Trakhtenbrot's theorem [SEP] named after
Trakhtenbrot's theorem
In logic, finite model theory, and computability theory, Trakhtenbrot's theorem (due to Boris Trakhtenbrot) states that the problem of validity in first-order logic on the class of all finite models is undecidable. In fact, the class of valid sentences over finite models is not recursively enumerable (though it is co-recursively enumerable).
Trakhtenbrot's theorem implies that Gödel's completeness theorem (that is fundamental to first-order logic) does not hold in the finite case. Also it
Trakhtenbrot and Allan Borodin.
Although Trakhtenbrot's derivation preceded Borodin's by several years, it was not known nor recognized in the West until after Borodin's work was published.
Gap theorem.
The general form of the theorem is as follows.
The theorem can be proved by using the Blum axioms without any reference to a concrete computational model, so it applies to time, space, or any other reasonable complexity measure.
For the special case of time complexity, this can be stated more simply as