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Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation f ξ ξ∈On of a normal function f is a sequence of normal functions so that f 0 = id, f 1 = f and for all α, β we have that f α+β = f α f β. These conditions do not determine f α uniquely; in addition, we require that f α α∈On be minimal in an appropriate sense. We study hyperations systematically and show that they are a natural refinement of Veblen progressions. Next, we define cohyperations, very similar to hyperations except that they are left-additive: given α, β, f α+β = f β f α. Cohyperations iterate initial functions which are functions that map initial segments to initial segments. We systematically study co-hyperations and see how they can be employed to define left inverses to hyperations. Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more fine-grained analysis of such sequences is called for. They are very amenable to algebraic manipulation and hence are convenient to work with. 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if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions. Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation f ξ ξ∈On of a normal function f is a sequence of normal functions so that f 0 = id, f 1 = f and for all α, β we have that f α+β = f α f β. These conditions do not determine f α uniquely; in addition, we require that f α α∈On be minimal in an appropriate sense. We study hyperations systematically and show that they are a natural refinement of Veblen progressions. Next, we define cohyperations, very similar to hyperations except that they are left-additive: given α, β, f α+β = f β f α. Cohyperations iterate initial functions which are functions that map initial segments to initial segments. We systematically study co-hyperations and see how they can be employed to define left inverses to hyperations. Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more fine-grained analysis of such sequences is called for. They are very amenable to algebraic manipulation and hence are convenient to work with. Cohyperations, meanwhile, give a novel way to describe slowly increasing functions as often appear, for example, in proof theory.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":113160830,"attachmentType":"pdf","workUrl":"https://www.academia.edu/117255558/Hyperations_Veblen_progressions_and_transfinite_iterations_of_ordinal_functions"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":113160830,"attachmentType":"pdf","workUrl":"https://www.academia.edu/117255558/Hyperations_Veblen_progressions_and_transfinite_iterations_of_ordinal_functions"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div></div><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="113160830" data-landing_url="https://www.academia.edu/117255558/Hyperations_Veblen_progressions_and_transfinite_iterations_of_ordinal_functions" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="49777198" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/49777198/Hyperations_Veblen_progressions_and_transfinite_iteration_of_ordinal_functions">Hyperations, Veblen progressions and transfinite iteration of ordinal functions</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="51116229" href="https://independent.academia.edu/JoostJoosten">Joost Joosten</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Annals of Pure and Applied Logic, 2013</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions. Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation f ξ ξ∈On of a normal function f is a sequence of normal functions so that f 0 = id, f 1 = f and for all α, β we have that f α+β = f α f β. These conditions do not determine f α uniquely; in addition, we require that f α α∈On be minimal in an appropriate sense. We study hyperations systematically and show that they are a natural refinement of Veblen progressions. Next, we define cohyperations, very similar to hyperations except that they are left-additive: given α, β, f α+β = f β f α. Cohyperations iterate initial functions which are functions that map initial segments to initial segments. We systematically study co-hyperations and see how they can be employed to define left inverses to hyperations. Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more fine-grained analysis of such sequences is called for. They are very amenable to algebraic manipulation and hence are convenient to work with. Cohyperations, meanwhile, give a novel way to describe slowly increasing functions as often appear, for example, in proof theory.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Hyperations, Veblen progressions and transfinite iteration of ordinal functions","attachmentId":68014216,"attachmentType":"pdf","work_url":"https://www.academia.edu/49777198/Hyperations_Veblen_progressions_and_transfinite_iteration_of_ordinal_functions","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/49777198/Hyperations_Veblen_progressions_and_transfinite_iteration_of_ordinal_functions"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="69310070" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/69310070/Iterations_of_Ordinal_Functions">Iterations of Ordinal Functions</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="51116229" href="https://independent.academia.edu/JoostJoosten">Joost Joosten</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2014</p><p class="ds-related-work--abstract ds2-5-body-sm">In this paper we introduce hyperations and cohyperations, which are forms of transfinite iteration of ordinal functions. Hyperations are iterations of normal functions. Unlike iteration by pointwise convergence, hyperation preserves normality. The hyperation f ξ ξ∈On of a normal function f is a sequence of normal functions so that f 0 = id, f 1 = f