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By proof of a statement S we mean any process which the subject
of knowledge accepts as the evidence that S is true and is ready to
use S for predictions on the basis of which to make decisions.
There are two cases of proof which can never arise any doubt because
they do not base on any assumptions: verification and
direct refutation.
When a statement cannot be directly verified or refuted, it is still
possible that we take is as true relying on our intuition.
For example, consider a test process T the stages of which can be
represented by single symbols in some language.
Let further the process T develop in such
a manner that if the current stage is represented by the symbol A, then
the next stage is also A, and the process comes to a successful end
when the current stage is a different symbol, say B. This definition
leaves us absolutely certain that T is infinite, even though this cannot
be either verified, or refuted. In logic we call some most
immediate of such statements axioms and use as a basis for
establishing the truth of other statements. In natural sciences
some of the self-evident truths serve as the very beginning of the
construction of theories.
