Geometric solution to the Russel Paradox

Welcome everybody to another episode of Solving Infinity! I’m Colin Power, your host, and today we are going to delve into the mind-blowing world of the Russell Paradox in mathematics. This paradox has remained unsolved for a long time, challenging our understanding of infinity.

The Russell Paradox revolves around the concept that there are more infinite numbers between zero and one than there are whole numbers in existence. This idea creates a paradox because traditional thinking would suggest that all whole numbers can be contained within the space of zero and one, as each whole number has a corresponding reciprocal value when divided.

However, upon closer examination, we discover that the numbers between zero and one do not follow the usual pattern of increasing towards larger numbers. Instead, they diminish towards zero. This introduces the notion of zero as a boundless concept, with infinity extending inward.

Furthermore, there are different types of infinity. While some sets of infinity explode and grow larger, there is also an infinite set of fractions that diminish towards zero. Our previous post explored Aleph 0.5, where we demonstrated that the sum of these fractions forms an infinite series exactly half the size of the series of whole numbers.

This intriguing finding indicates a greater density of infinity between zero and one. Now, let’s explore how we can visualize and understand this concept using just a square piece of paper and a bit of origami.

Imagine folding the paper and opening it out again, creating four distinct spaces. By labelling one space as zero and another as infinity, we can see a number line that spans from zero to one and includes all the numbers falling between them. Any number greater than one, including decimal fractions, would fall beyond this range.

By folding the paper across its axis, we create a new number line, which encompasses all the numbers from zero to infinity. Here, the main difference lies in the way the lines cross. On the number line representing whole numbers, each point is formed by a single line crossing. On the number line representing fractions, there is a difference in the way each point is formed, as multiple lines intersect.

This geometric model provides a resolution to the Russell Paradox, explaining how there can be more numbers between zero and one than the combined set of all whole numbers. The numbers between zero and one can be represented as a different quality or type of dot, resulting in a distinct pattern on the number line.

To better understand the resolution of the paradox, let’s consider a simple example with a card. If we write the statement “On the other side of this card is true” on one side and “On the other side of this card is false” on the other side, we encounter a logical paradox. However, when we fold the card in half, both statements become true, eliminating the contradiction.

This folding of number space and the card analogy demonstrate the resolution of such paradoxes that have baffled mathematicians for centuries. By comprehending the structure of number space, we can unlock a new understanding of mathematics and its infinite complexity.