User blog comment:Vel!/Call for math facts/@comment-1605058-20140730170530


 * Langton's ant is known to escape to infinity on every finite initial configuration of cells. What isn't known is in how many ways it can do so - for all known starting positions ant eventually starts building out configuration known as "highway".


 * Gödel's incompleteness theorems are well known to apply to every theory at least as strong as Peano arithmetic. It also holds for some weaker theories, including Robinson's Q, which doesn't include any induction axiom.


 * Von Neumann-Bernays-Gödel (NBG) set theory is an extension of ZFC allowing some limited operations on proper classes. However, if we limit ourselves to sentences involving sets only, then NBG proves exactly the theorems of ZFC. This is why NBG is called conservative extension of ZFC.


 * One cannot exhibit a definition of a "small" subset of natural numbers which satisfies: 1. Every finite set is small, 2. Union of two small sets is small, 3. Subset of small set is small, 4. Set is small iff its complement isn't. Set of these subsets, which aren't small, would create a nonprincipal ultrafilter, existence of which is unprovable in ZF.


 * If P and NP are not equal, then P and NP-complete are disjoint. It's also known that if that's the case, there exist problems in NP which are of intermediate complexity - not in P nor NP-complete.


 * There are undecidable problems which are strictly easier than halting problem. There are no natural examples of such problems known.


 * Because forcing techniques are working with countable models, main difficulty in adding sets is taking care of not adding any information which would reveal the truth about size of sets. The way around that is adding only sets which are generic, i.e. they behave like an "average" set of its type.