and for all α, β we have that f α+β = f α f β. These conditions do not determine f α uniquely; in addition, we require that f α α∈On be minimal in an appropriate sense. We study hyperations systematically and show that they are a natural refinement of Veblen progressions. Next, we define cohyperations, very similar to hyperations except that they are left-additive: given α, β, f α+β = f β f α. Cohyperations iterate initial functions which are functions that map initial segments to initial segments. We systematically study co-hyperations and see how they can be employed to define left inverses to hyperations. Hyperations provide an alternative presentation of Veblen progressions and can be useful where a more fine-grained analysis of such sequences is called for. They are very amenable to algebraic manipulation and hence are convenient to work with. Cohyperations, meanwhile, give a novel way to describe slowly increasing functions as often appear, for example, in proof theory.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Iterations of Ordinal Functions","attachmentId":79455380,"attachmentType":"pdf","work_url":"https://www.academia.edu/69310070/Iterations_of_Ordinal_Functions","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/69310070/Iterations_of_Ordinal_Functions"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="1397009" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/1397009/A_uniform_approach_to_fundamental_sequences_and_hierarchies">A uniform approach to fundamental sequences and hierarchies</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="597738" href="https://ugent.academia.edu/AndreasWeiermann">Andreas Weiermann</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="253964796" href="https://independent.academia.edu/BuchholzWilfried">Wilfried Buchholz</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Mathematical Logic …, 1994</p><p class="ds-related-work--abstract ds2-5-body-sm">In this article we give a unifying approach to the theory of fundamental sequences and their related Hardy hierarchies of number-theoretic functions and we show the equivalence of the new approach with the classical one. Mathematics Subject Classification: 03D20, 03F15, 03E10. tal sequences, Descent functions. 4 ) A similar (but not equivalent) definition is contained in FRIEDMAN-SHEARD [7, Lemma 1.311 ')The new approach has recently also been proved useful for bounding derivation lengths of rewrite systems with slow growing functions (see WEIERMANN [ZO]) and for investigations on slow versus fast growing for proof-theoretic ordinals larger than the first subrecursive ordinal (see WEIERMANN [21]).</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A uniform approach to fundamental sequences and hierarchies","attachmentId":9019093,"attachmentType":"pdf","work_url":"https://www.academia.edu/1397009/A_uniform_approach_to_fundamental_sequences_and_hierarchies","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/1397009/A_uniform_approach_to_fundamental_sequences_and_hierarchies"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="101406062" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/101406062/Review_of_subrecursion_Functions_and_hierarchies">Review of subrecursion: Functions and hierarchies</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="33264119" href="https://bc.academia.edu/PeterClote">Peter Clote</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Advances in Mathematics, 1985</p><p class="ds-related-work--abstract ds2-5-body-sm">This book studies hierarchies of recursive functions which for the most part are related to the extended Grzegorczyk (or Grzegorczyk-Wainer) hierarchy. Such work is of general interest to mathematicians and computer scientists for the following two reasons. First, questions in low level computational complexity theory are related to questions about low level Grzegorczyk classes (described in Chapter 5). Secondly, fast growing recursive functions are related to the Giidel incompleteness theorem, which has received attention of late due to the concrete combinatorial examples of true yet "unprovable" formulas (cf. the Paris-Harrington modification of the finite Ramsey theorem in "Handbook of Mathematical Logic," (North-Holland). Ketonen-Solvay gave a combinatorial proof of the Paris-Harrington theorem by studying the rate of growth of the Pari-Harrington function with respect to the rate of growth of functions ffi in the extended Grzegorczyk hierarchy. This incompleteness phenomenon has even entered computer science with the proof by Fortune-Leivant-O'Donnell that certain loop-free and recursion-free programs can be given in certain polymorphic languages which certainly terminate on all inputs, but for which the termination property cannot be proved in Peano arithmetic. Similarly, Fortune-Leivant-O'Donnell showed that f,, a function which grows faster than all functions provably recursive in Peano arithmetic, could be computed by a loop-free and recursion-free program in a polymorphic language. There is thus a natural relevance for the study of functions in the extended Grzegorczyk hierarchy. The book under review, "Subrecursion: Functions and Hierarchies," is a carefully written work, whose principal aim is to study various classes of recursive functions which can be arranged in a natural hierarchy, and as such should be of interest to students and researchers in mathematical logic and theoretical computer science and specifically in recursion theory, proof theory, models of arithmetic, and in computational complexity theory. To have a better idea of the scope of this book, we introduce some notation conerning ordinals and the extended Grzegorczyk hierarchy of 205</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Review of subrecursion: Functions