The post on two easy to grasp explanations on Gödel's theorem and the Banach-Tarski paradox made me think of other explanations that I found easy or insightful and that I could share them as well.

1) Here is a nice proof of the Pythagorean theorem:

2) An easy and concise explanation of expected utility calculations by Luke Muehlhauser:

Decision theory is about choosing among possible actions based on how much you desire the possible outcomes of those actions.

How does this work? We can describe what you want with something called a utility function, which assigns a number that expresses how much you desire each possible outcome (or “description of an entire possible future”). Perhaps a single scoop of ice cream has 40 “utils” for you, the death of your daughter has -⁠274,000 utils for you, and so on. This numerical representation of everything you care about is your utility function.

We can combine your probabilistic beliefs and your utility function to calculate the expected utility for any action under consideration. The expected utility of an action is the average utility of the action’s possible outcomes, weighted by the probability that each outcome occurs.

Suppose you’re walking along a freeway with your young daughter. You see an ice cream stand across the freeway, but you recently injured your leg and wouldn’t be able to move quickly across the freeway. Given what you know, if you send your daughter across the freeway to get you some ice cream, there’s a 60% chance you’ll get some ice cream, a 5% your child will be killed by speeding cars, and other probabilities for other outcomes.

To calculate the expected utility of sending your daughter across the freeway for ice cream, we multiply the utility of the first outcome by its probability: 0.6 × 40 = 24. Then, we add to this the product of the next outcome’s utility and its probability: 24 + (0.05 × -⁠274,000) = -⁠13,676. And suppose the sum of the products of the utilities and probabilities for other possible outcomes was 0. The expected utility of sending your daughter across the freeway for ice cream is thus very low (as we would expect from common sense). You should probably take one of the other actions available to you, for example the action of not sending your daughter across the freeway for ice cream — or, some action with even higher expected utility.

A rational agent aims to maximize its expected utility, because an agent that does so will on average get the most possible of what it wants, given its beliefs and desires.

3) Micro- and macroevolution visualized.

4) Slopes of Perpendicular Lines.

5) Proof of Euler's formula using power series expansions.

6) Proof of the Chain Rule.

7) Multiplying Negatives Makes A Positive.

8) Completing the Square and Derivation of Quadratic Formula.

9) Quadratic factorization.

10) Remainder Theorem and Factor Theorem.

11) Combinations with repetitions.

12) Löb's theorem.

I want to share the following explanations that I came across recently and which I enjoyed very much. I can't tell and don't suspect that they come close to an understanding of the original concepts but that they are so easy to grasp that it is worth the time if you don't already studied the extended formal versions of those concepts. In other words, by reading the following explanations your grasp of the matter will be less wrong than before but not necessarily correct.

World's shortest explanation of Gödel's theorem

by Raymond Smullyan, '5000 BC and Other Philosophical Fantasies' via Mark Dominus (ask me for the PDF of the book)

We have some sort of machine that prints out statements in some sort of language. It needn't be a statement-printing machine exactly; it could be some sort of technique for taking statements and deciding if they are true. But let's think of it as a machine that prints out statements.

In particular, some of the statements that the machine might (or might not) print look like these:

P*x	(which means that	the machine will print x)
NP*x	(which means that	the machine will never print x)
PR*x	(which means that	the machine will print xx)
NPR*x	(which means that	the machine will never print xx)

For example, NPR*FOO means that the machine will never print FOOFOO. NP*FOOFOO means the same thing. So far, so good.

Now, let's consider the statement NPR*NPR*. This statement asserts that the machine will never print NPR*NPR*.

Either the machine prints NPR*NPR*, or it never prints NPR*NPR*.

If the machine prints NPR*NPR*, it has printed a false statement. But if the machine never prints NPR*NPR*, then NPR*NPR* is a true statement that the machine never prints.

So either the machine sometimes prints false statements, or there are true statements that it never prints.

So any machine that prints only true statements must fail to print some true statements.

Or conversely, any machine that prints every possible true statement must print some false statements too.

Mark Dominus further writes,

The proof of Gödel's theorem shows that there are statements of pure arithmetic that essentially express NPR*NPR*; the trick is to find some way to express NPR*NPR* as a statement about arithmetic, and most of the technical details (and cleverness!) of Gödel's theorem are concerned with this trick. But once the trick is done, the argument can be applied to any machine or other method for producing statements about arithmetic.

The conclusion then translates directly: any machine or method that produces statements about arithmetic either sometimes produces false statements, or else there are true statements about arithmetic that it never produces. Because if it produces something like NPR*NPR* then it is wrong, but if it fails to produce NPR*NPR*, then that is a true statement that it has failed to produce.

So any machine or other method that produces only true statements about arithmetic must fail to produce some true statements.

The Banach-Tarski Paradox

by MarkCC

Suppose you have a sphere. You can take that sphere, and slice it into a finite number of pieces. Then you can take those pieces, and re-assemble them so that, without any gaps, you now have two spheres of the exact same size as the original.

[...]

How about this? Take the set of all natural numbers. Divide it into two sets: the set of even naturals, and the set of odd naturals. Now you have two infinite sets, the set {0, 2, 4, 6, 8, ...}, and the set {1, 3, 5, 7, 9, ...}. The size of both of those sets is the ω - which is also the size of the original set you started with.

Now take the set of even numbers, and map it so that for any given value i, f(i) = i/2. Now you've got a copy of the set of natural numbers. Take the set of odd naturals, and map them with g(i) = (i-1)/2. Now you've got a second copy of the set of natural numbers. So you've created two identical copies of the set of natural numbers out of the original set of natural numbers.

[...] math doesn't have to follow conservation of mass [...]. A sphere doesn't have a mass. It's just an uncountably infinite set of points with a particular collection of topological relationship and geometric relationships.