Solving the Russell Paradox: 2D geometric solution to the continuum hypothesis


When Bertrand Russell posed a mathematical paradox, it seemed like an impossible contradiction. The outcome concluded that the Continuum Hypothesis could not be solved using conventional Mathematical Axioms. This led mathematicians to believe it could never be solved. We offer a simple solution that folds number space, and has vast implications for modern mathematics.


At the heart of modern mathematics is set theory. A set can be viewed as a group of objects. If an object appears inside a set, then it is concluded that it cannot be a part of that set. This seems to be quite logical, however when considering the nature of infinity and numbers, it leads us to a paradox. Let us consider the complete set of infinite real numbers, ranging from zero to infinity. The infinite set contains the number one. Yet, all numbers above one exhibit a reciprocal fraction between zero and one.

How can we qualify the infinite set of numbers when that infinite set is contained within the numerical space between zero and one, as those numbers are also part of the set? All attempts to resolve this mathematical paradox have failed, leading mathematicians to believe that the continuum hypothesis is not resolvable in terms of a mathematical proof.

Our solution suggests we can resolve this paradox by folding numerical space. This has rather interesting implications for not just the nature of mathematics, but also the very reality we are experiencing.

Watch the Video

This concept of folding reciprocal number space into whole number space is explained in this video by Colin Power.



A part of the whole

Let us begin by considering the number line. Beginning from zero, the numbers extend towards infinity. This can be considered as an infinite set. Mathematicians often refer to the set as Aleph 0. It contains all whole numbers, as well as all fractional values. So how many of these numbers are there?

infinite set of whole numbers

In between each number is an infinite set of fractional numbers, so how many numbers are there?

The simple answer might be an infinite number, however, in between each number unit there also exists an infinite number of fractional values. Now here comes the problem. All the numbers above 1, including all fractions, are found to exist in the space between zero and one. Yet, the numbers between zero and one also contain all the infinite set of numbers above one. This creates a paradox. How can the infinite set of all real numbers, of which numbers between zero and one, also be contained within the space between zero and one? As there are the same number of infinite fractions between each number, surly the whole infinite set must be larger than the numbers between zero and one? This problem has never been full resolved by current mathematics. In fact, it has a mathematical proof that it cannot be solved using our current modern mathematical system. However, 4D and Geometric Maths do not comply to the standard axioms of traditional mathematics, and therefore, the solution is not only possible, but incredibly simple to understand.

Key point

4D and geometric maths can solve the Russell paradox in a simple to understand manner.

number Uints

The first point to consider, if we wish to solve the Russell Paradox, is the nature of whole numbers.

Whole numbers are an infinite set. We can carry on counting to any number, and we always find that there is another number beyond this count than the one reached thus far. It is this nature of numbers that leads to the infinite solution, 0 ±1 = ±∞. Whilst at first impressions this appears to be a relatively simple expression, the implications are quite profound. Infinity is intrinsically linked to our nature of consciousness. We are not counting numbers, we are calculating them based of a series. Notice that we only need the numbers zero and one. 

However, what terminates the infinite is the reduction of a number. In other words, when we count to one, then the number two is already insinuated. 2 – 1 = 1, which allows the number one to ‘exist’. Even if we are not aware of the number two as we have not counted it, we still are aware of its existence, otherwise we could never extend the number count to the next number. The same can be said of all numbers. 3-1=2 for example. Whilst at first this might seem like a moot point, it is in fact quite a deep metaphysical concept.

The universe is constructed from an infinite set of number. When we observe countable objects in reality, we are distinguishing them from an infinite number of points in ‘space’. At a deep level, this is how were are able to observe reality. We can create a boundary around recognisable objects and define them as units.

A bowl of apples can be counted. Yet, that does not mean the apples in the bowl are the only ones to exist. We can add another apple to the bowl and so on. The bowl itself is a container. If we cut each apple into two, then we double the number of pieces. That does not mean we double the number of apples. Moreover, we have performed an action of consciousness that allows us to redefine the numbers of countable apple pieces in the bowl. We can in theory divide a single apple into an infinite number of smaller pieces, and count them. We are not adding more apples, we are redefining the boundary.

Similarly, a single unit on a number line can be divided into an infinite number of smaller fractions without changing the nature of the unit itself.

Numbers and MEasurment

We have covered the nature of reciprocal space in terms of the boundary conditions relating to mathematical functions. The quality of the number ONE acts as a boundary unlike any other number through reiterative roots and powers.

We can therefore distinguish reciprocal space as exhibiting a different quality compares to all other numbers greater than one. This is easily provable, as any number above ONE will have a reciprocal value. If we mark out a distance measure of one, then the second distance measure of the same size will contain all numbers above ONE. At first, this seems improbable, as we use measurement in our investigation of the material universe. However, this is resolved when we realise that we are not actually measuring distances. This is where all scientist begin to fail in their appreciation of how ‘space’ actually works. Any distance between an observer and an object is actually only a distance of ONE. This distance is then divided into units based on the scale employed as a measuring device. This might seem completely illogical. However, it is the nature of observation. Whenever we take a measurement we are observing a unit distance of ONE which is then divided into smaller units. I.E: we ONLY ever extrapolate from the infinite.

