Implicit Definability in Number Theories
In this chapter we consider several ways in which number-theoretic functions and relations can be implicitly defined in number theories. We do not mean elementarily definable as in Chapter 11; the present notions of definability are expressed in terms of theories and not of structures. As we shall see, the notions lead to new equivalents of the notion of recursiveness; see 14.12, 14.20, and 14.26. They also form the basis for diagonalization procedures which produce many undecidable theories (see the next chapter). We shall be concerned with two types of implicit definability. The first, syntactic definability, follows; the second, spectral representability, is given in 14.22.
KeywordsNumber Theory Recursive Function Relation Symbol Finite Model Operation Symbol
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