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u-fw700 u-mr5x u-pr5x u-borderColorGrayLight u-borderRight1"><a href="https://ugent.academia.edu/"><span class="u-linkUnstyled u-tcGrayDarkest">Ghent University</span></a></span><h1 class="u-m0x u-fw300 u-fs24 u-displayInline">Applied Mathematics and Computer Science</h1></div><div class="u-floatRight u-mt1x"></div></div></div></div></div><div class="TabbedNavigation"><div class="container"><div class="row"><div class="col-xs-12 clearfix"><ul class="nav u-m0x u-p0x list-inline"><li class="u-floatLeft u-pr5x u-mr5x u-borderColorGrayLight u-borderRight1"><a href="https://ugent.academia.edu/"><span><i class="fa fa-arrow-left"></i>  All Departments</span></a></li><li class="u-floatLeft active"><a href="https://ugent.academia.edu/Departments/Applied_Mathematics_and_Computer_Science/Documents">129 Papers</a></li><li class="u-floatLeft"><a href="https://ugent.academia.edu/Departments/Applied_Mathematics_and_Computer_Science">45 Researchers</a></li></ul></div></div></div></div><div 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href="https://www.academia.edu/3006948/Phase_transition_thresholds_for_some_natural_subclasses_of_the_computable_functions">Phase transition thresholds for some natural subclasses of the computable functions</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Abstract. In this paper we first survey recent advances on phase transition phenomena which are related to natural subclasses of the recursive functions. Special emphasis is put on descent recursive functions, witness bounding functions... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_3006948" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Abstract. In this paper we first survey recent advances on phase transition phenomena which are related to natural subclasses of the recursive functions. Special emphasis is put on descent recursive functions, witness bounding functions for well-partial orders and Ramsey functions. In the last section we prove in addition some results which show how the asymptotic of the standard Ramsey function is affected by phase transitions for associated parameterized Ramsey functions.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3006948" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="6fba7e3d288f86b869b246d0a336ee87" rel="nofollow" data-download="{"attachment_id":30953413,"asset_id":3006948,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/30953413/download_file?st=MTczMjgzODk4Myw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="597738" href="https://ugent.academia.edu/AndreasWeiermann">Andreas Weiermann</a><script data-card-contents-for-user="597738" type="text/json">{"id":597738,"first_name":"Andreas","last_name":"Weiermann","domain_name":"ugent","page_name":"AndreasWeiermann","display_name":"Andreas Weiermann","profile_url":"https://ugent.academia.edu/AndreasWeiermann","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_3006948 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3006948"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3006948, container: ".js-paper-rank-work_3006948", }); });</script></li><li class="js-percentile-work_3006948 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 3006948; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_3006948"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_3006948 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="3006948"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 3006948; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=3006948]").text(description); $(".js-view-count-work_3006948").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_3006948").removeClass('hidden') })</script></div></li></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_3006951" data-work_id="3006951" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/3006951/Classifying_the_phase_transition_threshold_for_Ackermannian_functions">Classifying the phase transition threshold for Ackermannian functions</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">It is well known that the Ackermann function can be defined via diagonalization from an iteration hierarchy (of Grzegorczyk type) which is built on a start function like the successor function. In this paper we study for a given start... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_3006951" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">It is well known that the Ackermann function can be defined via diagonalization from an iteration hierarchy (of Grzegorczyk type) which is built on a start function like the successor function. In this paper we study for a given start function g iteration hierarchies with a sub-linear modulus h of iteration. In terms of g and h we classify the phase transition for the resulting diagonal function from being primitive recursive to being Ackermannian.