This essay’s content is rendered by the titles of the successive sections. 1. Effective solvability versus intuitive solvability. — 2. Decidability, i.e. effective solvability, in predicate logic. The speedup phenomenon — 3. Contributions of the second-order logic to the problems of solvability — 4. The infinite progress of science in the light of Turing’s idea of the oracle. The term “oracle” is a technical counterpart of the notion of mathematical intuition.
A more detailed summary can be obtained through juxtaposing the textboxes labelled with letters A...F. Conclusion: in the progress of science an essential role is played by the feedback between intellectual intuitions (intuitive solvability) and algorithmic procedures (effective solvability).
: computability, decidability, tractability, prov-
ability, testability. Each of this notions refers to a specific kind of solvability.
In yet other dimension, rather epistemological than logical, we have to deal
with intuitivesolvability. Each of these points requires an attentive reflec-
Attention is desirable to avoid a superficial stereotype, i.e. oversimplified
opinions, concerning Gödel’s and Hilbert’s attitudes to the issue of solvabil-
ity. A common view, even in some professional approaches, is to the effect
that Gödel’s incompleteness result refutes Hilbert