Based on the new geometric Axioms of the Math of Infinity, we can finally propose a viable framework the can solve the infinite.

Overview

The problem posed by the Continuum Hypothesis have remained unsolved to this day. What has been established it that our current mathematical axioms are not adequate enough to resolve the nature of infinity.

By answering the questions that lie at the foundations of the number line itself, and the nature of the origin of mathematical operators, In2infinity claim to have produced axiomatic system that is able to answer the seemingly unanswerable.

What is the density of the infinite set of real numbers? Is there an infinite set that has greater density than the whole numbers, but less density than the real?

The answer to these question digs deep into the heart of what mathematics actually is. Rocking the foundations of everything we thought we knew about numerical processes. From the nature of division and the number Zero, a geometric mathematics can be constructed, that not only solves the Continuum Hypothesis, but also challenges the entire nature of the number i and the complex number plane.

THE

Concept

mathematical Origins?

The first step towards resolving the Continuum Hypothesis come about from asking a very profound question. How did mathematics come about in the first place? At first this seems like an impossible question to answer. However, as mathematics only has four main types of mathematical operator we can formulate the philosophical framework simply by a process of elimination.

The number line is constructed from an infinite set of positive and negative numbers. We suggest that it is this feature that enable the process of addition, subjection, and multiplication to occur. However, division does not need a negative and positive concept for its operation. Let’s explore at this concept in more detail.

Addition and subtraction

As the first thing we are taught we we begin to learn about maths is the process of addition. From the perspective of the 3D world this makes perfect sense, we see and count an object. Alternatively we remove an object and which subjects the number from the total count. simple right?

But when we are dealing with the uncountable, i.e infinity,, addition and subtraction take on a different quality. It is only by virtue of positive and negative number space that a value can be subtracted. At first this might seem strange. However, by we placing a + or – sign next to each number, we can see that calculations are just a combination of a particular value. It is our convention to remove the + sign from positive numbers. Yet the correct representation should be +1, +2, +3, and so on.

Therefore by its very nature addition and subtraction require the concept of negative and positive numbers in order to be enacted.

Multiplication

Often multiplication is assumed to be the opposite of division. It is easy to see why. If we multiply 2 by 3 we get 6. If we divide 6 by 2 we get 3, and if we divided 6 by 3 we get 2. That seems to be like fairly straight forward expression and it works mathematically. But only for countable numbers, not infinity without any notion of positive of negative.

For example, the number ZERO can be considered non-polar, any value multiplied by zero equals zero. The number one has both a positive and negative expression, yes when a number is multiplied by one, it the result remains the same. 1 x 2 = 2 and 1 x 3 = 3 and so on.

As the number ONE has a positive and negative expression, we can multiply from either the positive side or the negative. This will ‘switch’ the polarity of the result, but the number will remain the same.

Division

If we apply the same equation used for multiplication above but swap the operator with division. If we divide 2 by 1 then the result is the same, just like multiplication. But when we swap the equation around, then 1 / 2 = 0.5. A reciprocal value. This demonstrates that division is not the ‘opposite’ of multiplication. Moreover, it is unique, for out of all the four operators, it is the only one that can create a fraction from whole numbers.

As fractions make up the class of rational numbers, and can also generate all whole numbers, it seems like division is the only candidate for the seed of all other mathematical operations.

We can also examine the nature division in terms of the number ZERO. At first is appears that dividing any number by zero has no effect. The result is always zero. This occurrences regardless of the orientation of the numbers. Yet appearances can be deceiving.

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Infinity and Zero

We have seen that the order by which the number appear in a calculation involving division determine the result. If the larger number is placed at the top then the result will be 1 or greater. If we invert the fraction, the result will be a reciprocal value, between zero and one. This is actually one of the keys to understanding the nature of infinity. That all real numbers greater than one are contained within the space between zero and one. Let now look at why this should occur from the perspective of the infinite.