For example, let’s say we measure the distance from the Earth to the Sun. We can call it an ‘Astronomical Mile’ or we can call it 150,000 KM. both are correct based on the measurement. The observer creates the scale.

From the perspective of the observable universe (93 billion light years in diameter) we can also set the scale to be ONE. The distance from the earth to the sun now becomes a fraction that is so small it becomes completely impractical. We therefore change the scale to suit the needs.

Our capacity to alter this scale to suit our own needs is therefore derived from the nature of our consciousness. The implications of these statements are quite terrifying for mathematicians and scientists, who have predicated their whole understanding of number and the universe on systems of algebra and measurement, based in the nature of 3D. Yet 4D and geometric maths show that systems of measurement are human inventions. Numbers are different from mathematics. Numbers are expressed in geometry (space), time (music), which can transcend dimension. Once we begin to adapt our consciousness towards 4th dimensional awareness, the universe changes in character. The unfathomable becomes comprehensible. Yet, this shift is by no means easy. It requires insight, a change of perception, that begins with comprehending the true nature of infinity.

Large within the small

To begin adapting our perception to the infinite, we must accept that all numbers beyond ONE are contained within the unit space between ONE and TWO. Whilst at first this may seem to defy logic, the model is actually very simple. We can begin by drawing a line and dividing it in half. On the one side we have WHOLE number space and on the other we have RECIPROCAL number space. In our mathematical system, we define these two spaces by the letter R (for reciprocal) and the letter ᴙ for whole number space. When placed back to back, we create a glyph that represent the combination of the two spaces.

For those who are interested in the deeper meaning of the glyph can examine the contents of the ‘KNOWLEDGE BOOK‘.

The fact that whole and reciprocal number space are ‘mirror reflections’ of each other can be clearly demonstrated mathematically. Any real number above ONE, which includes all fractions, can be expressed as a reciprocal value and vice versa. Using a simple calculator, we can enter any value above ONE. By placing the number one over the it, we create its reciprocal fraction, between zero and one.

any whole number above two has a reciprocal value
Any number (including fractions) has a reciprocal between zero and one. TRY IT FOR YOURSELF!

Therefore we can represent this as line divided into two equal spaces. The implications for this are that the number ONE and TWO are different from what is normally believed. In traditional  mathematics numbers are a product of a number line, whereas what we have insinuated here is that this is not the case. All numbers are reflected from reciprocal space into whole space. The implication is that ‘number units’ as we commonly conceive them are only an aspect of consciousness, and not an actual truth or reality!

whole and reciprocal numbers within two number units
Whole and reciprocal numbers are contained within the number2 ZERO, ONe, and TWO.

In Geometric and 4D Maths, we often write the numbers ZERO, ONE and TWO in capital letters in order to differentiate them from our standard notion of numbers on a number line. These capitalised numbers are not just unit measures. They are the boundaries that contain all other numbers. This can be demonstrated on the 4D Calculator by setting the time and dimensional numbers to 2. Now, when the space number is changed to any other value, the result will always devolve towards 2.

d calculator prooves all numbers devolve towards TWO
Download our 4D CALCULATOR and try it for yourself!

No such thing as a number line

Throughout our mathematical history, it has always been assumed that number appear on a line. The implications of our solution to the Russell Paradox and the truth expressed by the nature of ᴙR and the 4D calculator severely undermines this assumption. The number line is just a mathematical figment of human consciousness. In truth, numbers cannot be expressed in a 1 dimensional form. This realisation hold deep ramifications for both mathematics and science. The truth is that numbers, as we commonly perceive them, can only be express in 2 dimensions or higher. Only be moving into the 3rd and 4th dimension are we able to solve infinity.

A 2D number plain can be expressed as a single square. The bottom side represents all whole numbers. We divide the line in half, then the first section expresses the numbers zero and one. The second half expresses all other whole numbers up to infinity. Each whole number unit is expressed as a line, instead of the traditional notion of a dot. Whilst the first section has no lines as there are no whole numbers between zero and one, the second half contains an infinite set of numbers 2 to infinity. In our theory of geometric maths, an infinite set of parallel lines forms a 2-dimensional surface. The result is that one half is formed of an empty surface, whereas the other forms a fully filled surface. We represent this as a series of veritcle lines.

In the above image we have included the idea of the fractional numbers appearing in the first section. This is a guide that helps us to visualise the nature of the numbers reflecting from reciprocal into whole number space. However, in the final model, the fractional numbers are placed on the vertical axis.


Here we can see that the infinite set of whole numbers above ONE are represented within the same ‘sized’ numerical space as their reciprocals. Each whole number exhibits a line instead of a dot, forming a 2D plane that gets denser as the whole numbers increase.