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3006951" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="4192d867e391e3041b47d762d62c3a4d" rel="nofollow" data-download="{"attachment_id":30953411,"asset_id":3006951,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/30953411/download_file?st=MTczMjgzODk4Myw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="597738" href="https://ugent.academia.edu/AndreasWeiermann">Andreas Weiermann</a><script data-card-contents-for-user="597738" type="text/json">{"id":597738,"first_name":"Andreas","last_name":"Weiermann","domain_name":"ugent","page_name":"AndreasWeiermann","display_name":"Andreas Weiermann","profile_url":"https://ugent.academia.edu/AndreasWeiermann","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_3006951 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3006951"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3006951, container: ".js-paper-rank-work_3006951", }); });</script></li><li class="js-percentile-work_3006951 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 3006951; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_3006951"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_3006951 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="3006951"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 3006951; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=3006951]").text(description); $(".js-view-count-work_3006951").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_3006951").removeClass('hidden') })</script></div></li></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_3006952" data-work_id="3006952" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/3006952/Sharp_phase_transition_thresholds_for_the_Paris_Harrington_Ramsey_numbers_for_a_fixed_dimension">Sharp phase transition thresholds for the Paris Harrington Ramsey numbers for a fixed dimension</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest">Abstract: This article is concerned with investigations on a phase transition which is related to the (finite) Ramsey theorem and the Paris-Harrington theorem.</div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3006952" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="5307d77864c79d902841c186fa251e5b" rel="nofollow" data-download="{"attachment_id":30953409,"asset_id":3006952,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/30953409/download_file?st=MTczMjgzODk4Myw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="597738" href="https://ugent.academia.edu/AndreasWeiermann">Andreas Weiermann</a><script data-card-contents-for-user="597738" type="text/json">{"id":597738,"first_name":"Andreas","last_name":"Weiermann","domain_name":"ugent","page_name":"AndreasWeiermann","display_name":"Andreas Weiermann","profile_url":"https://ugent.academia.edu/AndreasWeiermann","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_3006952 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3006952"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3006952, container: ".js-paper-rank-work_3006952", }); });</script></li><li class="js-percentile-work_3006952 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 3006952; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_3006952"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_3006952 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="3006952"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 3006952; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=3006952]").text(description); $(".js-view-count-work_3006952").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_3006952").removeClass('hidden') })</script></div></li></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_3006953" data-work_id="3006953" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/3006953/Slow_Consistency">Slow Consistency</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Abstract. The fact that “natural” theories, ie theories which have something like an “idea” to them, are almost always linearly ordered with regard to logical strength has been called one of the great mysteries of the foundation of... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_3006953" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Abstract. The fact that “natural” theories, ie theories which have something like an “idea” to them, are almost always linearly ordered with regard to logical strength has been called one of the great mysteries of the foundation of mathematics. However, one easily establishes the existence of theories with incomparable logical strengths using self-reference (Rosser-style). As a result, PA+ Con (PA) is not the least theory whose strength is greater than that of PA.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3006953" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="978b2937b83ebb56fd0ad756d115d859" rel="nofollow" data-download="{"attachment_id":30953408,"asset_id":3006953,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/30953408/download_file?st=MTczMjgzODk4Myw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="597738" href="https://ugent.academia.edu/AndreasWeiermann">Andreas Weiermann</a><script data-card-contents-for-user="597738" type="text/json">{"id":597738,"first_name":"Andreas","last_name":"Weiermann","domain_name":"ugent","page_name":"AndreasWeiermann","display_name":"Andreas Weiermann","profile_url":"https://ugent.academia.edu/AndreasWeiermann","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_3006953 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3006953"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3006953, container: ".js-paper-rank-work_3006953", }); });</script></li><li class="js-percentile-work_3006953 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 3006953; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_3006953"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_3006953 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="3006953"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 3006953; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=3006953]").text(description); $(".js-view-count-work_3006953").