Axioms: Sets (Russell’s Paradox)

Bertrand Russell (1872-1970) It isn’t too often that we think about or even explore the fundamental building blocks of mathematics. These building blocks are called axioms. Axioms are statements taken to be true, i.e. they cannot be proven. This causes mathematicians, whether they know it or not, to take a lot of things on faith. All of […]

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Banach-Tarski Paradox

It is a common occurrence in mathematics that when something does go wrong, it goes terribly wrong. This exact phenomenon occurs with the Banach-Tarski Paradox. Informally, it says that one can take a sphere (in 3 or more dimensional space)  can be split into finitely many pieces and, using only rigid motions, can be rearranged […]

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The Prisoner’s Dilemma

The prisoner’s dilemma is a very popular situation in crime fiction, and can be found in many areas of real life. It originated from Merrill Flood and Melvin Dresher, however Albert Tucker is credited for formalizing it into its current form. So let us take a look. Two criminal partners, A and B, are arrested […]

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Coordinates in 3-Space

This post was inspired by a calculus student and will be in three parts: This on the development of different coordinate systems, one with calculus and one without. We are taught in school that the volume of a sphere with radius is . In this post we shall look at a development in calculus that will not […]

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