and hierarchies","attachmentId":101956613,"attachmentType":"pdf","work_url":"https://www.academia.edu/101406062/Review_of_subrecursion_Functions_and_hierarchies","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/101406062/Review_of_subrecursion_Functions_and_hierarchies"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="8769278" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/8769278/Built_up_systems_of_fundamental_sequences_and_hierarchies_of_number_theoretic_functions">Built-up systems of fundamental sequences and hierarchies of number-theoretic functions</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="18861306" href="https://hs-heilbronn.academia.edu/DianaSchmidt">Diana Schmidt</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Archive for Mathematical Logic, 1977</p><p class="ds-related-work--abstract ds2-5-body-sm">In [2], L6b and Wainer introduced a general procedure for generating hierarchies which can be used for classifying a wide variety of classes of number-theoretic functions. Similar hierarchies were also studied by Robbin [3], Rose [4] and Schwichtenberg . The basic ingredient in all these cases was a transfinite sequence (F~)~ A of number-theoretic functions, indexed by an initial segment A of the second number class and defined inductively as follows: F0 = some strictly monotonic function; F~÷I is defined from F~ so that F~÷I is strictly monotonic if F, is and grows faster than F~;</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Built-up systems of fundamental sequences and hierarchies of number-theoretic functions","attachmentId":48009311,"attachmentType":"pdf","work_url":"https://www.academia.edu/8769278/Built_up_systems_of_fundamental_sequences_and_hierarchies_of_number_theoretic_functions","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/8769278/Built_up_systems_of_fundamental_sequences_and_hierarchies_of_number_theoretic_functions"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="83915727" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/83915727/Transfinite_Induction_on_Ordinal_Configurations">Transfinite Induction on Ordinal Configurations</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="44734508" href="https://independent.academia.edu/LuizPaulodeAlcantara">Luiz Paulo de Alcantara</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, 1981</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Transfinite Induction on Ordinal Configurations","attachmentId":89109221,"attachmentType":"pdf","work_url":"https://www.academia.edu/83915727/Transfinite_Induction_on_Ordinal_Configurations","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/83915727/Transfinite_Induction_on_Ordinal_Configurations"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="55250202" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/55250202/The_proof_theoretic_analysis_of_transfinitely_iterated_quasi_least_fixed_points">The proof-theoretic analysis of transfinitely iterated quasi least fixed points</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="28760565" href="https://independent.academia.edu/AntonSetzer">Anton Setzer</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The Journal of Symbolic Logic, 2006</p><p class="ds-related-work--abstract ds2-5-body-sm">The starting point of this article is an old question asked by Feferman in his paper on Hancock&#39;s conjecture [6] about the strength of . This theory is obtained from the well-known theory ID1 by restricting fixed point induction to formulas that contain fixed point constants only positively. The techniques used to perform the proof-theoretic analysis of also permit to analyze its transfinitely iterated variants . Thus, we eventually know that</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The proof-theoretic analysis of transfinitely iterated quasi least fixed points","attachmentId":71209238,"attachmentType":"pdf","work_url":"https://www.academia.edu/55250202/The_proof_theoretic_analysis_of_transfinitely_iterated_quasi_least_fixed_points","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/55250202/The_proof_theoretic_analysis_of_transfinitely_iterated_quasi_least_fixed_points"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="12541833" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/12541833/Ordinal_notations_and_well_orderings_in_bounded_arithmetic_vol_120_pg_197_2003_">Ordinal notations and well-orderings in bounded arithmetic (vol 120, pg 197, 2003)</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="31440864" href="https://independent.academia.edu/ArnoldBeckmann">Arnold Beckmann</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Ordinal notations and well-orderings in bounded arithmetic (vol 120, pg 197, 2003)","attachmentId":46102372,"attachmentType":"pdf","work_url":"https://www.academia.edu/12541833/Ordinal_notations_and_well_orderings_in_bounded_arithmetic_vol_120_pg_197_2003_","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/12541833/Ordinal_notations_and_well_orderings_in_bounded_arithmetic_vol_120_pg_197_2003_"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="69310071" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/69310071/MATHEMATICAL_COMMUNICATIONS_109_Math_Commun_18_2013_109_121_%CE%A001_ordinal_analysis_beyond_first_order_arithmetic">MATHEMATICAL COMMUNICATIONS 109 Math. Commun. 