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This talk examines the arithmetical hierarchy of classical principles from a constructive perspective, focusing on limitations and implications. Classical principles such as Post's Theorem, Markov's Principle, and Excluded Middle are analyzed in the context of Heyting's Intuitionistic Arithmetic (HA). Theorems are presented to establish the provability of these principles within HA, showcasing their relative strengths and weaknesses. The generalization to higher degrees of principles is also discussed, providing insights into the classification of principles based on their degree.
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An Arithmetical Hierarchy of the Laws of Excluded Middle and Related PrinciplesLICS 2004, Turku, Finland Yohji Akama (Tohoku University) Stefano Berardi (Turin University) Susumu Hayashi (Kobe University) Ulrich Kohlenbach (Darmstadt University)
Acknoledgements Our research was supported by: • the Grant in Aid for Scientific Research of Japan Society of the Promotion of Science • the McTati Research Project (constructive methods in Topology, Algebra and Computer Science). • the Grant from the Danish Natural Science Research Council.
The subject of this talk We are concerned with classifying classical principles from a constructive viewpoint.
Some motivations for our research work • Limit Interpretation for non-constructive proofs: see Susumu Hayashi’s homepage. http://www.shayashi.jp/PALCM/index-eng.html • Effective Bound Extraction from partially non-constructive proofs: (see Ulrich Kohlenbach’s homepage. http://www.mathematik.tu-darmstadt.de/~kohlenbach/novikov.ps.gz
Some Classical Principles we are concerned with We compare up to provability in HA (Heyting’s Intuitionistic Arithmetic): • Post’s Theorem • Markov’s Principle • 01-Lesser Limited Principle of Omniscience. • Excluded Middle for 01-predicates • Excluded Middle for 01-predicates
Theorem 1.The only implications provable in HA are: 01-Ex. Middle 01-Ex. Middle Markov Principle No principle in this picture is provable in HA 01-L.L.P.O. Post’s Theorem
Post’s Theorem “Any subset of N which both positively and negatively decidable is decidable” • Equivalently, in HA: for any P,Q01 z: ( x.P(x,z) y.Q(y,z) ) x.P(x,z) x. P(x,z) • Post’s Theorem is not derivable in HA. It is strictly weaker in HA than any other classical principle we considered.
Markov’s Principle “Any computation which does not run foverer eventually stops” • Equivalently, in HA: for any P01 z: x.P(x,z) x.P(x,z) • Markov’s Principle is independent from 01-Lesser Limited Principles of Omniscience in HA.
01-Lesser Limited Principles of Omniscience “If two positively decidable statements are not both true, then some of them is false” • Equivalently, in HA: for any P,Q01 z: x,y.(P(x,z) Q(y,z)) x.P(x,z) y.Q(y,z)
01- L.L.P.O andWeak Koenig’s Lemma 01- L.L.P.O is equivalent, in HA+Choice, to: Weak Koenig’s Lemma for recursive trees “any infinite binary recursive tree has some infinite branch”
Excluded Middle for 01-predicates “Excluded Middle holds for all negatively decidable statements” • Equivalently, in HA: for any P01 z: x.P(x,z) x.P(x,z) • 01-E.M. is, in HA, stronger than 01-LLPO (i.e., than Koenig’s Lemma), but weaker than 01-E.M..
Excluded Middlefor 01-predicates “Excluded Middle holds for all positively decidable statements” • Equivalently, in HA: for any P01 z: x.P(x,z) x.P(x,z) • 01-E.M. is stronger, in HA, than all classical principles we considered until now.
Generalizing to higher degrees • For each principle there is a degree n version, for degree n formulas. • For degree n principles we proved the same classification results we proved for the originary principles.
Theorem 2.For all n, the only implications provable in HA are: 0n-Ex. Middle 0n-Ex. Middle n-Markov’s Principle n-Koenig’s Lemma n-Post’s Theorem 0n-1-Ex. Middle … …
