Mortgage loan

From formulasearchengine
Revision as of 06:59, 4 February 2014 by en>Nueragroup (Recent trends: Fixed broken link and found corresponding pdf on bbc website)
Jump to navigation Jump to search

In mathematical logic, Craig's theorem states that any recursively enumerable set of well-formed formulas of a first-order language is (primitively) recursively axiomatizable. This result is not related to the well-known Craig interpolation theorem.

Recursive axiomatization

Let A1,A2, be an enumeration of the axioms of a recursively enumerable set T of first-order formulas. Construct another set T* consisting of

AiAii

for each positive integer i. The deductive closures of T* and T are thus equivalent; the proof will show that T* is a decidable set. A decision procedure for T* lends itself according to the following informal reasoning. Each member of T* is either A1 or of the form

BjBjj.

Since each formula has finite length, it is checkable whether or not it is A1 or of the said form. If it is of the said form and consists of j conjuncts, it is in T* if it is the expression Aj; otherwise it is not in T*. Again, it is checkable whether it is in fact An by going through the enumeration of the axioms of T and then checking symbol-for-symbol whether the expressions are identical.

Primitive recursive axiomatizations

The proof above shows that for each recursively enumerable set of axioms there is a recursive set of axioms with the same deductive closure. A set of axioms is primitive recursive if there is a primitive recursive function that decides membership in the set. To obtain a primitive recursive aximatization, instead of replacing a formula Ai with

AiAii

one instead replaces it with

AiAif(i) (*)

where f(x) is a function that, given i, returns a computation history showing that Ai is in the original recursively enumerable set of axioms. It is possible for a primitive recursive function to parse an expression of form (*) to obtain Ai and j. Then, because Kleene's T predicate is primitive recursive, it is possible for a primitive recursive function to verify that j is indeed a computation history as required.

References

  • William Craig. On Axiomatizability Within a System, The Journal of Symbolic Logic, Vol. 18, No. 1 (1953), pp. 30-32.