On the logical structure of choice and bar induction principles
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...
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| Vydané v: | Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science s. 1 - 13 |
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| Hlavní autori: | , |
| Médium: | Konferenčný príspevok.. |
| Jazyk: | English |
| Vydavateľské údaje: |
IEEE
29.06.2021
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| Shrnutí: | 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 GDC ABT of the axiom of dependent choice and dually a generalised bar induction principle GBI ABT such that:GDC ABT intuitionistically captures the strength of*the general axiom of choice expressed as ∀a∃bR(a,b) ⇒ ∃α∀aR(a,α(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 \mathbb{B} (for a constructive definition of prime filter),*the axiom of dependent choice if A = \mathbb{N},*Weak Kőnig's Lemma if A = \mathbb{N} and B = \mathbb{B} (up to weak classical reasoning).GBI ABT intuitionistically captures the strength of*Gödel's completeness theorem in the form validity implies provability for entailment relations if B = \mathbb{B} (for a constructive definition of validity),*bar induction if A = \mathbb{N},*the Weak Fan Theorem if A = \mathbb{N} and B = \mathbb{B}.Contrastingly, even though GDC ABT and GBI ABT smoothly capture several variants of choice and bar induction, some instances are inconsistent, e.g. when A is {\mathbb{B}^{\mathbb{N}}} and B is \mathbb{N}. |
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| DOI: | 10.1109/LICS52264.2021.9470523 |