By John N. Crossley, Michael Dummett (editors)
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Extra resources for Formal Systems and Recursive Functions: Proceedings of the Eighth Logic Colloquium, Oxford, July 1963
Ar; for some rio We can still use A 1, ••• , for vafs, there being no need for them to be finite in number. Let (i' R; denote respectively the ith axiom scheme and the ith rule of J (for appropriate values of i). An example of a system of type J which illustrates the differences which can exist between such a system and one of type K is the following: propositional variables - just a i ; propositional constants - none; connectives - just ~ (binary); axiom scheme A 1 ~ Ai; rules Ai (A 3 ~ ~ (A 3 A z) ~ ~ A z) Ai It is immediate that the set of provable formulae of the system is the set of formulae of the form U ~ V where U, V are formulae, that is, it is the set of instances of Ai ~ A z .
The last result shows that T is a reduction type for validity. 3. Construction of the formula ct s ;w, w' Let S be a Semi-Thue-system over the alphabet at> ... , aN' Let the rules of S be (L k , R k ) (k = 1, ... , M), where L k , R k are words in the alphabet of S. One can assume that none of the L k , R k are empty and that every word LkR k consists of at least three letters (because all SM can be chosen in this manner). The formula ct s; w, w' contains at most the following N + 2 predicate variables: Singulary predicate variables E j (j = 1, ...
PROOF. This is similar to the proof of Theorem 2* preceded by the proof of a result which corresponds to Lemma 10*. Suppose X E S, and that X is provable in L i, then TX . Z)i and is provable in Ltz by Theorem 6. Suppose now that X E S, and that TX . =:> ai is provable in Ltz. /TX'. /X' and therefore X is provable in L; Hence, for i = 1, 2, if X E Si then X is provable in L, if and only if TX . z. z is primitive recursive in those of Lj , L z can be obtained, using Theorem 6, by trivial modification of the corresponding part of the proof of Theorem 2*.