Bulletin of Symbolic Logic: Volume 30 -

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class="more-by-this-author ">DAVID FERNÁNDEZ-DUQUE</a><span class="separator">, </span><a href="/core/search?filters%5BauthorTerms%5D=ANDREAS WEIERMANN&eventCode=SE-AU" class="more-by-this-author ">ANDREAS WEIERMANN</a> </li> <li> <dl> <dt class="published dl-inline">Published online by Cambridge University Press: </dt> <dd> <span class="date"> 17 January 2024</span><span class="pages">, pp. 1-19</span> </dd> </dl> </li> <li> <ul class="access links"> <li class="type article">Article</li> <li> <ul class="access-modal"> <li class="status entitled"> <span class="sr-only">You have access </span> <span class="entitled status-text" title="You have access to this product" aria-hidden="true">Access</span> <span class="icon access-icon cursorDefault" title="You have access to this product" aria-hidden="true"> <svg id="access-FA21BEB7DF1B04340FBD0E962A7327B4" xmlns="http://www.w3.org/2000/svg" viewBox="-701.2 242.7 206.9 328.3"> <circle class="st0" fill="#229619" cx="-597.9" cy="413.8" 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number-theoretic statement known to be independent of Peano arithmetic. It involves sequences of natural numbers which at first appear to diverge, but eventually decrease to zero. These sequences are defined relative to a notation system based on exponentiation for the natural numbers. In this article, we provide a self-contained and modern analysis of Goodstein’s principle, obtaining some variations and improvements. We explore notions of optimality for notation systems and apply them to the classical Goodstein process and to a weaker variant based on multiplication rather than exponentiation. In particular, we introduce the notion of <span class='italic'>base</span>-<span class='italic'>change maximality,</span> and show how it leads to far-reaching extensions of Goodstein’s result. We moreover show that by varying the initial base of the Goodstein process, one readily obtains independence results for each of the fragments <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240402024606446-0816:S1079898624000015:S1079898624000015_inline1.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathsf {I}\Sigma _n$</span></span></span></span> of Peano arithmetic.</p></div> </div> </div> </li> </ul> </div> </div> </div> </div> <div class="representation overview search" data-prod-id="58AED5F57C778A768E3087C32EFD576E" data-mathjax-disabled="false"> <div class="row"> <div class="small-1 columns checklist-col"> <label class="product-listing-with-inputs-input"> <input type="checkbox" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"58AED5F57C778A768E3087C32EFD576E","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}' name="productParts" style="display: inline-block" data-prod-id="58AED5F57C778A768E3087C32EFD576E" data-prod-source="listing" data-prod-type="JOURNAL_ARTICLE" data-file-size="3176" data-has-entitlement="true" data-has-pdf="true" /> </label> </div> <div class="product-listing-with-inputs-content"> <div class="small-10 columns margin-bottom"> <ul class="details"> <li class="title"><h3><a class="part-link" href="/core/journals/bulletin-of-symbolic-logic/article/universal-theory-of-the-hyperfinite-ii1-factor-is-not-computable/58AED5F57C778A768E3087C32EFD576E"> THE UNIVERSAL THEORY OF THE HYPERFINITE II<span class="inlineFormula"><span class="alternatives"><img class="inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off" src="//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000076:S1079898624000076_inline1.png" /><span class="mathjax-tex-wrapper"><span class="tex-math mathjax-tex-math mathjax-on">$_1$</span></span></span></span> FACTOR IS NOT COMPUTABLE </a></h3></li> <li></li> <li class="author"> <a href="/core/search?filters%5BauthorTerms%5D=ISAAC GOLDBRING&eventCode=SE-AU" class="more-by-this-author ">ISAAC GOLDBRING</a><span class="separator">, </span><a href="/core/search?filters%5BauthorTerms%5D=BRADD HART&eventCode=SE-AU" class="more-by-this-author ">BRADD HART</a> </li> <li> <dl> <dt class="published dl-inline">Published online by Cambridge University Press: </dt> <dd> <span class="date"> 16 February 2024</span><span class="pages">, pp. 181-198</span> </dd> </dl> </li> <li> <ul class="access links"> <li class="type article">Article</li> <li> <ul class="access-modal"> <li class="status entitled"> <span class="sr-only">You have access </span> <span class="entitled status-text" title="You have access to this product" aria-hidden="true">Access</span> <span class="icon access-icon cursorDefault" title="You have access to this product" aria-hidden="true"> <svg id="access-58AED5F57C778A768E3087C32EFD576E" xmlns="http://www.w3.org/2000/svg" viewBox="-701.2 242.7 206.9 328.3"> <circle class="st0" fill="#229619" cx="-597.9" cy="413.8" r="102.5"/> <path class="st1" fill="#fff" d="M-616 472.4l-49.6-49.6 26.5-26.5 23.1 23 59.3-59.3 26.5 26.5z"/> </svg> </span> </li> <li class="status open-access"> </li> </ul> </li> <li> <a href="/core/services/aop-cambridge-core/content/view/58AED5F57C778A768E3087C32EFD576E/S1079898624000076a.pdf/universal_theory_of_the_hyperfinite_ii1_factor_is_not_computable.pdf" data-pdf-content-id="58AED5F57C778A768E3087C32EFD576E" target="_blank" aria-label="Download PDF for THE UNIVERSAL THEORY OF THE HYPERFINITE II$_1$ FACTOR IS NOT COMPUTABLE [Opens in a new window]" > <span class="icon icn pdf"> <span class="download-label">PDF </span> </span> </a> </li> <li> <a href="/core/product/58AED5F57C778A768E3087C32EFD576E/core-reader" target="_blank" aria-label="View HTML full text for THE UNIVERSAL THEORY OF THE HYPERFINITE II$_1$ FACTOR IS NOT COMPUTABLE [Opens in a new window]" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"58AED5F57C778A768E3087C32EFD576E","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}'> <span class="icon icn html wide"> <span class="download-label">HTML</span> </span> </a> </li> <li> <script> $(document).ready(function () { /** * Disable tooltip aria label attributes, so the screen reader can correctly read out the core * reader tab navigation. * Set a short delay, so we can be sure the tooltips have rendered, before updating the DOM. */ setTimeout(function () { var $downloadLinks = $('.download-types li a'); 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The proof uses the recent result that MIP*=RE. Combined with an earlier observation of the authors, this yields a proof that the Connes Embedding Problem has a negative solution that avoids the equivalences with Kirchberg’s QWEP Conjecture and Tsirelson’s Problem.