User blog comment:Vel!/Call for math facts/@comment-1605058-20140730095534


 * Three-dimensional analogue of Sierpińki's carpet is called Menger's sponge.


 * It has been proven that, for a real number x, if we take geometric mean of first n numbers in continued fraction expansion of x, then as n tends to infinity this mean will almost always converge to universal constant called Khinchin's constant. Interestingly, even though this holds for almost all numbers, we don't have any specific example for which it holds.


 * It is known that, for any infinite cardinal A, if there is no cardinals strictly between A and 2^A nor between 2^A and 2^2^A then A can be well-ordered. It's unknown if the first condition alone is enough.


 * Axiom of determinacy states that in every 2-player game where players in turn choose natural numbers, there is countably many turns and there are no ties, one of the players has a winning strategy. This axiom sits in contradiction with axiom of choice, thus leading to many counterintuitive results, but is still extensively analysed.


 * "There is more real numbers than definitions, thus there are undefinable reals" - this is so called math-tea argument, standard for showing existence of undefinable objects. As convincing as it sounds, it's actually wrong - formal language can't define what "definable" means, and thus we can't argue about it. Model-theoretic techniques can be used to exhibit counterexamples to this statement (so called pointwise definable models have all of its elements definable).


 * There exists an everywhere differentiable function which is not monotonic on any interval. It's derivative is continuous function which on every interval takes both positive and negative values.


 * If we color natural numbers using finite number of colours, then there will exist singly-colored arithmetic progressions of any length. This result is known as Van der Waerden's theorem. Its strengthening, Szemeredi's theorem, states that if set has positive upper density among natural numbers, then it will contain arithmetic progressions of arbitrary length as well. A result of Green and Tao states that similar result holds for subsets of prime numbers, provided we use relative density.

That's it for now
 * There exist nonstandard models of Peano arithmetic. In these there are numbers beyond 1,2,... we are familiar with, but axioms of arithmetic cannot differ between these new numbers and standard numbers.