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 2ℵ0 = ℵ1, where ℵ0 is the cardinal number of an infinite countable set (such as the set of natural numbers), and the cardinal numbers of larger “well-orderable sets” are ℵ1, ℵ2, …, ℵα, …, indexed by the ordinal numbers. The cardinality of the continuum can be shown to equal 2ℵ0; thus, the continuum hypothesis rules out the existence of a set of size intermediate between the natural numbers and the continuum.
KEy Points
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Whole number fractions when added together produce a number series that progresses at half the rate of the whole numbers.
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This demonstrates a specific relatipnship between whole numbers and the sum of fractions created at each step
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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
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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.
At each stage the whole number integer is found below the factional line, whereas the cardinal numbers form the top part, in sequential order. When we sum up the total fractions in each row and double the result, we create the whole number value for the next cardinal step.
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
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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.
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