</p></div> </div> </div> </li> </ul> </div> </div> </div> </div> <div class="representation overview search" data-prod-id="84EA24D938C0775C59BE4D54E5E645B5" data-mathjax-disabled="false"> <div class="row"> <div class="small-1 columns checklist-col"> <label class="product-listing-with-inputs-input"> <input type="checkbox" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"84EA24D938C0775C59BE4D54E5E645B5","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}' name="productParts" style="display: inline-block" data-prod-id="84EA24D938C0775C59BE4D54E5E645B5" data-prod-source="listing" data-prod-type="JOURNAL_ARTICLE" data-file-size="3739" data-has-entitlement="true" data-has-pdf="true" /> </label> </div> <div class="product-listing-with-inputs-content"> <div class="small-10 columns margin-bottom"> <ul class="details"> <li class="title"><h3><a class="part-link" href="/core/journals/bulletin-of-symbolic-logic/article/on-the-existence-of-strong-proof-complexity-generators/84EA24D938C0775C59BE4D54E5E645B5"> ON THE EXISTENCE OF STRONG PROOF COMPLEXITY GENERATORS </a></h3></li> <li> Part of: <ul class="partOfCollection"> <li><a href="/core/product/identifier/MSC_2010_THEORY_OF_COMPUTING/type/BESPOKE_COLLECTION" class="">Theory of computing</a></li> <li><a href="/core/product/identifier/MSC_2010_PROOF_THEORY_AND_CONSTRUCTIVE/type/BESPOKE_COLLECTION" class="">Proof theory and constructive mathematics</a></li> </ul> </li> <li class="author"> <a href="/core/search?filters%5BauthorTerms%5D=JAN KRAJÍČEK&eventCode=SE-AU" class="more-by-this-author ">JAN KRAJÍČEK</a> </li> <li> <dl> <dt class="published dl-inline">Published online by Cambridge University Press: </dt> <dd> <span class="date"> 22 November 2023</span><span class="pages">, pp. 20-40</span> </dd> </dl> </li> <li> <ul class="access links"> <li class="type article">Article</li> <li> <ul class="access-modal"> <li class="status entitled"> <span class="sr-only">You have access </span> <span class="entitled status-text" title="You have access to this product" aria-hidden="true">Access</span> <span class="icon access-icon cursorDefault" title="You have access to this product" aria-hidden="true"> <svg id="access-84EA24D938C0775C59BE4D54E5E645B5" xmlns="http://www.w3.org/2000/svg" viewBox="-701.2 242.7 206.9 328.3"> <circle class="st0" fill="#229619" cx="-597.9" cy="413.8" r="102.5"/> <path class="st1" fill="#fff" d="M-616 472.4l-49.6-49.6 26.5-26.5 23.1 23 59.3-59.3 26.5 26.5z"/> </svg> </span> </li> <li class="status open-access"> <span class="icon access open-access cursorDefault"></span>Open access </li> </ul> </li> <li> <a href="/core/services/aop-cambridge-core/content/view/84EA24D938C0775C59BE4D54E5E645B5/S1079898623000409a.pdf/on_the_existence_of_strong_proof_complexity_generators.pdf" data-pdf-content-id="84EA24D938C0775C59BE4D54E5E645B5" target="_blank" aria-label="Download PDF for ON THE EXISTENCE OF STRONG PROOF COMPLEXITY GENERATORS [Opens in a new window]" > <span class="icon icn pdf"> <span class="download-label">PDF </span> </span> </a> </li> <li> <a href="/core/product/84EA24D938C0775C59BE4D54E5E645B5/core-reader" target="_blank" aria-label="View HTML full text for ON THE EXISTENCE OF STRONG PROOF COMPLEXITY GENERATORS [Opens in a new window]" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"84EA24D938C0775C59BE4D54E5E645B5","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}'> <span class="icon icn html wide"> <span class="download-label">HTML</span> </span> </a> </li> <li> <script> $(document).ready(function () { /** * Disable tooltip aria label attributes, so the screen reader can correctly read out the core * reader tab navigation. * Set a short delay, so we can be sure the tooltips have rendered, before updating the DOM. */ setTimeout(function () { var $downloadLinks = $('.download-types li a'); if ($downloadLinks.length) { $downloadLinks.attr('role', 'tab'); $downloadLinks.attr('aria-describedby', ''); // Remove the aria alerts on the tool tips $('.qtip').attr('role', 'none'); } }, 500); }); </script> </li> <li><a href="#" class="export-citation-component" data-prod-id="84EA24D938C0775C59BE4D54E5E645B5" data-container="representation" aria-label="Export citation for ON THE EXISTENCE OF STRONG PROOF COMPLEXITY GENERATORS">Export citation </a></li> </ul> </li> <li class="accordion no-margin-top abstract" data-accordion> <div class="accordion-navigation"> <button onclick="toggleViewHide(this, 'abstractS1079898623000409')" class="button-no-transition transparent-no-border" aria-expanded="false" aria-label="View Abstract for ON THE EXISTENCE OF STRONG PROOF COMPLEXITY GENERATORS" data-cup-event='{"eventCode":"IN-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","usedIdentities":[],"authenticationMethods":[],"referringURL":"","eventType":"VADE","searchScope":"PRODUCT","productId":"84EA24D938C0775C59BE4D54E5E645B5"}' data-cup-event-disable-on-hide="true" > <div class="icon open-drawer small"> <svg xmlns="http://www.w3.org/2000/svg" width="15" height="15" viewBox="0 0 17 17" class="is-opened"> <circle fill="#696969" cx="8.5" cy="8.5" r="7.5"/> <circle fill="#FFF" cx="8.5" cy="8.5" r="7"/> <path fill="#696969" d="M12.6 7.3H4.4c-.8 0-1.4.5-1.4 1.2s.6 1.2 1.4 1.2h8.3c.8 0 1.4-.5 1.4-1.2-.1-.7-.7-1.2-1.5-1.2z"/> </svg> <svg xmlns="http://www.w3.org/2000/svg" width="15" height="15" viewBox="0 0 15 15" class="is-closed"> <g transform="translate(1 1)" fill="none" fill-rule="evenodd"> <circle fill="#696969" cx="6.5" cy="6.5" r="6.5"/> <circle fill="#FFF" cx="6.5" cy="6.5" r="6"/> <path d="M9.855 5.518H7.482V3.145C7.482 2.491 7.072 2 6.5 2c-.573 0-.982.49-.982 1.145V5.518H3.145C2.491 5.518 2 5.928 2 6.5c0 .573.49.982 1.145.982H5.518v2.372c0 .655.41 1.146.982 1.146.573 0 .982-.49.982-1.145V7.482h2.372c.655 0 1.146-.41 1.146-.982 0-.573-.49-.982-1.145-.982z" fill="#696969"/> </g> </svg> <span class="toggle-text">View </span> <span class="excerpt-type">abstract</span> </div> </button> <div id="abstractS1079898623000409" class="content"> <div class="abstract" data-abstract-type="normal"><p>Cook and Reckhow [5] pointed out that <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240402024606446-0816:S1079898623000409:S1079898623000409_inline1.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathcal {N}\mathcal {P} \neq co\mathcal {N}\mathcal {P}$</span></span></span></span> iff there is no propositional proof system that admits polynomial size proofs of all tautologies. The theory of proof complexity generators aims at constructing sets of tautologies hard for strong and possibly for all proof systems. We focus on a conjecture from [16] in foundations of the theory that there is a proof complexity generator hard for all proof systems. This can be equivalently formulated (for p-time generators) without a reference to proof complexity notions as follows:</p><p><ul><li><p><span class='abs-list-label'>•</span> There exists a p-time function <span class='italic'>g</span> stretching each input by one bit such that its range <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240402024606446-0816:S1079898623000409:S1079898623000409_inline2.