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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 intuitive solvability. Each of these points requires an attentive reflec- tion. 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