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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Paradoxes: Gabriels Horn

Gabriel’s horn is the function on the interval rotated around the x-axis. Intuitively, it should require less paint to fill the horn than to paint the surface (imagine painting a box or a ball). With Gabriel’s Horn this is not the case. In fact, There is an infinite surface area and only a finite volume. […]

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Paradoxes: Zeno of Elea

Zeno of Elea was a Greek philosopher famous for paradoxes which assert that motion is nothing but an illusion. It is assumed that Zeno created the paradoxes in support of Plato’s Parminedes.  We will present two of these paradoxes along with a refutation.   1. Achilles and the Tortoise Achilles was challenged to a race by […]

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