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$rng(g)$</span></span></span></span> intersects all infinite <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240402024606446-0816:S1079898623000409:S1079898623000409_inline3.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathcal {N}\mathcal {P}$</span></span></span></span> sets.</p></li></ul>We consider several facets of this conjecture, including its links to bounded arithmetic (witnessing and independence results), to time-bounded Kolmogorov complexity, to feasible disjunction property of propositional proof systems and to complexity of proof search. We argue that a specific gadget generator from [18] is a good candidate for <span class='italic'>g</span>. We define a new hardness property of generators, the <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240402024606446-0816:S1079898623000409:S1079898623000409_inline4.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\bigvee $</span></span></span></span>-hardness, and show that one specific gadget generator is the <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240402024606446-0816:S1079898623000409:S1079898623000409_inline5.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\bigvee $</span></span></span></span>-hardest (w.r.t. any sufficiently strong proof system). We define the class of feasibly infinite <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240402024606446-0816:S1079898623000409:S1079898623000409_inline6.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathcal {N}\mathcal {P}$</span></span></span></span> sets and show, assuming a hypothesis from circuit complexity, that the conjecture holds for all feasibly infinite <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240402024606446-0816:S1079898623000409:S1079898623000409_inline7.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathcal {N}\mathcal {P}$</span></span></span></span> sets.</p> </div> </div> </div> </li> </ul> </div> </div> </div> </div> <div class="representation overview search" data-prod-id="015C78E98FF1ABA01AEA5F15C1EC0844" data-mathjax-disabled="false"> <div class="row"> <div class="small-1 columns checklist-col"> <label class="product-listing-with-inputs-input"> <input type="checkbox" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"015C78E98FF1ABA01AEA5F15C1EC0844","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}' name="productParts" style="display: inline-block" data-prod-id="015C78E98FF1ABA01AEA5F15C1EC0844" data-prod-source="listing" data-prod-type="JOURNAL_ARTICLE" data-file-size="3367" data-has-entitlement="true" data-has-pdf="true" /> </label> </div> <div class="product-listing-with-inputs-content"> <div class="small-10 columns margin-bottom"> <ul class="details"> <li class="title"><h3><a class="part-link" href="/core/journals/bulletin-of-symbolic-logic/article/iterated-priority-arguments-in-descriptive-set-theory/015C78E98FF1ABA01AEA5F15C1EC0844"> ITERATED PRIORITY ARGUMENTS IN DESCRIPTIVE SET THEORY </a></h3></li> <li> Part of: <ul class="partOfCollection"> <li><a href="/core/product/identifier/MSC_2010_SET_THEORY/type/BESPOKE_COLLECTION" class="">Set theory</a></li> <li><a href="/core/product/identifier/COMPUTABILITY_AND_RECURSION_THEORY/type/BESPOKE_COLLECTION" class="">Computability and recursion theory</a></li> </ul> </li> <li class="author"> <a href="/core/search?filters%5BauthorTerms%5D=ADAM DAY&eventCode=SE-AU" class="more-by-this-author ">ADAM DAY</a><span class="separator">, </span><a href="/core/search?filters%5BauthorTerms%5D=NOAM GREENBERG&eventCode=SE-AU" class="more-by-this-author ">NOAM GREENBERG</a><span class="separator">, </span><a href="/core/search?filters%5BauthorTerms%5D=MATTHEW HARRISON-TRAINOR&eventCode=SE-AU" class="more-by-this-author ">MATTHEW HARRISON-TRAINOR</a><span class="separator">, </span><a href="/core/search?filters%5BauthorTerms%5D=DAN TURETSKY&eventCode=SE-AU" class="more-by-this-author ">DAN TURETSKY</a> </li> <li> <dl> <dt class="published dl-inline">Published online by Cambridge University Press: </dt> <dd> <span class="date"> 08 April 2024</span><span class="pages">, pp. 199-226</span> </dd> </dl> </li> <li> <ul class="access links"> <li class="type article">Article</li> <li> <ul class="access-modal"> <li class="status entitled"> <span class="sr-only">You have access </span> <span class="entitled status-text" title="You have access to this product" aria-hidden="true">Access</span> <span class="icon access-icon cursorDefault" title="You have access to this product" aria-hidden="true"> <svg id="access-015C78E98FF1ABA01AEA5F15C1EC0844" 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viewBox="0 0 17 17" class="is-opened"> <circle fill="#696969" cx="8.5" cy="8.5" r="7.5"/> <circle fill="#FFF" cx="8.5" cy="8.5" r="7"/> <path fill="#696969" d="M12.6 7.3H4.4c-.8 0-1.4.5-1.4 1.2s.6 1.2 1.4 1.2h8.3c.8 0 1.4-.5 1.4-1.2-.1-.7-.7-1.2-1.5-1.2z"/> </svg> <svg xmlns="http://www.w3.org/2000/svg" width="15" height="15" viewBox="0 0 15 15" class="is-closed"> <g transform="translate(1 1)" fill="none" fill-rule="evenodd"> <circle fill="#696969" cx="6.5" cy="6.5" r="6.5"/> <circle fill="#FFF" cx="6.5" cy="6.5" r="6"/> <path d="M9.855 5.518H7.482V3.145C7.482 2.491 7.072 2 6.5 2c-.573 0-.982.49-.982 1.145V5.518H3.145C2.491 5.518 2 5.928 2 6.5c0 .573.49.982 1.145.982H5.518v2.372c0 .655.41 1.146.982 1.146.573 0 .982-.49.982-1.145V7.482h2.372c.655 0 1.146-.41 1.146-.982 0-.573-.49-.982-1.145-.982z" fill="#696969"/> </g> </svg> <span class="toggle-text">View </span> <span class="excerpt-type">abstract</span> </div> </button> <div id="abstractS1079898624000234" class="content"> <div class="abstract" data-abstract-type="normal"><p>We present the true stages machinery and illustrate its applications to descriptive set theory. We use this machinery to provide new proofs of the Hausdorff–Kuratowski and Wadge theorems on the structure of <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000234:S1079898624000234_inline1.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathbf {\Delta }^0_\xi $</span></span></span></span>, Louveau and Saint Raymond’s separation theorem, and Louveau’s separation theorem.