Shows how the P=NP? problem has a strange, self-referential character that’s not quite like anything previously encountered in mathematics, including in the work of Gödel and Turing.

PTTP is an implementation of the model elimination theorem-proving procedure that extends Prolog to the full first-order predicate calculus.

One of the justifications for the use of Bayesian probability theory. Probability is interpreted as a formal system of logic, the natural extension of Aristotelian logic into the realm of reasoning in the presence of uncertainty.

Implies that every arbitrarily long proof for any statement in propositional logic can be formalized (in polynomial time), so that one can check whether it is correct or not by only reading a constant number of letters from it!

A fixed-point theorem is a result saying that a function F will have at least one fixed point (a point x for which F(x) = x), under some conditions on F that can be stated in general terms. Results of this kind are amongst the most useful in mathematics.

Theorem : Every Goodstein sequence eventually terminates at 0. Unprovable in Peano arithmetic. In contrast to the Collatz conjecture, Goodstein's theorem has been proven using the axioms of set theory.