We can write a computer program that will successively print out the digits of the decimal expansion of Pi. We...

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## Ordinal Numbers

One of the most useful properties of the whole numbers is that every non-empty subset has a least element; this...

Continue reading...## All Horses are the Same Color

If you know how to prove things by induction, then here is an amazing fact: Theorem. All horses are the...

Continue reading...## Continuum Hypothesis

We have seen in the Fun Fact Cantor Diagonalization that the real numbers (the “continuum”) cannot be placed in 1-1 correspondence with...

Continue reading...## Equidecomposability

Two sets A and B are said to be equidecomposable if you can partition set A into a finite number of subsets...

Continue reading...## Envy-free Cake Division

Say you and a friend wish to share a cake. What is a “fair” way to split it? Probably you...

Continue reading...## Banach-Tarski Paradox

Did you know that it is possible to cut a solid ball into 5 pieces, and by re-assembling them, using...

Continue reading...## Sierpinski-Mazurkiewicz Paradox

If you’ve seen the Banach-Tarski paradox, you know that it is possible to cut a solid 3-dimensional ball into 5 pieces...

Continue reading...## Arrow’s Impossibility Theorem

Elections are democracy in action. People go to polls and express their preferences, and somehow we must aggregate the preferences...

Continue reading...## All Numbers are Interesting

There are clearly many interesting whole numbers. For instance, 2 is the only even prime number, 3 is the first...

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