</p></div> </div> </div> </li> </ul> </div> </div> </div> </div> <div class="representation overview search" data-prod-id="F52356239F0EB220539582607C3F8803" data-mathjax-disabled="false"> <div class="row"> <div class="small-1 columns checklist-col"> <label class="product-listing-with-inputs-input"> <input type="checkbox" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"F52356239F0EB220539582607C3F8803","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}' name="productParts" style="display: inline-block" data-prod-id="F52356239F0EB220539582607C3F8803" data-prod-source="listing" data-prod-type="JOURNAL_ARTICLE" data-file-size="3266" data-has-entitlement="true" data-has-pdf="true" /> </label> </div> <div class="product-listing-with-inputs-content"> <div class="small-10 columns margin-bottom"> <ul class="details"> <li class="title"><h3><a class="part-link" href="/core/journals/bulletin-of-symbolic-logic/article/categorical-quantification/F52356239F0EB220539582607C3F8803"> CATEGORICAL QUANTIFICATION </a></h3></li> <li> Part of: <ul class="partOfCollection"> <li><a href="/core/product/identifier/MSC_2010_PHILOSOPHICAL_ASPECTS_OF_LOGIC_AND/type/BESPOKE_COLLECTION" class="">Philosophical aspects of logic and foundations</a></li> <li><a href="/core/product/identifier/MSC_2010_PROOF_THEORY_AND_CONSTRUCTIVE/type/BESPOKE_COLLECTION" class="">Proof theory and constructive mathematics</a></li> <li><a href="/core/product/identifier/MSC_2010_MODEL_THEORY/type/BESPOKE_COLLECTION" class="">Model theory</a></li> <li><a href="/core/product/identifier/MSC_2010_GENERAL_LOGIC/type/BESPOKE_COLLECTION" class="">General logic</a></li> </ul> </li> <li class="author"> <a href="/core/search?filters%5BauthorTerms%5D=CONSTANTIN C. BRÎNCUŞ&eventCode=SE-AU" class="more-by-this-author ">CONSTANTIN C. BRÎNCUŞ</a> </li> <li> <dl> <dt class="published dl-inline">Published online by Cambridge University Press: </dt> <dd> <span class="date"> 24 January 2024</span><span class="pages">, pp. 227-252</span> </dd> </dl> </li> <li> <ul class="access links"> <li class="type article">Article</li> <li> <ul class="access-modal"> <li class="status entitled"> <span class="sr-only">You have access </span> <span class="entitled status-text" title="You have access to this product" aria-hidden="true">Access</span> <span class="icon access-icon cursorDefault" title="You have access to this product" aria-hidden="true"> <svg id="access-F52356239F0EB220539582607C3F8803" xmlns="http://www.w3.org/2000/svg" viewBox="-701.2 242.7 206.9 328.3"> <circle class="st0" fill="#229619" cx="-597.9" cy="413.8" r="102.5"/> <path class="st1" fill="#fff" d="M-616 472.4l-49.6-49.6 26.5-26.5 23.1 23 59.3-59.3 26.5 26.5z"/> </svg> </span> </li> <li class="status open-access"> </li> </ul> </li> <li> <a href="/core/services/aop-cambridge-core/content/view/F52356239F0EB220539582607C3F8803/S1079898624000039a.pdf/categorical_quantification.pdf" data-pdf-content-id="F52356239F0EB220539582607C3F8803" target="_blank" aria-label="Download PDF for CATEGORICAL QUANTIFICATION [Opens in a new window]" > <span class="icon icn pdf"> <span class="download-label">PDF </span> </span> </a> </li> <li> <a href="/core/product/F52356239F0EB220539582607C3F8803/core-reader" target="_blank" aria-label="View HTML full text for CATEGORICAL QUANTIFICATION [Opens in a new window]" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"F52356239F0EB220539582607C3F8803","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}'> <span class="icon icn html wide"> <span class="download-label">HTML</span> </span> </a> </li> <li> <script> $(document).ready(function () { /** * Disable tooltip aria label attributes, so the screen reader can correctly read out the core * reader tab navigation. * Set a short delay, so we can be sure the tooltips have rendered, before updating the DOM. */ setTimeout(function () { var $downloadLinks = $('.download-types li a'); if ($downloadLinks.length) { $downloadLinks.attr('role', 'tab'); $downloadLinks.attr('aria-describedby', ''); // Remove the aria alerts on the tool tips $('.qtip').attr('role', 'none'); } }, 500); }); </script> </li> <li><a href="#" class="export-citation-component" data-prod-id="F52356239F0EB220539582607C3F8803" data-container="representation" aria-label="Export citation for CATEGORICAL QUANTIFICATION">Export citation </a></li> </ul> </li> <li class="accordion no-margin-top abstract" data-accordion> <div class="accordion-navigation"> <button onclick="toggleViewHide(this, 'abstractS1079898624000039')" class="button-no-transition transparent-no-border" aria-expanded="false" aria-label="View Abstract for CATEGORICAL QUANTIFICATION" data-cup-event='{"eventCode":"IN-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","usedIdentities":[],"authenticationMethods":[],"referringURL":"","eventType":"VADE","searchScope":"PRODUCT","productId":"F52356239F0EB220539582607C3F8803"}' data-cup-event-disable-on-hide="true" > <div class="icon open-drawer small"> <svg xmlns="http://www.w3.org/2000/svg" width="15" height="15" viewBox="0 0 17 17" class="is-opened"> <circle fill="#696969" cx="8.5" cy="8.5" r="7.5"/> <circle fill="#FFF" cx="8.5" cy="8.5" r="7"/> <path fill="#696969" d="M12.6 7.3H4.4c-.8 0-1.4.5-1.4 1.2s.6 1.2 1.4 1.2h8.3c.8 0 1.4-.5 1.4-1.2-.1-.7-.7-1.2-1.5-1.2z"/> </svg> <svg xmlns="http://www.w3.org/2000/svg" width="15" height="15" viewBox="0 0 15 15" class="is-closed"> <g transform="translate(1 1)" fill="none" fill-rule="evenodd"> <circle fill="#696969" cx="6.5" cy="6.5" r="6.5"/> <circle fill="#FFF" cx="6.5" cy="6.5" r="6"/> <path d="M9.855 5.518H7.482V3.145C7.482 2.491 7.072 2 6.5 2c-.573 0-.982.49-.982 1.145V5.518H3.145C2.491 5.518 2 5.928 2 6.5c0 .573.49.982 1.145.982H5.518v2.372c0 .655.41 1.146.982 1.146.573 0 .982-.49.982-1.145V7.482h2.372c.655 0 1.146-.41 1.146-.982 0-.573-.49-.982-1.145-.982z" fill="#696969"/> </g> </svg> <span class="toggle-text">View </span> <span class="excerpt-type">abstract</span> </div> </button> <div id="abstractS1079898624000039" class="content"> <div class="abstract" data-abstract-type="normal"><p>Due to Gödel’s incompleteness results, the categoricity of a sufficiently rich mathematical theory and the semantic completeness of its underlying logic are two mutually exclusive ideals. For first- and second-order logics we obtain one of them with the cost of losing the other. In addition, in both these logics the rules of deduction for their quantifiers are non-categorical. In this paper I examine two recent arguments—Warren [43] and Murzi and Topey [30]—for the idea that the natural deduction rules for the first-order universal quantifier are categorical, i.e., they uniquely determine its semantic intended meaning. Both of them make use of McGee’s open-endedness requirement and the second one uses in addition Garson’s [19] local models for defining the validity of these rules. I argue that the success of both these arguments is relative to their semantic or infinitary assumptions, which could be easily discharged if the introduction rule for the universal quantifier is taken to be an infinitary rule, i.e., non-compact. Consequently, I reconsider the use of the <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000039:S1079898624000039_inline1.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\omega $</span></span></span></span>-rule and I show that the addition of the <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000039:S1079898624000039_inline2.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\omega $</span></span></span></span>-rule to the standard formalizations of first-order logic is categorical. In addition, I argue that the open-endedness requirement does not make the first-order Peano Arithmetic categorical and I advance an argument for its categoricity based on the inferential conservativity requirement.