logic

Computability of Real Numbers

We can write a computer program that will successively print out the digits of the decimal expansion of Pi. We can also write one that will list the digits of the decimal expansion of the square root of 2. Similarly, there are computer programs that...

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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 same color. Proof. We’ll induct on the number of horses. Base case: 1 horse. Clearly with just 1 horse, all horses have the same color....

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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 the rational numbers. So they form an infinite set of a different “size” than the rationals, which are countable. It is not hard to show that...

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