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_3006953").removeClass('hidden') })</script></div></li></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_3006954" data-work_id="3006954" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/3006954/Reverse_mathematics_and_well_ordering_principles">Reverse mathematics and well-ordering principles</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Chapter 11 Reverse Mathematics and Well-ordering Principles Michael Rathjen∗ and Andreas Weiermann Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, UK E-mail: rathjen@ maths. leeds. ac. uk Vakgroep Zuivere Wiskunde en... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_3006954" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Chapter 11 Reverse Mathematics and Well-ordering Principles Michael Rathjen∗ and Andreas Weiermann Department of Pure Mathematics, University of Leeds Leeds LS2 9JT, UK E-mail: rathjen@ maths. leeds. ac. uk Vakgroep Zuivere Wiskunde en Computeralgebra, Ghent University Krijgslaan 281-Gebouw S22, B9000 Gent, Belgium E-mail: Andreas. Weiermann@ ugent.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3006954" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="8ea786ce319e0119a6d8092530251bf5" rel="nofollow" data-download="{"attachment_id":30953407,"asset_id":3006954,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/30953407/download_file?st=MTczMjgzODk4Myw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="597738" href="https://ugent.academia.edu/AndreasWeiermann">Andreas Weiermann</a><script data-card-contents-for-user="597738" type="text/json">{"id":597738,"first_name":"Andreas","last_name":"Weiermann","domain_name":"ugent","page_name":"AndreasWeiermann","display_name":"Andreas Weiermann","profile_url":"https://ugent.academia.edu/AndreasWeiermann","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_3006954 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3006954"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3006954, container: ".js-paper-rank-work_3006954", }); });</script></li><li class="js-percentile-work_3006954 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 3006954; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_3006954"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_3006954 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="3006954"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 3006954; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=3006954]").text(description); $(".js-view-count-work_3006954").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_3006954").removeClass('hidden') })</script></div></li></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_3006955" data-work_id="3006955" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/3006955/Characterizing_the_elementary_recursive_functions_by_a_fragment_of_G%C3%B6dels_T">Characterizing the elementary recursive functions by a fragment of Gödel's T</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Abstract. Let T be Gödel's system of primitive recursive functionals of finite type in a combinatory logic formulation. Let be the subsystem of T in which the iterator and recursor constants are permitted only when immediately applied to... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_3006955" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Abstract. Let T be Gödel's system of primitive recursive functionals of finite type in a combinatory logic formulation. Let be the subsystem of T in which the iterator and recursor constants are permitted only when immediately applied to type 0 arguments. By a Howard-Schütte-style argument the-derivation lengths are classified in terms of an iterated exponential function. As a consequence a constructive strong normalization proof for is obtained.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3006955" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="1d2592a883152cda39644046cf51864b" rel="nofollow" data-download="{"attachment_id":30953406,"asset_id":3006955,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/30953406/download_file?st=MTczMjgzODk4NCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="597738" href="https://ugent.academia.edu/AndreasWeiermann">Andreas Weiermann</a><script data-card-contents-for-user="597738" type="text/json">{"id":597738,"first_name":"Andreas","last_name":"Weiermann","domain_name":"ugent","page_name":"AndreasWeiermann","display_name":"Andreas Weiermann","profile_url":"https://ugent.academia.edu/AndreasWeiermann","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_3006955 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3006955"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3006955, container: ".js-paper-rank-work_3006955", }); });</script></li><li class="js-percentile-work_3006955 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 3006955; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_3006955"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_3006955 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="3006955"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 3006955; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=3006955]").text(description); $(".js-view-count-work_3006955").