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Fundamental sequences and fast-growing hierarchies for the Bachmann-Howard ordinal

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Τίτλος: Fundamental sequences and fast-growing hierarchies for the Bachmann-Howard ordinal
Συγγραφείς: Fernández Duque, David, Weiermann, Andreas
Πηγή: Articles publicats en revistes (Filosofia)
Στοιχεία εκδότη: Elsevier B.V.
Έτος έκδοσης: 2024
Συλλογή: Dipòsit Digital de la Universitat de Barcelona
Θεματικοί όροι: Successions espectrals (Matemàtica), Anàlisi combinatòria, Spectral sequences (Mathematics), Combinatorial analysis
Περιγραφή: Hardy functions are defined by transfinite recursion and provide upper bounds for the growth rate of the provably total computable functions in various formal theories, making them an essential ingredient in many proofs of independence. Their definition is contingent on a choice of fundamental sequences, which approximate limits in a ‘canonical’ way. In order to ensure that these functions behave as expected, including the aforementioned unprovability results, these fundamental sequences must enjoy certain regularity properties. In this article, we prove that Buchholz’s system of fundamental sequences for the ϑ function enjoys such conditions, including the Bachmann property. We partially extend these results to variants of the ϑ function, including a version without addition for countable ordinals. We conclude that the Hardy functions based on these notation systems enjoy natural monotonicity properties and majorize all functions defined by primitive recursion along ϑ(εΩ+1).
Τύπος εγγράφου: article in journal/newspaper
Περιγραφή αρχείου: 1 p.; application/pdf
Γλώσσα: English
Relation: Versió postprint del document publicat a; Annals of Pure and Applied Logic, 2024, vol. 175, num.8; https://hdl.handle.net/2445/224563; 754681
Διαθεσιμότητα: https://hdl.handle.net/2445/224563
Rights: cc-by-nc-nd (c) Elsevier B.V., 2024 ; http://creativecommons.org/licenses/by-nc-nd/4.0/ ; info:eu-repo/semantics/openAccess
Αριθμός Καταχώρησης: edsbas.44FD5689
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  Data: Fundamental sequences and fast-growing hierarchies for the Bachmann-Howard ordinal
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  Data: <searchLink fieldCode="AR" term="%22Fernández+Duque%2C+David%22">Fernández Duque, David</searchLink><br /><searchLink fieldCode="AR" term="%22Weiermann%2C+Andreas%22">Weiermann, Andreas</searchLink>
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  Data: 2024
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  Data: Hardy functions are defined by transfinite recursion and provide upper bounds for the growth rate of the provably total computable functions in various formal theories, making them an essential ingredient in many proofs of independence. Their definition is contingent on a choice of fundamental sequences, which approximate limits in a ‘canonical’ way. In order to ensure that these functions behave as expected, including the aforementioned unprovability results, these fundamental sequences must enjoy certain regularity properties. In this article, we prove that Buchholz’s system of fundamental sequences for the ϑ function enjoys such conditions, including the Bachmann property. We partially extend these results to variants of the ϑ function, including a version without addition for countable ordinals. We conclude that the Hardy functions based on these notation systems enjoy natural monotonicity properties and majorize all functions defined by primitive recursion along ϑ(εΩ+1).
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  Data: Versió postprint del document publicat a; Annals of Pure and Applied Logic, 2024, vol. 175, num.8; https://hdl.handle.net/2445/224563; 754681
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