Dividing Zero

Zero sits right at the centre of the number line. As such it is considered to be non-polar. yet is that true? Actually no. There exists a form of zero that is polarised. In the previous example we divided 1 by 2. The result 0.5 is a positive number. The initial digit is a +zero. if we divide -1 by 2 we get -0.5. Again the first digit is a -zero. In fact all reciprocal numbers begin with either + or – zero. So ± zero exist, and yet the zero at the centre of the number line is non-polar.

As we have seen, it is division that generates all reciprocal fractions, so what happens when we divide zero into 2 parts? We get a +0 and a -0, which is the whole number part of all reciprocals.

Notice that on the standard number line we only have a positive and negative choice. This duality means that the number 2 is the specific divisor that can create a +0 and -0.

As a point to note, in order to divide zero, we need to arrange the calculation as 0/2 = +0/-0. This notion of division also challenges the idea the 2/0=0.

If the pattern of fractions is that whenever the smaller value is placed above it creates a numbers reciprocal, then this indicates that zero might not have been correct interpreted by current mathematical axioms.

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Dividing Infinity

Having examined the notion of the +0 and -0, we can see that the same polarity concept is applied to infinity itself. The infinite set of whole numbers expand into infinity in both the negative and positive direction. As infinity has only be defined in terms of a set, this point is often omitted. Yet the two opposite forms of infinity are clearly apparent , and can be placed at either end of the number line.

With the definition of ±∞ established, we can now ask the question, how did it get there?

Just as ZERO exhibits a non-polar quality, so the same can be said of infinity. There is no need for an infinite line to exhibit a positive or negative. Only if we divide the line in line into 2 parts can we now apply a dualistic nature to the line. Just as zero can be divided to create ±0 so the same priciple can be applied to infinity itself.

As the positive and negative numbers form a perfect bijection, we know that each exhibits exactly the same density. Infinity, can only be divided into equal parts. As the only mathematical function that can achieve this result is the process of division, we are left to conclude that it is division that must be responsible for the formation of the number line itself. Establishing this mechanism of formation is a crucial step in solving the Continuum Hypothesis, as it is upon the number line that the conjecture has been constructed.

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ONE and TWO

Creation of the number line

With the nature of division established as the prime mover in the act of the manifestation of mathematics and indeed the number line itself, we can begin to piece together a logical construct that explains why we have positive and negative numbers, and why the non-polar version of ZERO appear exactly at its centre.

To begin we can imagine an infinite line that exists without any kind of distinguishing features, running into infinity in two directions. If we mark a point on the line, then we can now distinguish one part on the line from the other. As the line is infinite, it does not matter where we place the dot. Any dot placed on any kind of infinite space will be at the centre of that space.

Both spaces will be equal in terms of number density, as they are the mirror image of each other.

So what happens if we repeat the process to each of the infinite sides. We know they must create an equally dense infinite set. As in the previous example we need to mark a point midway between zero and infinity. Choosing two random points at an equal distance away from zero on the positive and negative side on the line, we are defining a unit measurement. This we call ±ONE.

It does not matter how far the point is marked, as no other units exist on the line. All that matters is that the two dots are marked at the same distance away from the ZERO point, in order to maintain an equilibrium in their density.

With the number ONE defined we have manifested a reciprocal space, into which all real numbers can be manifested as a reciprocal, which will be mirrored on the opposite side of the number one as a defined unit distance.

Did you know there a only 8 types of infinity that form the number line?

ThE Infinity Equation

With the appearance of the number ONE we complete all of the element needed for the infinity equation to be enacted

0 ± 1 = ±∞

FIND OUT MORE ABOUT THE INFINITY EQUATION

Doubling Sequence

The process of the division of a non-polar infinite line gives rise to the structure of the number line. This is a reiterative process of division into 2. But what happens if we continue to reiterate this same simple pattern.

What we find is that a sequence of numbers arises that we call the doubling sequence.

In reciprocal space, each successive iteration generates a value closer to zero, which in whole number space is reflected at a point closer to infinity.

This sequence is well recognised as it appears in the foundation of all life. The division of cells for example. From this view, it is the what we call the ‘primary’ sequence, which structures reciprocal space. This in turn can be reflected as a mirrored image into whole number space, beyond the number ONE.

