Abstract
We present an overview of the development of mathematics (and logic) in connection with the two celebrated results by Kurt Gödel, i.e. completeness (of calculi for the first order predicate logic) and incompleteness (of all sufficiently strong axiomatic theories), and we discuss the impact of these results. The research in foundations of mathematics at the beginning of the 20th century revealed the need of exact ways how to formulate mathematical statements as well as proofs.
Hence the quest for completeness originated. But then, Gödel's theorem, the most important result in logic of the 20th century, forced logicians to change the opinion on incompleteness.
While before 1931 it was possible to understand incompleteness as a lack of axioms and incomplete theory as a semi-finished product, afterwards there was no other way but to accept that many theories are incomplete in principal. Moreover, these theories are not special, they are regular part of ordinary mathematical practice.
Detailed explanation of the change in mathematical practice caused by Gödel's results is accompanied by the new and revisited critical edition of Czech translation of the two famous papers 1. Die Vollstandigkeit der Axiome des logischen Funktionenkalkuls (1930), and 2.
Uber formal unentscheidbare Satze der Principia Mathematica und verwandter Systeme I (1931). based on edition: Kurt Gödel (1986). Collected works, Vol.
I, Publications 1929-1936, editor in chief Solomon Feferman, edited by John W. Dawson, Jr., Stephen C.
Kleene, Gregory H. Moore, Robert M.
Solovay and Jean van Heijenoort. The Czech translation of Gödel's two papers now allows Czech readers to trace that unique and enigmatic experience of the completeness and incompleteness of deductive method.