</p> </div> </div> </div> </li> </ul> </div> </div> </div> </div> <div class="representation overview search" data-prod-id="F904806EA3032814085F696922A2DCB5" data-mathjax-disabled="false"> <div class="row"> <div class="small-1 columns checklist-col"> <label class="product-listing-with-inputs-input"> <input type="checkbox" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"F904806EA3032814085F696922A2DCB5","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}' name="productParts" style="display: inline-block" data-prod-id="F904806EA3032814085F696922A2DCB5" data-prod-source="listing" data-prod-type="JOURNAL_ARTICLE" data-file-size="3251" data-has-entitlement="true" data-has-pdf="true" /> </label> </div> <div class="product-listing-with-inputs-content"> <div class="small-10 columns margin-bottom"> <ul class="details"> <li class="title"><h3><a class="part-link" href="/core/journals/bulletin-of-symbolic-logic/article/poincareweyls-predicativity-going-beyond-gamma-0/F904806EA3032814085F696922A2DCB5"> POINCARÉ–WEYL’S PREDICATIVITY: GOING BEYOND <span class="inlineFormula"><span class="alternatives"><img class="inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off" src="//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240402024606446-0816:S1079898624000027:S1079898624000027_inline1.png" /><span class="mathjax-tex-wrapper"><span class="tex-math mathjax-tex-math mathjax-on">$\Gamma _{0}$</span></span></span></span> </a></h3></li> <li> Part of: <ul class="partOfCollection"> <li><a href="/core/product/identifier/MSC_2010_PHILOSOPHICAL_ASPECTS_OF_LOGIC_AND/type/BESPOKE_COLLECTION" class="">Philosophical aspects of logic and foundations</a></li> <li><a href="/core/product/identifier/MSC_2010_SET_THEORY/type/BESPOKE_COLLECTION" class="">Set theory</a></li> <li><a href="/core/product/identifier/MSC_2010_GENERAL_LOGIC/type/BESPOKE_COLLECTION" class="">General logic</a></li> <li><a href="/core/product/identifier/JLU_MSCCLASSIFICATIONS_MATHEMATICAL_LOGIC_AND/type/BESPOKE_COLLECTION" class="">Mathematical Logic and Foundations</a></li> <li><a href="/core/product/identifier/MSC_2010_GENERAL_AND_MISCELLANEOUS_SPECIFIC/type/BESPOKE_COLLECTION" class="">General and miscellaneous specific topics</a></li> </ul> </li> <li class="author"> <a href="/core/search?filters%5BauthorTerms%5D=ARNON AVRON&eventCode=SE-AU" class="more-by-this-author ">ARNON AVRON</a> </li> <li> <dl> <dt class="published dl-inline">Published online by Cambridge University Press: </dt> <dd> <span class="date"> 19 January 2024</span><span class="pages">, pp. 41-91</span> </dd> </dl> </li> <li> <ul class="access links"> <li class="type article">Article</li> <li> <ul class="access-modal"> <li class="status entitled"> <span class="sr-only">You have access </span> <span class="entitled status-text" title="You have access to this product" aria-hidden="true">Access</span> <span class="icon access-icon cursorDefault" title="You have access to this product" aria-hidden="true"> <svg id="access-F904806EA3032814085F696922A2DCB5" xmlns="http://www.w3.org/2000/svg" viewBox="-701.2 242.7 206.9 328.3"> <circle class="st0" fill="#229619" cx="-597.9" cy="413.8" r="102.5"/> <path class="st1" fill="#fff" d="M-616 472.4l-49.6-49.6 26.5-26.5 23.1 23 59.3-59.3 26.5 26.5z"/> </svg> </span> </li> <li class="status open-access"> <span class="icon access open-access cursorDefault"></span>Open access </li> </ul> </li> <li> <a 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data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240402024606446-0816:S1079898624000027:S1079898624000027_inline2.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\Gamma _0$</span></span></span></span> and much bigger ordinals can be defined as von Neumann ordinals. This refutes the accepted view of <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20240402024606446-0816:S1079898624000027:S1079898624000027_inline3.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\Gamma _0$</span></span></span></span> as the “limit of predicativity”.</p></div> </div> </div> </li> </ul> </div> </div> </div> </div> <h4 class="journal-article-listing-type heading_12 margin-bottom">Review</h4> <div class="representation overview search" data-prod-id="5A23F271418DB785E2CA65CAEB36B9A0" data-mathjax-disabled="false"> <div class="row"> <div class="small-1 columns checklist-col"> <label class="product-listing-with-inputs-input"> <input type="checkbox" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"5A23F271418DB785E2CA65CAEB36B9A0","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}' name="productParts" style="display: inline-block" data-prod-id="5A23F271418DB785E2CA65CAEB36B9A0" data-prod-source="listing" data-prod-type="JOURNAL_ARTICLE" data-file-size="46038" data-has-entitlement="true" data-has-pdf="true" /> </label> </div> <div class="product-listing-with-inputs-content"> <div class="small-10 columns margin-bottom"> <ul class="details"> <li class="title"><h3><a class="part-link" href="/core/journals/bulletin-of-symbolic-logic/article/sam-roberts-pluralities-as-nothing-over-and-above-journal-of-philosophy-vol-cxix-2022-no-8-pp-405424/5A23F271418DB785E2CA65CAEB36B9A0"> Sam Roberts. 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if ($downloadLinks.length) { $downloadLinks.attr('role', 'tab'); $downloadLinks.attr('aria-describedby', ''); // Remove the aria alerts on the tool tips $('.qtip').attr('role', 'none'); } }, 500); }); </script> </li> <li><a href="#" class="export-citation-component" data-prod-id="1E6962B8F1CBC095ED8C2E1492BCE2AB" data-container="representation" aria-label="Export citation for ONE-VARIABLE FRAGMENTS OF FIRST-ORDER LOGICS">Export citation </a></li> </ul> </li> <li class="accordion no-margin-top abstract" data-accordion> <div class="accordion-navigation"> <button onclick="toggleViewHide(this, 'abstractS1079898624000222')" class="button-no-transition transparent-no-border" aria-expanded="false" aria-label="View Abstract for ONE-VARIABLE FRAGMENTS OF FIRST-ORDER LOGICS" data-cup-event='{"eventCode":"IN-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","usedIdentities":[],"authenticationMethods":[],"referringURL":"","eventType":"VADE","searchScope":"PRODUCT","productId":"1E6962B8F1CBC095ED8C2E1492BCE2AB"}' data-cup-event-disable-on-hide="true" > <div class="icon open-drawer small"> <svg xmlns="http://www.w3.org/2000/svg" width="15" height="15" viewBox="0 0 17 17" class="is-opened"> <circle fill="#696969" cx="8.5" cy="8.5" r="7.5"/> <circle fill="#FFF" cx="8.5" cy="8.5" r="7"/> <path fill="#696969" d="M12.6 7.3H4.4c-.8 0-1.4.5-1.4 1.2s.6 1.2 1.4 1.2h8.3c.8 0 1.4-.5 1.4-1.2-.1-.7-.7-1.2-1.5-1.2z"/> </svg> <svg xmlns="http://www.w3.org/2000/svg" width="15" height="15" viewBox="0 0 15 