Fractional Numbers

In mathematics, there is a differentiation between the Whole (Natural) Numbers and the complete set of Infinite fractions. In our solution to the Russell Paradox, the next step is to place the fractional numbers on a vertical axis. Each now represents a line that extends in the horizontal.

The blue lines in the vertical represent any whole numbers, whereas the red line in the horizontal represent all fractional values.

There are in infinite number of fractional values for each number unit. As there are an infinite number of whole numbers, there will be infinity² fractional numbers. This creates a completely filled surface. We have limited the number of red horizontal lines in order to make the concept clearer. Note that the same is not true of the infinite set of whole numbers, that get closer together as each step, and yet will always exhibit an ’empty’ number space from one unit to the next. Within each number space, there is always an infinite number of fractional values that can be found.

This takes a little bit of contemplation in order for us to begin to truly perceive this novel number space. Infinity can be infinity big and infinity small, but it can also be expressed between two finite number points. For this reason, we often consider infinity in terms of density, rather than as a number that is countable.


The REAL number line.

The next step to solving the Russell Paradox is to draw a diagonal line from the ZERO across the square number plain. When we do, we see that the number line ends at ∞².

The green line represents the ‘REAL’ number line the ranges from ZERO to Infinity²

What we see is that each ‘point’ on the green line is defined by a red fractional value. This is true of all numbers, even those that fall between ZERO and ONE. However, the whole number values above one are defined by an additional blue line in the vertical. This differentiates numbers in the reciprocal space to those that are found in whole number space.

Now here comes the interesting part. If we cut out the square, we can fold the zero point over to meet ∞². When we do, we find that the horizontal red lines change orientation to become vertically aligned. The whole numbers now have reciprocal fractions between zero and one that represent each unit.

We have solved the Russell Paradox!

That which is reciprocal now becomes whole. Notice that any reciprocal value found on the green line is identified by a red line crossing it, forming a node. Whereas in the Whole number space, each point on the green line where the whole numbers fall is defined by 3 intersecting lines.

As we consider this concept in greater detail we begin to realise that numbers can only ‘exist’ if they are defined by a numerical line, Any real number smaller than ONE is defined by a single fractional line, that when folded becomes a whole number line by nature of its change in orientation.

Whilst this is a simple model of numerical space, it begins to solve the most challenging problem of the last century, the Continuum Hypothesis, and introduces the reader to a new concept of Geometric and 4D maths. Whilst we have only dealt with positive numbers, we can apply the same concept in the negative. However, in order to gain a fuller understanding of the structure of numbers, requires that we extend the model into 4D space, which introduces other types of infinite number sets not presently defined by standard mathematical axioms.

Why not try this at home

If you are struggling with some of the concepts present in this post, don’t worry. Remember, the greatest mathematical minds have pondered a solution to the Russell Paradox and all failed to resolve it. However, it does help it you can engage with the idea practically. Download the simple PDF and try cutting out the square and folding it, then hold it up to the light. See for yourself that the lines do change orientation.

Then think about it.

A Final Word - The Byde Effect

Whilst writing this post I received news that a good friend of mine, John Byde, passed away. As we have spent much time talking about new mathematical concepts, I wanted to dedicate this mathematical discovery to him. The ‘Byde Effect’ is my name for the folding of numerical space in the resolution of Infinity.

I am sure he would approve.



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So what does this tell us about the structure of numbers?

There is no such thing as a number line, in the traditional sense. Instead, numbers can only be constructed from an intersection of fractional and whole numbers. The implications for modern mathematics is that the complex plane is not a the ‘end of all numbers’. Instead, we find we are only beginning to understand the true nature of number, upon which we have created a particular mathematical system.

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What are the consequences of solving the Russell Paradox?

As its heart we are suggesting that the only ‘numbers’ that exist are ZERO, ONE and TWO, which are the boundaries needed to produce all the real numbers. We can identify that the Aleph 0 (reals) has infinity² compared to the whole numbers, Aleph 1 (natural). Moreover, we have begun to completely revolutionise our notion of number, in the same way that Quantum Physics revolutionised Newtonian science.

Carry On Learning

This article is part of our new theory, ‘Maths of Infinity
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Sanne Breimer

If numbers don’t exist, then how comes science can use calculations to predict experimental results?


The nature of numbers is geometric. The reason why science can produce consistent results is because they are measuring phenomena in ‘space’. In our new system of dimensionless science, we have translated over 40 constants into pure geometric ratio. This resolves many problems facing science today, that has based its reasoning on measuring reality, rather that comprehending it as a geometric ratio.

Hilary Faverman

If numbers do not exist, then how comes we can perform algebra and many other systems of mathematics with exact results?


In 4D maths algebra is only valid when times and dimension are set to 1. Outside of this all mathematical functions produce other results. As standard mathematics is perform under these ‘normal’ space-time conditions, we have never be able to examine mathematical operations outside of this time space awareness. 4D Maths can explore mathematical results outside of our normal space time condition.

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