1,2,4,8,16,32…

Did you know that the doubling sequence even structures the number of electrons in each atomic shell, up to the number 32, which is the maximum for a stable element.

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Infinity of TWO

It is well established that the total sum of the doubling sequence is limited by the number 2. We call this limit TWO in order to distinguish it from the ordinary number 2.

Another unique property of TWO is that it is the only even prime number. Due to this it is the root of all other even numbers. just as the number ONE is the root of all add numbers.

When the number TWO is defined we are dividing the infinite set into either an odd or even number. Just as the number line, divided into two generates positive and negative. This occurs as the odd numbers appear exactly in the middle of the numerical space between two sequential even numbers. By dividing this space into two equal parts, so the respective odd number can be found.

FIND OUT MORE ABOUT THE INFINITY OF TWO

→With the numbers +0, +1, +2 and -0, -1, -2 we are able to generate the entire set of whole numbers, and suddenly the other function such as addition, subtraction, multiplication, powers, and roots now have the foundations needed to produce mathematical results.

...-2 -1 -0 ← 0 → +0 +1 +2...

With the exception of the number phi, all other number series require at least three elements in order to predict the sequence, i.e 2,4,6 and the next number is 8, whereas 2,4,8 defines the doubling sequence whereby the next number is 16, notice we only needed to change the last number of the set in order to predict the difference between the two. Therefore 0,1,2 are enough to predict the complete infinite series of whole numbers.

4D squaring

When it comes to powers, our new rotational concept we call 4D squaring, and ZERO² can be enacted at this point. In our view the notion of powers is a transformation of dimensional number space. In the case of the 1D number line, it becomes transformed into an x.y axis on the 2D plain.

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Number Categories

Units & Numbers

The framework for the continuum hypothesis is predicated on the assumptions based of the particular way our present mathematics has been constructed. However, as it does not question the very nature of numbers on the number line, these classification are only based on whether a number can be calculated into existence. Under the current system, geometric ratio, roots, and powers are all classified as either rational or irrational, with no distinction made between the space between the reciprocal space between zero and one and the rest of the number line. As such resolving the continuum hypothesis is completely impossible.

As we have restructures the classification to incorporate the concept of the mathematical operators themselves, we find that the philosophical model is exactly inverse to current axioms in set theory. These suggest that the Natural numbers (ℵ0) are contained within the set of Real numbers (ℵ1), which is true. However we suggest the the formation of numbers arises out of the process of division, which is not accommodated by current classifications.

Just as the great mathematician Euclid suggested that the number 1 is not a number, but a unit measure that creates all other numbers, we suggest the same.

ZERO and ±ONE are the ‘fence posts’ that hold up the line of infinite numbers. Just like a musical instrument keeps a string taught. We cannot sat the instrument is the same thing as a string.

The key difference then is that our unit measure contains the entire set of real numbers, with the exclusion of ZERO and ONE. Is is not as Euclid suggested tha the number ONE is the Unit, rather the space between ZERO and ONE. inside of which infinity is contained.

FIND OUT MORE ABOUT THE INFINITY EQUATION

Geometric Ratios

The second important point in terms of classification is that we suggest number such as root and powers are functions that transform the dimensionality of the number line. Whilst these can be represent, it is inaccurate to say, ‘they emanate from the same class as all other numbers’. As these special calculation always involve equal quantities, the law of equilibrium is in effect. This is apparent by the fact that roots and powers are contained within reciprocal space. and cannot cross over the boundary of ZERO or ONE.

Similar ratios, such a the transcendental number pi are also not part of the set. As pi is a geometric constant, applied to circles, its estimated appearance on the number line is only a projection. just as we can make a shadow projection in 3D space into 2, the same can be said of the 1D number line.

In reciprocal space, each successive iteration generates a value closer to zero, which in whole number space is reflected at a point closer to infinity.

having removed these from the classifications of ‘real numbers, we find that we are left with rational fraction, and completely irrational numbers that have no apparent mathematical origins. As infinity contains all possibility of number all can exist, however only through mathematical logic can the universe be structure through relationship.

