The real elements in a consistency proof for simple type theory I
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Simple type theory can be reduced to arithmetic over the primitive recursive functionals plus the quantifierfree rule of extensionality plus one comprehension over a special Λ-property on objects of type ≠ O.
Consistency is fully describable by the notion “continuity indicator”, a generalization of the modulus of continuity with respect to types and extensionality. Especially:
A necessary condition for consistency is that the theory T of the primitive recursive functionals contains no continuity indicator for type O(Oα) at definable points.
The μ-operator based on continuity gives no proof-theoretic strengthening; it rather leads to the negations of the above mentioned comprehension instances and to continuity indicators. Therefore a consistency proof has to make very fine distinctions.
ω-rules for types greater than zero cannot be added consistently to theories of classical and intuitionistic analysis of a certain strength. Moreover Markov's principle (MP) and Kreisel-Troelstra's Λα⌝⌝Va·α=a imply intuitionistically that the existence of a non-recursive constructive function is not absurd. From this it follows that ω-rules for lawlike objects of type O are intuitionistically incompatible with (MP) and Λα⌝⌝Va·α=a.
The second paper makes the failure of the functional interpretation of higher comprehensions-even with functionals which have only a description by intuitionistic means-explicit. Significant results on the second question above are obtained by pursuing Hilbert's fundamental concept of the real elements, which was developed in full for arithmetic and analysis by the author in a previous paper on the basis of functional interpretation.
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