15" class="is-closed"> <g transform="translate(1 1)" fill="none" fill-rule="evenodd"> <circle fill="#696969" cx="6.5" cy="6.5" r="6.5"/> <circle fill="#FFF" cx="6.5" cy="6.5" r="6"/> <path d="M9.855 5.518H7.482V3.145C7.482 2.491 7.072 2 6.5 2c-.573 0-.982.49-.982 1.145V5.518H3.145C2.491 5.518 2 5.928 2 6.5c0 .573.49.982 1.145.982H5.518v2.372c0 .655.41 1.146.982 1.146.573 0 .982-.49.982-1.145V7.482h2.372c.655 0 1.146-.41 1.146-.982 0-.573-.49-.982-1.145-.982z" fill="#696969"/> </g> </svg> <span class="toggle-text">View </span> <span class="excerpt-type">abstract</span> </div> </button> <div id="abstractS1079898624000222" class="content"> <div class="abstract" data-abstract-type="normal"><p>The one-variable fragment of a first-order logic may be viewed as an “S5-like” modal logic, where the universal and existential quantifiers are replaced by box and diamond modalities, respectively. Axiomatizations of these modal logics have been obtained for special cases—notably, the modal counterparts <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000222:S1079898624000222_inline1.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathrm {S5}$</span></span></span></span> and <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000222:S1079898624000222_inline2.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathrm {MIPC}$</span></span></span></span> of the one-variable fragments of first-order classical logic and first-order intuitionistic logic, respectively—but a general approach, extending beyond first-order intermediate logics, has been lacking. To this end, a sufficient criterion is given in this paper for the one-variable fragment of a semantically defined first-order logic—spanning families of intermediate, substructural, many-valued, and modal logics—to admit a certain natural axiomatization. More precisely, an axiomatization is obtained for the one-variable fragment of any first-order logic based on a variety of algebraic structures with a lattice reduct that has the superamalgamation property, using a generalized version of a functional representation theorem for monadic Heyting algebras due to Bezhanishvili and Harding. An alternative proof-theoretic strategy for obtaining such axiomatization results is also developed for first-order substructural logics that have a cut-free sequent calculus and admit a certain interpolation property.</p></div> </div> </div> </li> </ul> </div> </div> </div> </div> <h4 class="journal-article-listing-type heading_12 margin-bottom">Review</h4> <div class="representation overview search" data-prod-id="6F1966780E332F5EB0C97ABA1C2617EF" data-mathjax-disabled="false"> <div class="row"> <div class="small-1 columns checklist-col"> <label class="product-listing-with-inputs-input"> <input type="checkbox" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"6F1966780E332F5EB0C97ABA1C2617EF","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}' name="productParts" style="display: inline-block" data-prod-id="6F1966780E332F5EB0C97ABA1C2617EF" data-prod-source="listing" data-prod-type="JOURNAL_ARTICLE" data-file-size="3290" data-has-entitlement="true" data-has-pdf="true" /> </label> </div> <div class="product-listing-with-inputs-content"> <div class="small-10 columns margin-bottom"> <ul class="details"> <li class="title"><h3><a class="part-link" href="/core/journals/bulletin-of-symbolic-logic/article/compactness-ofand-strong-axioms-of-determinacy-n-trang-structure-theory-ofand-its-applications-journal-of-symbolic-logic-vol-80-2015-no-1-pp-2955-n-trang-supercompactness-can-be-equiconsistent-with-measurability-notre-dame-journal-of-formal-logic-vol-62-2021-no-4-pp-593618-n-trang-and-t-wilson-determinacy-from-strong-compactness-of-annals-of-pure-and-applied-logic-vol-172-2021-no-6-article-no-102944-30pp-d-ikegami-and-n-trang-on-supercompactness-of-omega-1-advances-in-mathematical-logic-t-arai-m-kikuchi-s-kuroda-m-okada-t-yorioka-editors-springer-proceedings-mathematics-statistics-singapore-369-2021-pp-2745/6F1966780E332F5EB0C97ABA1C2617EF"> COMPACTNESS OFAND STRONG AXIOMS OF DETERMINACY - N. Trang, <span class="italic">Structure theory of</span><span class="italic">and its applications</span><span class="italic">.</span> <span class="italic"><span class="noitalic">Journal of Symbolic Logic</span></span>, vol. 80 (2015), no. 1, pp. 29–55. - N. Trang, <span class="italic">Supercompactness can be equiconsistent with measurability</span>. <span class="italic"><span class="bold"><span class="italic">Notre Dame Journal of Formal Logic</span></span></span>, vol. 62 (2021), no. 4, pp. 593–618. - N. Trang and T. Wilson, <span class="italic">Determinacy from strong compactness of</span> . <span class="italic"><span class="bold"><span class="italic">Annals of Pure and Applied Logic</span></span></span>, vol. 172 (2021), no. 6, Article no. 102944, 30pp. - D. Ikegami and N. Trang, <span class="italic">On supercompactness of</span> $\omega _1$, <span class="italic"><span class="bold"><span class="italic">Advances in Mathematical Logic</span></span></span> (T. Arai, M. Kikuchi, S. 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Yorioka, editors), Springer, Proceedings Mathematics & Statistics, Singapore, 369, 2021, pp. 27–45. </a></h3></li> <li></li> <li class="author"> <a href="/core/search?filters%5BauthorTerms%5D=Takehiko Gappo&eventCode=SE-AU" class="more-by-this-author ">Takehiko Gappo</a> </li> <li> <dl> <dt class="published dl-inline">Published online by Cambridge University Press: </dt> <dd> <span class="date"> 11 November 2024</span><span class="pages">, pp. 279-282</span> </dd> </dl> </li> <li> <ul class="access links"> <li class="type article">Article</li> <li> <ul class="access-modal"> <li class="status entitled"> <span class="sr-only">You have access </span> <span class="entitled status-text" title="You have access to this product" aria-hidden="true">Access</span> <span class="icon access-icon cursorDefault" title="You have access to this product" aria-hidden="true"> <svg id="access-6F1966780E332F5EB0C97ABA1C2617EF" xmlns="http://www.w3.org/2000/svg" viewBox="-701.2 242.7 206.9 328.3"> <circle class="st0" fill="#229619" cx="-597.9" cy="413.8" r="102.5"/> <path class="st1" fill="#fff" d="M-616 472.4l-49.6-49.6 26.5-26.5 23.1 23 59.3-59.3 26.5 26.5z"/> </svg> </span> </li> <li class="status open-access"> </li> </ul> </li> <li> <a href="/core/services/aop-cambridge-core/content/view/6F1966780E332F5EB0C97ABA1C2617EF/S1079898624000052a.pdf/compactness_ofand_strong_axioms_of_determinacy_n_trang_structure_theory_ofand_its_applications_journal_of_symbolic_logic_vol_80_2015_no_1_pp_2955_n_trang_supercompactness_can_be_equiconsistent_with_measurability_notre_dame_journal_of_formal_logic_vol_62_2021_no_4_pp_593618_n_trang_and_t_wilson_determinacy_from_strong_compactness_of_annals_of_pure_and_applied_logic_vol_172_2021_no_6_article_no_102944_30pp_d_ikegami_and_n_trang_on_supercompactness_of_omega_1_advances_in_mathematical_logic_t_arai_m_kikuchi_s_kuroda_m_okada_t_yorioka_editors_springer_proceedings_mathematics_statistics_singapore_369_2021_pp_2745.pdf" data-pdf-content-id="6F1966780E332F5EB0C97ABA1C2617EF" target="_blank" aria-label="Download PDF for COMPACTNESS OFAND STRONG AXIOMS OF DETERMINACY - N. 