The Continuum Hypothesis was the number one mathematical challenge set by Hilbert at the start of the 1900s. It was only ever solved in the negative. Yet, our discovery of Aleph 0.5 offers a simple solution with deep implications.
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.
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 there are exactly double the number (infinite dencity) of infinity whole number factions, as there are whole numbers.
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.
We call this infinite set of fractional numbers Aleph 0.5 ( ℵ0.5), as it is exactly double the density of the cardinal numbers.
Watch the Video
In this video, Colin Power explains the mathematical process that generates Aleph 0.5 from the infinite set of whole number fractions, revealing that they exhibit a double density of number, compared to the Whole numbers from 1 to infinity
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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.
divisions of reciprocal number space
The numbers that constitute aleph 0.5 are found as the infinite set of whole number fractions, each found in reciprocal number space, between zero and one. They begin with 1/1 which establishes the unit measure of reciprocal space. The next number divides the line into 2 parts. 1/2 is added to 2/2 to produce 1.5. In the next step, reciprocal space is divided into 3 parts, and 1/3 + 2/3 + 3/3 = 2, and so on. As each step, reciprocal space is divided into the next number of equal parts, and they are added together in sequence from the first part up to the last. As the whole numbers increases, so the size of each part between zero and one gets smaller.
Triangle and square numbers
Notice that Aleph 0.5 adds up the entire set of number fractions to produce a final result, which advances in half steps. Similarly, the number of whole numbers are each based on the preceding ones, but only the final result is counted if each row were added together then it would create the triangular number series, 1, 3, 6,10 and so on. Adding together pairs of the numbers creates the square number series, 1, 4, 9, 16, and so on.
Bijection and the silver/golden ratio
We can also express Aleph 0.5 on a square plain by employing Cantor’s method of bijection to produce a geometric comprehension of this infinite set. To begin, we can set the 1:1 bijection of the whole numbers. To do this, we set up an axis at 90° to the first. We can then draw diagonal lines from 1 to 1 correspondence. This creates a half filled square. When we perform the bijection between all whole numbers and aleph 0.5, the numbers start at 1 instead of zero, and fills a quarter of the square. This is because the number of Aleph 0.5 are all identified within the reciprocal space between zero and one. Thus, ONE becomes the boundary with divisions of Aleph 0.5 diminishing towards zero as the whole numbers increase towards infinity.
Left: the 1:1 bijection of whole numbers
Right: the bijection of whole numbers to Aleph 0.5, with whole numbers on the x-axis and reciprocal fractions on the y-axis.
Geometrically, the 1:1 bijection fills a surface which when filled represents the entire set of real numbers, including 1. Based on a unit of 1 the diagonal line is in ratio by √2. The bijection of Aleph 0.5 forms a triangle whose hypotenuse is √1.25. The geometric significance of this is that the former produced the Silver Ratio (√2±1) whereas the latter produces the Golden ratio (√1.25±0.5).
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‘
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YOUR QUESTIONS ANSWERED
Got a Question? Then leave a comment below.
I did not quite get the part where you say “double the result we create the whole number value for the next cardinal step”?
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.
Isn’t the result of 1/1 the same as 2/2?
Yes it is and no it isn’t. 2/2 is stil a line divided into 2 parts, whereas 1/1 is a whole line. In each case the number of units counted differs.
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