# Read e-book online Elementary Recursion Theory and its Applications to Formal PDF

By Kripke, Saul

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Example text

So far, we have been concentrating on interpreted languages whose domains include all the expressions of the language itself. However, we can generalize the discussion by considering languages that can talk about their own expressions indirectly, via coding. Let L be a language with a countable vocabulary and with an infinite domain D. Suppose we had a function f mapping the elements of some subset D1 of D onto the formulae, or at least onto the formulae with one free variable. Call this a coding function.

But if it turns out that there are some semi-computable sets and relations that are not expressible in RE, then it is quite conceivable that all semi-computable sets and relations are computable and that the enumeration theorem for semi-computability fails. The fact that the enumeration theorem is so fundamental to recursion theory, and that its proof for semi-computability requires Church's Thesis, indicates a limitation to how much recursion theory can be developed for the informal notion of computability by starting with intuitively true axioms about computability.

We will now begin our specification of the basis set, and later we will define the generating relations. e. It is important to stress at this stage a delicate point in our coding scheme. Remember that in our coding scheme, in order to code formulae we code terms first, by coding the sequence of numbers that code the individual symbols appearing in a term (in the same order). Thus, a term f11f11xi will be coded by any code of the sequence {[1,[4,[1,1]]], [2,[4,[1,1]]], [3,[1,i]]}, and, as a term, xi will be coded by any code of the sequence {[1,[1,i]]}.