# Boxing Pythagoras

## Intuitionism and the Excluded Middle

Introductory lessons on Logic often make note of three basic, but powerful, principles which are so universally recognized that they are commonly referred to as the Laws of Logic. The first is the Law of Identity which states something like, “A thing is equal to itself.” The second is the Law of Non-Contradiction, sometimes phrased as, “A proposition cannot be both true and false at the same time.” The third is known as the Law of the Excluded Middle which declares, “Either a given proposition is true or else its negation is true.”

A classic example of the Law of Identity might be, “Socrates is Socrates.” An illustration of Non-Contradiction could be, “Socrates cannot both be mortal and not be mortal at the same time.” For the Excluded Middle, we would say, “Either Socrates is mortal or else Socrates is not mortal.” This all seems perfectly obvious and simple, even to complete beginners in the study of Logic.

However, one might be surprised to learn that the Law of Excluded Middle is actually a source of some controversy in philosophy– particularly in the Philosophy of Mathematics, where there exists a small but strong community which rejects this principle vehemently.

## The Axiom of Infinity

In my previous post introducing the concept of Set Theory, we discussed one method for constructing the Natural numbers– a method often referred to as a Von Neumann construction. Using that method, we start with the Empty Set ($\emptyset$) and then systematically build the Natural numbers by following a rule. As described in that post, this was a step-wise process: look at a number, find its successor, look at the new number, find its successor, repeat ad infinitum. Now, obviously, given a finite amount of time there would be no way to perform this process enough times to generate every Natural number, since every new number we create would still have yet another number succeeding it.

But what if we want to discuss the whole set of Natural numbers?

As we just noted, we cannot construct the Natural numbers in a step-wise manner in order to get all of them. However, mathematicians like Ernst Zermelo, Abraham Fraenkel, and Thoralf Skolem devised a very clever way to take the very same ideas from our step-wise construction in order to discuss a whole, completed set. We refer to this notion as the Axiom of Infinity, and it is one of the premises which underlies the vast majority of modern mathematics.