Continuum Hypothesis – A Geometric Solution

In2Infinity

?? English
Summary

The number i has been traditionally assigned a vertical axis in the complex number plane. However if we transcend the notion of numbers only appearing as a point on a line, (1D Maths), then we can offer a new identity for the number i that expresses a flaw in the fundamentals of mathematical theory, which suggest that there is no such possibility of a negative square number.

KEY POINTS

✔All whole numbers are reflections of the numerical space between zero and one.
✔two number line combine at 90° to generate a new infinite set that has the length of √2.
✔√2 set is longer than 1 but shorter than 2 and contains the infinite set of all root numbers.
✔when a root number set transformed into a square number it creates the same sized is longer than 1 but shorter than 2 and contains the infinite set of all root numbers.
✔As this infinite set is larger in length than the integers and smaller than the infinite set of real numbers, the Continuum hypothesis must be false.

## The Proof

1. 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.
2. 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.
3. 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.
4. 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.
5. 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)
6. 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.
7. 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.
8. 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.