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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 {\\mathbf {P}} . For example: THEOREM 1. If a degree a {\\mathbf {a}} contains a complete extension of P {\\mathbf {P}} , then every countable partially ordered set can be embedded in the ordering of degrees ⩽ a \\leqslant {\\mathbf {a}} . (This strengthens a result of Scott and Tennenbaum that no such degree a {\\mathbf {a}} is a minimal degree.) THEOREM 2. If T {\\mathbf {T}} is an axiomatizable, essentially undecidable theory, and if { a n } \\{ {{\\mathbf {a}}_n}\\} is a countable sequence of nonzero degrees, then T {\\mathbf {T}} has continuum many complete extensions whose degrees are pairwise incomparable and incomparable with each a n {{\\mathbf {a}}_n} . THEOREM 3. 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Π⁰₁ classes and degrees of theories 论文

Π⁰₁ classes and degrees of theories · 相关文章