On the logical structure of choice and bar induction principles
Résumé
We develop an approach to choice principles and their contrapositive bar-induction principles as extensionality schemes connecting an "intensional" or "effective" view of respectively ill-and well-foundedness properties to an "extensional" or "ideal" view of these properties. After classifying and analysing the relations between different intensional definitions of ill-foundedness and well-foundedness, we introduce, for a domain A, a codomain B and a "filter" T on finite approximations of functions from A to B, a generalised form GDCA,B,T of the axiom of dependent choice and dually a generalised bar induction principle GBIA,B,T such that:
- GDCA,B,T intuitionistically captures the strength of
• the general axiom of choice expressed as ∀a∃βR(a,b)⇒∃α∀aR(α,(aα(a))) when T is a filter that derives point-wise from a relation R on A×B without introducing further constraints,
• the Boolean Prime Filter Theorem / Ultrafilter Theorem if B is the two-element set B (for a constructive definition of prime filter),
• the axiom of dependent choice if A=N,
• Weak König’s Lemma if A=N and B=B (up to weak classical reasoning)
- GBIA,B,T intuitionistically captures the strength of
• Gödel’s completeness theorem in the form validity implies provability for entailment relations if B=B,
• bar induction when A=N,
• the Weak Fan Theorem when A=N and B=B.
Contrastingly, even though GDCA,B,T and GBIA,B,T smoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when A is BN and B is N.
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