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Trang, Structure theory ofand its applications. Journal of Symbolic Logic, vol. 80 (2015), no. 1, pp. 29–55. - N. Trang, Supercompactness can be equiconsistent with measurability. Notre Dame Journal of Formal Logic, vol. 62 (2021), no. 4, pp. 593–618. - N. Trang and T. Wilson, Determinacy from strong compactness of . Annals of Pure and Applied Logic, vol. 172 (2021), no. 6, Article no. 102944, 30pp. - D. Ikegami and N. Trang, On supercompactness of $\omega _1$, Advances in Mathematical Logic (T. Arai, M. Kikuchi, S. Kuroda, M. Okada, T. 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Cambridge University Press, Cambridge, UK, 2023, 80 pp. </a></h3></li> <li></li> <li class="author"> <a href="/core/search?filters%5BauthorTerms%5D=Daniele Molinini&eventCode=SE-AU" class="more-by-this-author ">Daniele Molinini</a> </li> <li> <dl> <dt class="published dl-inline">Published online by Cambridge University Press: </dt> <dd> <span class="date"> 11 November 2024</span><span class="pages">, pp. 282-284</span> </dd> </dl> </li> <li> <ul class="access links"> <li class="type article">Article</li> <li> <ul class="access-modal"> <li class="status entitled"> <span class="sr-only">You have access </span> <span class="entitled status-text" title="You have access to this product" aria-hidden="true">Access</span> <span class="icon access-icon cursorDefault" title="You have access to this product" aria-hidden="true"> <svg id="access-1190E41A16D5BC0EFFE987F95CD90F93" xmlns="http://www.w3.org/2000/svg" viewBox="-701.2 242.7 206.9 328.3"> <circle class="st0" fill="#229619" cx="-597.9" cy="413.8" r="102.5"/> <path class="st1" fill="#fff" d="M-616 472.4l-49.6-49.6 26.5-26.5 23.1 23 59.3-59.3 26.5 26.5z"/> </svg> </span> </li> <li class="status open-access"> </li> </ul> </li> <li> <a href="/core/services/aop-cambridge-core/content/view/1190E41A16D5BC0EFFE987F95CD90F93/S1079898624000131a.pdf/christopher_pincock_mathematics_and_explanation_elements_in_the_philosophy_of_mathematics_cambridge_university_press_cambridge_uk_2023_80_pp.pdf" data-pdf-content-id="1190E41A16D5BC0EFFE987F95CD90F93" target="_blank" aria-label="Download PDF for Christopher Pincock. 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Classification problems in Descriptive Set Theory and regularity properties in Generalized Descriptive Set Theory </a></h3></li> <li> Part of: <ul class="partOfCollection"> <li><a href="/core/product/identifier/MSC_2010_LOW_DIMENSIONAL_TOPOLOGY/type/BESPOKE_COLLECTION" class="">Low-dimensional topology</a></li> <li><a href="/core/product/identifier/MSC_2010_PL_TOPOLOGY/type/BESPOKE_COLLECTION" class="">PL-topology</a></li> <li><a href="/core/product/identifier/MSC_2010_SET_THEORY/type/BESPOKE_COLLECTION" class="">Set theory</a></li> <li><a href="/core/product/identifier/MSC_2010_ORDERED_SETS/type/BESPOKE_COLLECTION" class="">Ordered sets</a></li> </ul> </li> <li class="author"> <a href="/core/search?filters%5BauthorTerms%5D=Martina Iannella&eventCode=SE-AU" class="more-by-this-author ">Martina Iannella</a> </li> <li> <dl> <dt class="published dl-inline">Published online by Cambridge University Press: </dt> <dd> <span class="date"> 11 November 2024</span><span class="pages">, pp. 285-286</span> </dd> </dl> </li> <li> <ul class="access links"> <li class="type article">Article</li> <li> <ul class="access-modal"> <li class="status entitled"> <span class="sr-only">You have access </span> <span class="entitled status-text" title="You have access to this product" aria-hidden="true">Access</span> <span class="icon access-icon cursorDefault" title="You have access to this product" aria-hidden="true"> <svg id="access-DEC8A5B4B4DE0BDA896BC5CC7272FB12" xmlns="http://www.w3.org/2000/svg" viewBox="-701.2 242.7 206.9 328.3"> <circle class="st0" fill="#229619" cx="-597.9" cy="413.8" r="102.5"/> <path class="st1" fill="#fff" d="M-616 472.4l-49.6-49.6 26.5-26.5 23.1 23 59.3-59.3 26.5 26.5z"/> </svg> </span> </li> <li class="status open-access"> </li> </ul> </li> <li> <a href="/core/services/aop-cambridge-core/content/view/DEC8A5B4B4DE0BDA896BC5CC7272FB12/S1079898624000106a.pdf/from_reallife_to_very_strong_axioms_classification_problems_in_descriptive_set_theory_and_regularity_properties_in_generalized_descriptive_set_theory.pdf" data-pdf-content-id="DEC8A5B4B4DE0BDA896BC5CC7272FB12" target="_blank" aria-label="Download PDF for From real-life to very strong axioms. 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Classification problems in Descriptive Set Theory and regularity properties in Generalized Descriptive Set Theory">Export citation </a></li> </ul> </li> <li class="accordion no-margin-top abstract" data-accordion> <div class="accordion-navigation"> <button onclick="toggleViewHide(this, 'abstractS1079898624000106')" class="button-no-transition transparent-no-border" aria-expanded="false" aria-label="View Abstract for From real-life to very strong axioms. Classification problems in Descriptive Set Theory and regularity properties in Generalized Descriptive Set Theory" data-cup-event='{"eventCode":"IN-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","usedIdentities":[],"authenticationMethods":[],"referringURL":"","eventType":"VADE","searchScope":"PRODUCT","productId":"DEC8A5B4B4DE0BDA896BC5CC7272FB12"}' data-cup-event-disable-on-hide="true" > <div class="icon open-drawer small"> <svg xmlns="http://www.w3.org/2000/svg" width="15" height="15" viewBox="0 0 17 17" class="is-opened"> <circle fill="#696969" cx="8.5" cy="8.5" r="7.5"/> <circle fill="#FFF" cx="8.5" cy="8.5" r="7"/> <path fill="#696969" d="M12.6 7.3H4.4c-.8 0-1.4.5-1.4 1.2s.6 1.2 1.4 1.2h8.3c.8 0 1.4-.5 1.4-1.2-.1-.7-.7-1.2-1.5-1.2z"/> </svg> <svg xmlns="http://www.w3.org/2000/svg" width="15" height="15" viewBox="0 0 15 15" class="is-closed"> <g transform="translate(1 1)" fill="none" fill-rule="evenodd"> <circle fill="#696969" cx="6.5" cy="6.5" r="6.5"/> <circle fill="#FFF" cx="6.5" cy="6.5" r="6"/> <path d="M9.855 5.518H7.482V3.145C7.482 2.491 7.072 2 6.5 2c-.573 0-.982.49-.982 1.145V5.518H3.145C2.491 5.518 2 5.928 2 6.5c0 .573.49.982 1.145.982H5.518v2.372c0 .655.41 1.146.982 1.146.573 0 .982-.49.982-1.145V7.482h2.372c.655 0 1.146-.41 1.146-.982 0-.573-.49-.982-1.145-.982z" fill="#696969"/> </g> </svg> <span class="toggle-text">View </span> <span class="excerpt-type">abstract</span> </div> </button> <div id="abstractS1079898624000106" class="content"> <div class="abstract" data-abstract-type="normal"><p>This thesis is divided into three parts, the first and second ones focused on combinatorics and classification problems on discrete and geometrical objects in the context of descriptive set theory, and the third one on generalized descriptive set theory at singular cardinals of countable cofinality.