18(2013), 109–121 Π01-ordinal analysis beyond first-order arithmetic</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="51116229" href="https://independent.academia.edu/JoostJoosten">Joost Joosten</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2012</p><p class="ds-related-work--abstract ds2-5-body-sm">Abstract. In this paper we give an overview of an essential part of a Π01 ordinal analysis of Peano Arithmetic (PA) as presented by Beklemishev ([2]). This analysis is mainly performed within the polymodal provability logic GLPω. We reflect on ways of extending this analysis beyond PA. A main difficulty in this is to find proper generalizations of the so-called Reduction Property. The Reduction Property relates reflection principles to reflection rules. In this paper we prove a result that simplifies the reflection rules. Moreover, we see that for an ordinal analysis the full Reduction Property is not needed. This latter observation is also seen to open up ways for applications of ordinal analysis relative to some strong base theory.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"MATHEMATICAL COMMUNICATIONS 109 Math. Commun. 18(2013), 109–121 Π01-ordinal analysis beyond first-order arithmetic","attachmentId":79455372,"attachmentType":"pdf","work_url":"https://www.academia.edu/69310071/MATHEMATICAL_COMMUNICATIONS_109_Math_Commun_18_2013_109_121_%CE%A001_ordinal_analysis_beyond_first_order_arithmetic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/69310071/MATHEMATICAL_COMMUNICATIONS_109_Math_Commun_18_2013_109_121_%CE%A001_ordinal_analysis_beyond_first_order_arithmetic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="47902881" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/47902881/Refined_hierarchy_of_formulas">Refined hierarchy of formulas</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="7408730" href="https://nsu-ru.academia.edu/VictorSelivanov">Victor Selivanov</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Algebra and Logic, 1991</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Refined hierarchy of formulas","attachmentId":66791352,"attachmentType":"pdf","work_url":"https://www.academia.edu/47902881/Refined_hierarchy_of_formulas","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/47902881/Refined_hierarchy_of_formulas"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":113160830,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":113160830,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_113160830" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="83456108" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/83456108/ORDINAL_NATURAL_NUMBERS_1_Alternative_algebraic_definitions_of_the_Hessenberg_natural_operations_in_the_ordinal_numbers">ORDINAL NATURAL NUMBERS 1. 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Free algebrae and alternative definitions of the Hessenberg operations in the ordinal numbers</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="21871778" href="https://uoi.academia.edu/ConstantineKyritsis">Constantine Kyritsis</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Conference: 1ST INTERNATIONAL CONFERENCE ON QUANTITATIVE, SOCIAL, BIOMEDICAL AND ECONOMIC ISSUES 2017 - ICQSBEI 2017 ,ATHENS, GREECEAt: ATHENS, GREECE Volume: PROCEEDINGS: https://books.google.gr/books?id=BSUsDwAAQBAJ&pg, 2017</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"ORDINAL NATURAL NUMBERS 2. 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data-collection-position="5" data-entity-id="12541806" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/12541806/Ordinal_notations_and_well_orderings_in_bounded_arithmetic">Ordinal notations and well-orderings in bounded arithmetic</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="31440864" href="https://independent.academia.edu/ArnoldBeckmann">Arnold Beckmann</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Annals of Pure and Applied Logic, 2003</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Ordinal notations and well-orderings in bounded 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href="https://www.academia.edu/55250165/Ordinal_systems_part_2_One_inaccessible">Ordinal systems part 2: One inaccessible</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="28760565" href="https://independent.academia.edu/AntonSetzer">Anton Setzer</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2000</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Ordinal systems part 2: One inaccessible","attachmentId":71209229,"attachmentType":"pdf","work_url":"https://www.academia.edu/55250165/Ordinal_systems_part_2_One_inaccessible","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link 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href="https://independent.academia.edu/Czes%C5%82awByli%C5%84ski">Czesław Byliński</a></div><p class="ds-related-work--metadata ds2-5-body-xs">1990</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The modification of a function by a function and the iteration of the composition of a function","attachmentId":111648371,"attachmentType":"pdf","work_url":"https://www.academia.edu/115158013/The_modification_of_a_function_by_a_function_and_the_iteration_of_the_composition_of_a_function","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" 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ds2-5-body-xs">Lecture Notes in Computer Science, 2001</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Ordinal Arithmetic: A Case Study for Rippling in a Higher Order Domain","attachmentId":47190317,"attachmentType":"pdf","work_url":"https://www.academia.edu/10705686/Ordinal_Arithmetic_A_Case_Study_for_Rippling_in_a_Higher_Order_Domain","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/10705686/Ordinal_Arithmetic_A_Case_Study_for_Rippling_in_a_Higher_Order_Domain"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" 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and the arithmetic hierarchy</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="102153699" href="https://unina.academia.edu/GiuseppeTrautteur">Giuseppe Trautteur</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Revue française d'automatique informatique recherche opérationnelle. 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