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$\prod^0_1$ Classes and Degrees of Theories

Carl G. Jockusch, Jr and Robert I. Soare
Transactions of the American Mathematical Society
Vol. 173 (Nov., 1972), pp. 33-56
DOI: 10.2307/1996261
Stable URL: http://www.jstor.org/stable/1996261
Page Count: 24
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$\prod^0_1$ Classes and Degrees of Theories
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Abstract

Using the methods of recursive function theory we derive several results about the degrees of unsolvability of members of certain $\prod^0_1$ classes of functions (i.e. degrees of branches of certain recursive trees). As a special case we obtain information on the degrees of consistent extensions of axiomatizable theories, in particular effectively inseparable theories such as Peano arithmetic, P. For example: THEOREM 1. If a degree a contains a complete extension of P, then every countable partially ordered set can be embedded in the ordering of degrees ≤ a. (This strengthens a result of Scott and Tennenbaum that no such degree a is a minimal degree.) THEOREM 2. If T is an axiomatizable, essentially undecidable theory, and if {an} is a countable sequence of nonzero degrees, then T has continuum many complete extensions whose degrees are pairwise incomparable and incomparable with each an. THEOREM 3. There is a complete extension T of P such that no nonrecursive arithmetical set is definable in T. THEOREM 4. There is an axiomatizable, essentially undecidable theory T such that any two distinct complete extensions of T are Turing incomparable. THEOREM 5. The set of degrees of consistent extensions of P is meager and has measure zero.

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