  Cook and Reckhow 1979 pointed out that NP is not closed under complementation
iff there is no propositional proof system that admits polynomial size proofs
of all tautologies. Theory of proof complexity generators aims at constructing
sets of tautologies hard for strong and possibly for all proof systems. We
focus at a conjecture from K.2004 in foundations of the theory that there is a
proof complexity generator hard for all proof systems. This can be equivalently
formulated (for p-time generators) without a reference to proof complexity
notions as follows:
  * There exist a p-time function $g$ stretching each input by one bit such
that its range intersects all infinite NP sets.
  We consider several facets of this conjecture, including its links to bounded
arithmetic (witnessing and independence results), to time-bounded Kolmogorov
complexity, to feasible disjunction property of propositional proof systems and
to complexity of proof search. We argue that a specific gadget generator from
K.2009 is a good candidate for $g$. We define a new hardness property of
generators, the $\bigvee$-hardness, and shows that one specific gadget
generator is the $\bigvee$-hardest (w.r.t. any sufficiently strong proof
system). We define the class of feasibly infinite NP sets and show, assuming a
hypothesis from circuit complexity, that the conjecture holds for all feasibly
infinite NP sets.
