Infinite Induction

an inference whereby from an infinite set of premises that exhaust all the particular cases of some general judgment (statement), one obtains this general judgment as a conclusion (consequence). For example, from the premises 0 + 0 = 0 + 0,0+1 = 1 + 0,0+2 = 2 + 0,1 + 1 = 1 + 1,0+3 = 3 + 0, 1 + 2 = 2+1,0 + 4 = 4 + 0, 1 + 3 = 3+1,2+2 = 2 + 2,0+5 = 5 + 0,1 + 4 = 4+1,2+3 = 3 + 2,. . . (where the dotted line denotes the assumption that the sums of the natural numbers on both sides of the equals signs successively run on through all the natural numbers) one obtains, according to infinite induction, the conclusion a + b = b + a which holds for any natural numbers a and b. Inasmuch as it is practically impossible to “enumerate” an infinite set of premises, in each such “application” of infinite induction, there is an element of idealization (which appears in the above example in the assumption of the validity of replacing the dotted line, which is a visible, finite, symbolic construction, by the purely imaginary, abstract form of the set of “all natural numbers”). Any expressions of the type “etc.,” which replace some infinite set (which does not necessarily consist of the natural numbers), have an ineffective and metaphorical character. By virtue of this ineffectiveness, infinite induction cannot be used directly either in deductive mathematical and logical theories or in the semiempirical constructions of natural science. In the first case it is often replaced by different forms of the principle of mathematical induction, in the second, by so-called natural-scientific (incomplete) induction. However, as an instrument of theoretical, methodological investigation, infinite induction (usually in the form of the so-called Carnaprule, named for the Austrian logician who proposed it in 1934) finds widespread and important application in mathematical logic. If the set of premises of infinite induction is assigned by a certain algorithm, then it can be used as a special deduction rule.