The Continuum Hypothesis was the number one mathematical challenge set by Hilbert at the start of the 1900’s. It was only ever solved in the negative. Yet our discovery of Aleph 0.5 offers a simple solution with deep implications.
Overview
In Georg Cantor’s notation, the continuum hypothesis can be stated by the simple equation 2^{ℵ}_{0} = ℵ_{1}, where ℵ_{0} is the cardinal number of an infinite countable set (such as the set of natural numbers), and the cardinal numbers of larger “wellorderable sets” are ℵ_{1}, ℵ_{2}, …, ℵ_{α}, …, indexed by the ordinal numbers. The cardinality of the continuum can be shown to equal 2^{ℵ}_{0}; thus, the continuum hypothesis rules out the existence of a set of size intermediate between the natural numbers and the continuum.
KEy Points

Whole number fractions when added together produce a number series that progresses at half the rate of the whole numbers.

This demonstrates a specific relatipnship between whole numbers and the sum of fractions created at each step

This established the quadratic nature of numbers, and allows us to reorientate the structure of the number line into a square, which radically shifts our understanding of the number ¡
THE
Concept
Aleph 0 (ℵ0)
The Division of Number Space between ZERO and ONE by whole number fractions creates a series of fractional parts that can be added together to produce a total that proceeds at the rate of half of the Whole Numbers (Aleph 0.5). As this density of infinity is exactly double that of the infinite set of whole numbers, we can establish that:
This establishes a bijection (one to one correspondence) between the whole numbers (left column above), and the total for the summed fractions (right column). The conjecture laid out by Cantor suggest that no infinite set exists between the cardinal and real number sets. Yet we can clearly see that the addition of whole fractions produces an infinite set that progresses at exactly half the rate of the cardinal numbers.
Therefore the continuum hypothesis is incorrect.
Key point
We call this infinite set of fractional numbers Aleph 0.5 ( ℵ0.5), as it is exactly double the density of the cardinal numbers.
REVIEW
Find out more about 4th dimensional mathematics by reading these interesting articles.
FIND OUT MORE ABOUT THE ZERO BOUNDARY
FIND OUT MORE ABOUT THE INFINITY OF ONE
FIND OUT MORE ABOUT RECIPROCAL SPACE
Aleph 0.5  Expressed in fractions
The notion of Aleph 0.5 can be expressed in terms of fractional values. By doing so it becomes clear that we are adding together sequencial values of cardinal numbers from which the next value is derived.
THE
Conclusion
What does this tell us about the Continuum Hypothesis?
Whole fractions of sequential cardinal number create a numerical series that progresses though half steps. As this is an ordered set it forms a one to one correspondence with the cardinal set, which proves that Cantor’s conjecture is incorrect.
Aleph 0.5 and the nature of infinity
This simple discovery of the infinite set of whole fractions offers new insights into the nature of infinity.Whilst it solves the most difficult mathematical problem of the 20th century, it also helps us to develop a deeper understanding of topic mathematical problems such as the Riemann Hypothesis.
Carry On Learning
This article is part of our new theory, ‘Maths of Infinity‘
Read the main article or browse more interesting post from the list below
Mathematics without algebra
Central to mathematics and science is algebra. The concept of x + y = z is so intrinsic to our mathematical understanding that it has been considered a foundation of much of our modern world. Yet, in 4D maths algebra fails to work, calculations produce unpredictable results. This bring in
The Theory of Effort
The Theory of Effort forms a part of our new concept of 4D maths. Using this powerful technology, we are able to track the number of numerical iterations, and quantify calculations that would otherwise appear to produce the same results. Overview The Theory of Effort is a key concept of
YOUR QUESTIONS ANSWERED
Got a Question? Then leave a comment below.
Question?
I did not quite get the part where you say “double the result we create the whole number value for the next cardinal step”?
ANSWER?
Beginning with 1/1=1, we double it to get 2. Next we add 1/2 + 2/2= 1.5. double it to get 3, the next cardinal number. Then 1/3 + 2/3 + 3/3 = 2, which when doubled creates the next cardinal number 4, and so on.
Question?
Isn’t the result of 1/1 the same as 2/2?
ANSWER?
Yes it is. However the calculation that generates the result is different. When we perform mathematical calculations we like to include the quality of the equation itself. In this way we can establish a pattern that becomes predictable, i.e: an ordered set.
STILL Looking for Answers?
Ask a question…