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.