagonal Argument and mathbbB  represent the base (∞) framework.

mathbbC :Cantor’s Diagonal Argument assumes the uncountability of real numbers.mathbbB :Base (∞) framework, where each real number has a unique symbol.

The formal implication is expressed as:

If mathbbB , then Cantor’s Diagonal Argument mathbbC is not applicable.

This statement asserts that within the base (∞) framework, the assumption of uncountability of real numbers, as per Cantor's Diagonal Argument, is not valid.

Connection to Existing Mathematics:

1. Convergence Patterns and Geometric Series:

The convergence patterns observed in the reciprocal tripling series mathbbS = 1/3 + 1/9 + 1/27 + ... align with the behavior of infinite geometric series. The convergence to a limit of 0.5 reflects the well-known property of geometric series.

2. Euclidean Geometry and Circle Circumference:

The geometric proof regarding concentric circles and the consistent passage of a line through their centers aligns with Euclidean geometry. This proof is reminiscent of the concept that all circles, regardless of size, share the same cardinality, similar to line segments converging to mathbbU in the framework.

3. Cardinality of Sets and Infinite Density:

The concept of infinite density ratios mathbb∞F(1:3 in the framework shares similarities with cardinality in set theory. While the infinite density ratio remains consistent for mathbbU , mathbbR ), and mathbb2R^2 , it contrasts with the decreasing density ratio observed in mathbb3R and mathbb3R^2 . This illustrates a nuanced relationship with cardinality.

Summery

In summary, this mathematical framework redefines the nature of numbers within an infinite set (I). Starting with a central point (P), continuous division creates polarities (P+ and P−), leading to the formation of a unit measure (U). Exploring infinite division through line segments (AB), an infinite reciprocal number space (R±) emerges.

The divergence of reciprocal powers and the introduction of an infinite density ratios provide unique insights. The framework challenges conventional views by proposing that each iteration in the reciprocal sequence (R, R2, R3, ...) adds a unit to density, offering a fresh perspective on the set of natural numbers (N).

This exploration presents a novel synthesis of traditional mathematical concepts with innovative ideas, paving the way for a deeper understanding of the intricate relationships and structures inherent in the realm of numbers.

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