We see the same process at wokr in the universe at large. Everything we are experiencing is made of 81 stable atoms. The possibility for more exist, but due to the limitations imposed by reality such manufactured artificial elements con only exist for a few micro seconds, in most cases.

We see mathematics as being the language of science, and we believe the the limitations such as the speed of light and other universal constants can all be explained through ratio.

Whilst irrational numbers exist, as they have no mathematical foundation, there is no possibility for them to manifest in our reality.

Based upon the idea that the universe is constructed from geometric ratio we are able t construct a new scientific language that provide the exact values of the plank constant (h) and the fine structure constant (α) to an infinite degree!

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Order out of chaos

Within the discipline of science the law of entropy is suggested to be one of the foundation concepts held true by most scientist. However our interpretation of numerical space suggest that this entropic nature is altered by mathematical logic, beyond that of our human imagination. The concept of negative entropy is the nature of order arising out of chaos.

From the infinite background energy of space, the fluctuations of quantum foam we find the paring of virtual particles, right at the foundations of our realty. The same can be said of electron pairing.

Mathematics and geometry are the only organising principles that can explain these and many other occurrences.

REAL and GEOMETRIC

The infinite set of real numbers are only comprised of two types; rational and irrational.

However the ‘numbers’ ZERO, ±ZERO, ±ONE, ±TWO, ±∞, x^1-∞, and ^1-∞√x, and others are from the geometric class, upon which numbers are formed and ordered. Notice that we use the capitalisation of the written number to distinguish them from the series of infinite numbers normally considered as whole numbers. Rational numbers can be formulated by mathematical operations. The irrational numbers derived from the spaces in-between.

There are an infinite number of possible irrational decimal numbers in between any two whole number fractions. This is because each single decimal fraction contains the infinite within it.

As the number i represents the √-1, it is included in this geometric set. This means that the complex number plain is not a correct view in terms of our conjecture.

This re-categorisation has deep implication for the both the Continuum Hypothesis, and indeed the whole of mathematics.

FIND OUT MORE ABOUT THE INFINITY OF TWO

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Countability

This view provides a different explanation to cantors explanation as to why numbers cannot be counted. Counting by trying to list all numbers on the number line is a proven impossibility. The same goes for all numbers. This is because the process of counting present by Cantor is additive in nature. As we have shown the addition and subtraction arise out of the division of infinite number space, we are not counting numbers. We are defining them by performing a mathematical operation. Division.

Decimal frcations

The fact that between two rational fractions there exists an infinite number of irrational numbers is demonstrated by the re-examination of decimal notation. We often write the number one as a single digit. However this is not always the case. for example 01 and 001 are also representative of the same number. In fact we can append an infinity number of zeros to the front of any number without changing its value. The same can be said of the decimal fraction that follows the decimal point. When we compile the ‘true’ picture, we see that the whole number and fractional parts are two sets of infinite number.

0 = ∞→…000.000→…∞
1 = ∞→…001.000→…∞
2 = ∞→…002.000→…∞

The decimal point divides whole a fractional space like zero divides positive and negative number space.

Just like the number line is composed of two equal sets of infinite positive and negative numbers that are in equilibrium in term of the number of digits. The possible form that any one particular digit is designated defined the base system of number. Binary is the most simple example and can switch a zero to a one or vice versa. Base number system run from 2 to ∞, whereas base 1 will only produce an infinite set of zero. when we reach base ∞ then each digit can be represented as an infinite possibility, and the infinity within the decimal system is revealed.

base 1 = ∞→…000.000→…∞

base 2 = ∞→…0101.0101→…∞

Base 3 = ∞→…012.012→…∞

base ∞= ∞→…∞∞∞.∞∞∞→…∞

Base ∞ gives each digit within a decimal number an infinite possible expression.

