Summary: Is P Versus NP Formally Independent?
Scott Aaronson #
University of California, Berkeley
This is a survey about the title question, written for people who (like the author) see logic as for
bidding, esoteric, and remote from their usual concerns. Beginning with a crash course on Zermelo
Fraenkel set theory, it discusses oracle independence; natural proofs; independence results of Razborov,
Raz, DeMilloLipton, Sazanov, and others; and obstacles to proving P vs. NP independent of strong
logical theories. It ends with some philosophical musings on when one should expect a mathematical
question to have a definite answer.
The P vs. NP problem has been called ``one of the most important problems in contemporary mathematics
and theoretical computer science'' . That is an understatement. Not only is P vs. NP the defining
question of our field; it's one of the deepest questions ever asked for which we'd know how to recognize an
answer. 1 (In other words, one of the deepest questions in NP.) If you doubt this, read the Clay Math
Institute's list of milliondollar prize problems , and notice how P vs. NP stands out, not merely as the
only problem of the seven relevant practically, but as the only one pregnant philosophically. Does the ability
to recognize an answer to the other six questions---or to P vs. NP, or to any question---entail the ability to
find an answer? We are after not projective algebraic varieties or zeros of the Riemann zeta function, but
the nature of mathematical thought itself.