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_3006955").removeClass('hidden') })</script></div></li></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_3006956" data-work_id="3006956" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/3006956/Phase_transitions_in_logic_and_combinatorics">Phase transitions in logic and combinatorics</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Abstract A major theme in proof theory consists in classifying the proof strength of mathematical frameworks for reasoning about mathematics. The resulting phase transitions from provability to unprovability are interesting from the... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_3006956" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Abstract A major theme in proof theory consists in classifying the proof strength of mathematical frameworks for reasoning about mathematics. The resulting phase transitions from provability to unprovability are interesting from the foundational as well as from the mathematical point of view. It is very surprising that during these investigations methods from analytic number theory, combinatorial probability theory, complex analysis and Ramsey theory enter the scene.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3006956" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="dae5cbf8c81f2a80dc654b18d5b7361b" rel="nofollow" data-download="{"attachment_id":30953405,"asset_id":3006956,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/30953405/download_file?st=MTczMjgzODk4NCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="597738" href="https://ugent.academia.edu/AndreasWeiermann">Andreas Weiermann</a><script data-card-contents-for-user="597738" type="text/json">{"id":597738,"first_name":"Andreas","last_name":"Weiermann","domain_name":"ugent","page_name":"AndreasWeiermann","display_name":"Andreas Weiermann","profile_url":"https://ugent.academia.edu/AndreasWeiermann","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_3006956 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3006956"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3006956, container: ".js-paper-rank-work_3006956", }); });</script></li><li class="js-percentile-work_3006956 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 3006956; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_3006956"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_3006956 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="3006956"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 3006956; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=3006956]").text(description); $(".js-view-count-work_3006956").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_3006956").removeClass('hidden') })</script></div></li></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_3006957" data-work_id="3006957" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/3006957/Stanford_University_Stanford_CA_March_19_22_2005">Stanford University, Stanford, CA March 19–22, 2005</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Since its introduction by Joyal and Moerdijk in [1], the algebraic approach to set theory has proven to be a flexible and powerful tool for constructing models of various elementary intuitionistic and classical set theories. The method... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_3006957" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Since its introduction by Joyal and Moerdijk in [1], the algebraic approach to set theory has proven to be a flexible and powerful tool for constructing models of various elementary intuitionistic and classical set theories. The method has been further advanced through several recent collaborations (cf.[2]), the results of which are summarized in this talk.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3006957" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="81e3baf4a99cec0956facecb922104bd" rel="nofollow" data-download="{"attachment_id":32402946,"asset_id":3006957,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/32402946/download_file?st=MTczMjgzODk4NCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="597738" href="https://ugent.academia.edu/AndreasWeiermann">Andreas Weiermann</a><script data-card-contents-for-user="597738" type="text/json">{"id":597738,"first_name":"Andreas","last_name":"Weiermann","domain_name":"ugent","page_name":"AndreasWeiermann","display_name":"Andreas Weiermann","profile_url":"https://ugent.academia.edu/AndreasWeiermann","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_3006957 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3006957"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3006957, container: ".js-paper-rank-work_3006957", }); });</script></li><li class="js-percentile-work_3006957 InlineList-item InlineList-item--bordered hidden u-tcGrayDark"><span class="percentile-widget hidden"><span class="u-mr2x percentile-widget" style="display: none">•</span><span class="u-mr2x work-percentile"></span></span><script>$(function () { var workId = 3006957; window.Academia.workPercentilesFetcher.queue(workId, function (percentileText) { var container = $(".js-percentile-work_3006957"); container.find('.work-percentile').text(percentileText.charAt(0).toUpperCase() + percentileText.slice(1)); container.find('.percentile-widget').show(); container.find('.percentile-widget').removeClass('hidden'); }); });</script></li><li class="js-view-count-work_3006957 InlineList-item InlineList-item--bordered hidden"><div><span><span class="js-view-count view-count u-mr2x" data-work-id="3006957"><i class="fa fa-spinner fa-spin"></i></span><script>$(function () { var workId = 3006957; window.Academia.workViewCountsFetcher.queue(workId, function (count) { var description = window.$h.commaizeInt(count) + " " + window.$h.pluralize(count, 'View'); $(".js-view-count[data-work-id=3006957]").text(description); $(".js-view-count-work_3006957").attr('title', description).tooltip(); }); });</script></span><script>$(function() { $(".js-view-count-work_3006957").removeClass('hidden') })</script></div></li></ul></li></ul></div></div><div class="u-borderBottom1 u-borderColorGrayLighter"><div class="clearfix u-pv7x u-mb0x js-work-card work_3006959" data-work_id="3006959" itemscope="itemscope" itemtype="https://schema.org/ScholarlyArticle"><div class="header"><div class="title u-fontSerif u-fs22 u-lineHeight1_3"><a class="u-tcGrayDarkest js-work-link" href="https://www.academia.edu/3006959/A_Simplified_Characterisation_of_Provably_Computable_Functions_of_the_System_ID1">A Simplified Characterisation of Provably Computable Functions of the System ID1</a></div></div><div class="u-pb4x u-mt3x"><div class="summary u-fs14 u-fw300 u-lineHeight1_5 u-tcGrayDarkest"><div class="summarized">Page 1. CiE 2012 - How the World Computes, June 22, 2012, Cambridge A Simplified Characterisation of Provably Computable Functions of the System ID1 Naohi Eguchi 1 and Andreas Weiermann 2 1 Mathematical Institute, Tohoku University, Japan... <a class="more_link u-tcGrayDark u-linkUnstyled" data-container=".work_3006959" data-show=".complete" data-hide=".summarized" data-more-link-behavior="true" href="#">more</a></div><div class="complete hidden">Page 1. CiE 2012 - How the World Computes, June 22, 2012, Cambridge A Simplified Characterisation of Provably Computable Functions of the System ID1 Naohi Eguchi 1 and Andreas Weiermann 2 1 Mathematical Institute, Tohoku University, Japan 2 Department of Mathematics, Ghent University, Belgium Page 2. Introduction Gödel's incompleteness theorem For any consistent system T of arithmetic, there exists a statement A st N |= A but T ̸⊢ A.</div></div></div><ul class="InlineList u-ph0x u-fs13"><li class="InlineList-item logged_in_only"><div class="share_on_academia_work_button"><a class="academia_share Button Button--inverseBlue Button--sm js-bookmark-button" data-academia-share="Work/3006959" data-share-source="work_strip" data-spinner="small_white_hide_contents"><i class="fa fa-plus"></i><span class="work-strip-link-text u-ml1x" data-content="button_text">Bookmark</span></a></div></li><li class="InlineList-item"><div class="download"><a id="4c2ddcebf09cfd9f20ac65646f325572" rel="nofollow" data-download="{"attachment_id":30953403,"asset_id":3006959,"asset_type":"Work","always_allow_download":false,"track":null,"button_location":"work_strip","source":null,"hide_modal":null}" class="Button Button--sm Button--inverseGreen js-download-button prompt_button doc_download" href="https://www.academia.edu/attachments/30953403/download_file?st=MTczMjgzODk4NCw4LjIyMi4yMDguMTQ2&s=work_strip"><i class="fa fa-arrow-circle-o-down fa-lg"></i><span class="u-textUppercase u-ml1x" data-content="button_text">Download</span></a></div></li><li class="InlineList-item"><ul class="InlineList InlineList--bordered u-ph0x"><li class="InlineList-item InlineList-item--bordered"><span class="InlineList-item-text">by <span itemscope="itemscope" itemprop="author" itemtype="https://schema.org/Person"><a class="u-tcGrayDark u-fw700" data-has-card-for-user="597738" href="https://ugent.academia.edu/AndreasWeiermann">Andreas Weiermann</a><script data-card-contents-for-user="597738" type="text/json">{"id":597738,"first_name":"Andreas","last_name":"Weiermann","domain_name":"ugent","page_name":"AndreasWeiermann","display_name":"Andreas Weiermann","profile_url":"https://ugent.academia.edu/AndreasWeiermann","photo":"/images/s65_no_pic.png"}</script></span></span></li><li class="js-paper-rank-work_3006959 InlineList-item InlineList-item--bordered hidden"><span class="js-paper-rank-view hidden u-tcGrayDark" data-paper-rank-work-id="3006959"><i class="u-m1x fa fa-bar-chart"></i><strong class="js-paper-rank"></strong></span><script>$(function() { new Works.PaperRankView({ workId: 3006959, container: ".js-paper-rank-work_3006959", }); 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