Base systems

a base system is a means of ordering the infinite. when we employ any base system we place a limitation around the types of possible number, for each increase in base the comminatory complexity also increases towards infinity, as base systems are a choice, the limitation produces a boundary. This creates a specific block size that can be reflected, (repeated), infinitely across the number line. What distinguishes a number is therefore a variation of a particular block from the infinite set of ZEROs that make up the complete picture of ant particular number.

The reason we cannot count whole numbers, was proven by Cantors diagonal solution. Here we can arrive at the same conclusion, as each digit is comprised of an infinite whole and an infinite fraction. From this view a whole number is defined when the fractional part consists of any type of infinite set of with the exception of an infinite set of ZEROs. This is because any reciprocal number is defined whenever the whole number part consists purely of an infinite set of ZEROs.

WHOLE NUMBER

∞→…00xx.000→…∞

RECIPROCAL NUMBER

∞→…000.xx00→…∞

In the above diagram XX is indicative of any infinite number combination

Solving Infinity

From this we are able to at last answer the unsolved question posed by Cantor who proved that the infinite set ℵ1 (real numbers) is denser than ℵ0 (integers), but could not ascertain by how much.

As the infinite sets on either side of the decimal point form a one-to-one correspondence, (bijection), we know they are equivalent. For each number that appears on the whole unit side, there are an infinite number of variations that appear on the decimal side. The ratio is therefore:

1 : ∞

This solution means that we are also able to ascertain the size of ℵ1 in the following way:

the set of whole numbers

1 x ∞ = ∞ = ℵ0

The set of real numbers

∞ x ∞ = ∞² = ℵ1

For the first time we can mathematically identify the specific density of the real and whole numbers, which is an essential step needed to solve the continuum hypothesis. For without knowing the density relationship of these two sets, we can never ascertain if there is another set that should exist in-between the two. With these two quantities identified the solution to infinite begins to take a more interesting shape

Harmonic
Philosophy

The identification of the relationship between ℵ0 and ℵ1 is very profound in the sphere of mathematical philosophy. Indeed numbers is the defining capacity that differentiates a species. Art, music, and science are just a few of the things in life that have their foundations in a notion of number. As part of our new theory, Harmonic Philosophy, investigates this nature of consciousness, relating the domain of spiritual wisdom, with the logic of mathematics

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The shape of infinity

From what we have discovered so far, we can construct a logical argument that indicates that there is no infinity between ℵ0 and ℵ1. This is due to the observation that whenever we divide infinity, two sets of infinity are created that are exactly equivalent. Whether we look at the positive and negative numbers, of the infinite of whole a fractions around a decimal point, the law of equivalence holds true.

It is true to say that there is no infinity between ℵ0 and ℵ1, as ℵ1 is constructed from an equivalent infinity as ℵ0. We have identified the constructing mechanism as ∞². Problem solved, right?

Not quite, there is one small problem that is more to do with how we frame the question and the methodologies, and philosophical background upon which the solution is formulated. As our formulation of the emergence of all mathematical phenomena it derived from the division of a non-polar infinity. Quite a novel idea for most people.

Whilst we can say that our presentation of whole and decimal numbers does solve the Continuum Hypothesis, the reasoning that lead us towards this conclusion involves a particular notion of the infinite, restructure the foundations of much of the thinking around infinity.

One key point is that we have demonstrated that the entire set of number is contained within the reciprocal space between ZERO and ONE. This means that our number classification system is different to the one that exists today. Under this system the number 1 is differentiated from the concept of ONE that forms the unit measure. The same can be applied to the number TWO, which we explore in more detail in our concept of 4D squaring.

This leads to a logic problem, which was identified by the Russel Paradox. As all numbers are contained within the number 1, how can numbers greater than one create a larger infinity. If we add a unit to the number one, it has the same infinite amount of numbers, yet as the same time each will have its reciprocal partner, between ZERO and ONE.

