About problems...

>>8514828
Yes anon it is time to realize your destiny is to solve all of the math problem and pull ridiculous engineer dick only to have them realize your social skills are sub par from being such a math experts that you can only have sex with the most autistic engineers.

>>8514828
Does the set of all sets contain itself?

>ZFCtards btfo

>>8514828
Yes and no.

Start with a set of axioms and a proof system. Enumerate all possible proofs until you find a proof of either A or ~A, where A is the proposition you want to examine. Thus method solves all solvable math problems.

However, by Gödel, not all math problems are solvable. There will be statements S such that you can neither prove S nor ~S.

>>8514896
Give me an example of such statement please.

>>8514908
Assuming your system is consistent: "This statement cannot be proven by your formal system"
Read up on Godels proof, it's fairly straightforward and pretty cool

>>8514828
No. Look into Turing's halting problem, which showed that there is no general algorithm that can answer the question of whether an arbitrary Turing machine halts. This was proven equivalent to Hilbert's Entscheidungsproblem, or "decision problem", which asks if there is any general procedure for deciding whether any statement in first-order logic is true or false; the answer is nope.

For a specific example of an unsolvable problem - what is the 8000th Busy Beaver number?
http://www.scottaaronson.com/blog/?p=2725

There is, provably, no algorithm that would allow you to compute the Busy Beaver function in general, and (within the standard ZFC axioms that underpin most of mathematics) the value of BB(8000) cannot even in principle be proven.

>>8514908
a generalized version of the collatz conjecture is undecidable.

Gödel.

>>8514842
Explain how to construct such a set using ZFC. >Protip: you can't.