 Mind Matters Natural and Artificial Intelligence News and Analysis

# TagCantor's theory of infinity

## Of Infinity and Beyond

What are the problems and solutions with infinity in mathematics?

The concept of infinity has plagued a great many proofs, both formal and informal. I think that there are two foundational problems at play in most people’s thinking about infinity that causes issues. The first problem people have with infinity is that they treat it as if it were a single value. Because infinity is bigger than all possible natural numbers, people assume that it is bigger than any number, and therefore there is nothing beyond infinity. Therefore, people have the concept that if I have two infinities, then I still have the same number.  They believe that 2 * infinity = infinity. However, using that logic can quickly lead to contradictions. This problem is exacerbated by much mathematical notation. People often will Read More ›

## 3. In Infinity, Lines and Squares Have an Equal Number of Points

We can demonstrate this fact with a simple diagram

In previous posts, we have established that two sets are of the same size if there is a one-to-one correspondence between the elements of both sets. Applying this principle to Cantor’s theory of infinity leads us to the weird but valid conclusion that the number of points on a line segment is the same as the number of points in a square. To show that this is true, here is a picture of a unit length line segment and a unit square. Let’s choose a point on the line segment. Let’s say 0.6917381276543… . It’s shown with a big blue dot on the line segment on the left. If this point corresponds to an irrational number, it goes on forever Read More ›