We overcome this problem by introducing the ‘Law of Equivalence’, which notes that a point placed in an infinite space will be at the centre of that infinite space. And secondly, that when infinity is divided the two infinities will be Equivalent. This in turn elevate the process of division to be the prime mover in the formation of mathematics, as it produces the notion of addition and subtraction through the generation of positive and negative infinite number space. It also reclassifies the number TWO as the ‘key divisor’ in the process. The numbers 0, 1, 2 are enough to generate the entire sequence of infinite whole numbers.What this means it that Whole number Space and reciprocal space are two equivalent infinities, as mathematical fact that can be demonstrated by the Infinity of ONE

Secondly, as our genesis of mathematics is

Well actually, that depends on how we pose the question. Originally mathematicians asked, “how many numbers are on the number line?” which is what lead to the logic behind the continuum hypothesis. However, that is very different from a statement that asks “How many numbers are there?” this statement is not confined by the limitation of a 1 dimensional number line. It invites us to look into a new kind of mathematical concept we call numerical space.

Numerical Space

Our traditional notion of number is predicated on the concept of a number line. In geometry a line is a one dimensional entity. when we square a number the result can be ‘shadow projected onto a 1D line and we see the result is the number of countable sections within a defined space on a 2D plain.

We can and do observe mathematical results and even reiterative functions, that can now be mapped by computer performing reiterative calculations.

The difference between a 1D line and a 2D plain are quite obvious. The addition of an exact replica of the original number line is rotated at 90°. The manifestation of this phenomena is covered in more detail in our post ‘Zero²‘.

Squaring is a mathematical function that increases dimension. If 1 is a line, then 1² creates a square or more accurately the 2D plain. Next 1³ a cube or 3D space. At each step dimensionality is increasing.

The nature of dimensional space is governed by geometric axioms. Each dimensional space has limitations placed on it by the very nature of regular shapes. As infinity operates through the laws of equivalence, the numerical space can only be constructed from regular forms.

The 1D number line, is limited in Euclidean Geometry as the shortest distance between two points. Similarly the second dimension can only be filled by either a triangle or square, using only two colours.In 3D we find that there are only 5 platonic solids, which are limited by the nature of the second dimension.

GEOMETRIC
MATHS

In our new theory of geometric maths, we challenge the conjecture of non-euclidean geometry, that suggests Euclid 5th postulate is not true.

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dimensional perspecitve

The 2nd dimension is characterised by an x,y axis, normally placed at 90° to the original number line. If we rotate the axis in 3D space along the horizontal axis,, we can see that the the vertical axis will begin to shrink, eventually disappearing into a dot at the centre of the number line. through rotation in a higher dimensional space, we have ‘flattened’ the 2D axis into dot at the centre of a 1D number line.

Hypothisis

There is no set whose cardinality is strictly between that of the integers and the real numbers.

Conjecture: FALSE

The infinite set of whole numbers is contained within the numbers 0 and 1, as all whole numbers have a reciprocal value. The naturals numbers are derived from the infinite division of this number space by the infinite series of prime numbers. This set can be represented as a line of unit one.
When the whole set of integers is doubled, then the second set can be placed at a 90°. In this case the line drawn between the two end points will be the bijection of the two infinite sets.
This forms a second type of infinite set whose length is √2. As this infinite defines the space of the bijection, it has to be different type of infinity from two infinite sets of cardinal numbers that create the bijection.
The ratio of 1:√2 remains consistent between the two sets on infinite integers. Sampling the set at any equal interval on the two axis always the same ratio, therefore, we can say that the line-length contains the infinite set of all the root numbers values for all real and subsequently natural numbers.
Root numbers are transformed into square numbers when they are multiplied by the same value, or infinite set. This is achieved when the set of integers that was doubled in the first instance is doubled again, to form a second identical triangle with of the same dimension (1 : 1: √2)
Again the two hypotenuse can be place at 90°, and a line drawn between them. This will exhibit a ration of 1:2 when compared with the integar sets. Again this ratio remain constants when any two equal points are sampled on the √2 (infinite root number) lines, and find the distance of its square number.
As the infinite set of integers is qualified is by the length of ONE, the size of the infinite set of square numbers is twice in length TWO, and and both contain the complete set of integers.
Therefore, as the square infinite set is larger in length than the integers and smaller than the infinite set of real numbers, the Continuum hypothesis must be false.

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Solving the Infinite