New formally undecidable propositions: non-trivial lower bounds on proof complexity and related theorems

H. Luckhardt
1991 Theoretical Computer Science
Luckhardt, H., New formally undecidable propositions: non-trivial lower bounds on proof complexity and related theorems, Theoretical Computer Science 83 (1991) 169-188. Within a formal theory T where a I-rule is provably valid and Gdel's second incompleteness theorem holds, it is not possible to prove any non-trivial lower bound LB, on proof complexity. Most calculi currently used in formalizing proofs are of this type. We give many sets of formulas where non-trivial lower bounds LB, are
more » ... in a simple way. Thus we have a new large class of formally undecidable mathematical propositions. To this we add the well-known theorems of recursive undecidability and proof speed-ups of T as well as examples resulting from prooftheoretic @-uniformity. Our examples also show that the formalistic goal of computing the "whole accessible mathematical world" is not attainable in a mathematically satisfactory way: the above mentioned LB&{ F,}) with F" provable in T can only be formally decided using known methods in extensions T' of T if almost all F" are assumed as axioms in T'. Such a completion T' practically amounts to listing and not proving theorems. Finally we see that Cook's thesis NP# P implies the existence of { F,} c TAUT having a lower bound LB,({F"}) of an order of growth comparable to that in our examples. This is new evidence for the validity of Cook's conjecture. A proof of this conjecture has to overcome the proof-theoretic difficulty that if validity + is replaced by provability t, in T, then this NP # P-variant cannot be proved in T. * The main results of this paper were obtained in 1984. I am obliged to A. Ferebee for help with the English version. 0304.3975/91/\$03.50 @ 1991-Elsevier Science Publishers B.V.