By Michael Zakharyaschev

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We describe below the basic model. There are many apparently more powerful variations (multitape, nondeterministic, two-way in nite tape, two-dimensional tape, : : : ) that can be simulated by this basic model. There are also many apparently less powerful variations (two-stack machines, two counter machines) that can simulate the basic model. All these models are equivalent in the sense that they compute all the same functions, although not with equal e ciency. One can include suitably abstracted versions of modern programming languages in this list.

E. if and only if there exists a decidable dyadic A = fx j 9y '(x y)g: Proof If A has such a representation, then we can construct a Turing machine M for A that enumerates all y in some order, and for each one tests whether '(x y), accepting if such a witness y is ever found. " This predicate is decidable, since its truth can be determined by running M on x for y steps. 5 are special cases of a more general relationship. Consider the following hierarchy of classes of sets, de ned inductively as follows.

It follows that any element of an ordinal is an ordinal. We use : : : to refer to ordinals. The class of all ordinals is denoted Ord. It is not a set, but a proper class. This rather neat but perhaps obscure de nition of ordinals has some far-reaching consequences that are not at all obvious. For ordinals , , de ne < if 2 . Then every ordinal is equal to the set of all smaller ordinals (in the sense of <). 3. If is an ordinal, then so is f g. The latter is called S the successor of and is denoted + 1.