Each new thought is a step down an energy ladder with a finite number of steps.
His famous Euclidean algorithm, when applied to a pair of natural numbers, leads in a finite number of steps to their greatest common divisor.
The idea is this: there are a number of basic mathematical truths, called axioms or postulates, from which other true statements may be derived in a finite number of steps.
If the number of existential instantiations required in the proof is known, the question of whether G follows from F can be decided in a finite number of steps.
(An effective procedure is one that is "mechanical" in nature and can always be relied on to give a definite result in a finite number of steps. The notion of effectiveness plays an important role in formal logic).
The theory had the revolutionary aspect of treating infinite sets as mathematical objects that are on an equal footing with those that can be constructed in a finite number of steps.
A striking mathematical theorem is that if a rule separating the two flowers exists (within the class of criteria we are considering, such as linear inequalities), then our algorithm will find it after a finite number of steps, no matter what the starting hypothesis was.
In this question the term computable sequence is defined to mean that the numbers ak are real and there is a Turing machine that, for any pair of positive integers A, B, will print out in order the first A digits of all ak, for k ranging from 1 to B, in a finite number of steps.
In one sense, decidability is a property of sets (of sentences): that of being subject (or not) to mechanical methods by which to decide in a finite number of steps, for any closed sentence of a given formal system (e.g., of N), whether it is true or not or as a different question whether it is a theorem or not.
There is a famous collection of arguments from the pioneering days of computer science to the effect that any device able to carry out every one of the entries on a certain relatively short list of elementary logical operations could, in some finite number of steps, calculate the value of any mathematical function that is calculable at all.
The algorithm consists of a finite number of steps.
