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It can be proved by induction e.g ., or by using the fact that they are 0-dimensional separable metric spaces in the order topology , and the Cantor set ( which is also an ordered space ) is universal for those . I am not quite sure how I would use the theorems proved by resolution provers unless they were translated into some foundational language . It is a classical result of Gentzen that the consistency of PA ( Peano arithmetic , or formal number theory ) can be proved by transfinite induction on ordinals below eps [ 0 ] . Let 's prove by induction that after the ith pass , the items are correctly sorted according to the least i digits . Because , as proven by Turing , you can do anything in any language .