</p><p>Descriptive Set Theory (briefly: DST) is the study of definable subsets of Polish spaces, i.e., separable completely metrizable spaces. One of the major branches of DST is Borel reducibility, successfully used in the last 30 years to solve and compare many classification problems. One of our goals is the classification of knots, very familiar and tangible objects in everyday life, which also play an important role in modern mathematics. The study of knots and their properties is known as knot theory. Our plan is to gain insight into knots using discrete objects, such as linear and circular orders. This approach was already exploited in [6]. The first part of this work is therefore devoted to countable linear orders and the study of the quasi-order of convex embeddability and its induced equivalence relation. We obtain both combinatorial and descriptive set-theoretic results. We also expand our research to the case of circular orders.</p><p>Another objective of this first part is to extend the notion of convex embeddability on countable linear orders. We provide a family of quasi-orders of which embeddability is a particular case as well. We study these quasi-orders from a combinatorial point of view and analyse their complexity with respect to Borel reducibility. Furthermore, we extend the analysis of these quasi-orders to the set of uncountable linear orders.</p><p>The second part of the project deals with classification problems on knots and <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000106:S1079898624000106_inline2.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$3$</span></span></span></span>-manifolds. The goal here is to apply the results obtained in the first part to the study of proper arcs and knots, establishing lower bounds for the complexity of some natural relations between these geometrical objects. We also obtain some combinatorial results which are particularly interesting when we restrict to the set of wild proper arcs and wild knots, classes which haven’t received much attention so far. These parts are included in the two preprints [4, 5] in collaboration with my supervisor Alberto Marcone, Luca Motto Ros (University of Torino), and Vadim Weinstein (University of Oulu).</p><p>The second part of this work also includes the classification of non-compact <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000106:S1079898624000106_inline3.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$3$</span></span></span></span>-manifolds up to homeomorphism (the case of compact <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000106:S1079898624000106_inline4.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$3$</span></span></span></span>-manifolds has already been solved: indeed, there are only countably many <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000106:S1079898624000106_inline5.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$3$</span></span></span></span>-manifolds up to homeomorphism; see [7]), and that of Cantor sets of <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000106:S1079898624000106_inline6.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathbb {R}^3$</span></span></span></span> up to conjugation (answering to Question 5.5 of [3]). Here we resort to algebraic tools. Stone duality gives a neat way to go back-and-forth between totally disconnected Polish spaces and countable Boolean algebras (see [1]). The main ingredient is the Stone space of all ultrafilters on a Boolean algebra. In this work we introduce a weaker concept which we call “blurry filter”. Using blurry filters instead of ultrafilters enables one to extend the class of spaces under consideration beyond totally disconnected. As an application of this method, we show that both homeomorphism on non-compact <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000106:S1079898624000106_inline7.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$3$</span></span></span></span>-manifolds and conjugation of Cantor sets in <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000106:S1079898624000106_inline8.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathbb {R}^3$</span></span></span></span> are completely classifiable by countable structures. These results are part of an upcoming paper in collaboration with Vadim Weinstein.</p><p>The last part of this thesis concerns the natural generalization of descriptive set theory that occurs when countable is replaced by uncountable, called Generalized Descriptive Set Theory (briefly: GDST). In particular, we focus on the case of GDST for a singular cardinal <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000106:S1079898624000106_inline9.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\kappa $</span></span></span></span> of countable cofinality. The goal here is to study the generalizations of the Perfect Set Property and the Baire Property to subsets of spaces of higher cardinalities, like the power set <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000106:S1079898624000106_inline10.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathcal {P}(\kappa )$</span></span></span></span>. We consider the question under which large cardinal hypotheses classes of definable subsets of these spaces possess such regularity properties, focusing on rank-into-rank axioms, like I2, and classes of sets definable by <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000106:S1079898624000106_inline11.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\Sigma _1$</span></span></span></span>-formulas with parameters from various collections of sets. The obtained results are included in [2], a joint work with my co-supervisor Vincenzo Dimonte and Philipp Lücke (University of Hamburg).</p><p>Abstract prepared by Martina Iannella</p><p><span class='italic'>E-mail</span>: <a href='mailto:martina.iannella@tuwien.ac.at'>martina.iannella@tuwien.ac.at</a>.</p><p><span class='italic'>URL</span>: <a href='https://air.uniud.it/handle/11390/1262824'>https://air.uniud.it/handle/11390/1262824</a>.</p></div> </div> </div> </li> </ul> </div> </div> </div> </div> <h4 class="journal-article-listing-type heading_12 margin-bottom">Meeting Report</h4> <div class="representation overview search" data-prod-id="3C9018E702BB8DBA19FAB9E43B1E97BC" data-mathjax-disabled="false"> <div class="row"> <div class="small-1 columns checklist-col"> <label class="product-listing-with-inputs-input"> <input type="checkbox" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"3C9018E702BB8DBA19FAB9E43B1E97BC","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}' name="productParts" style="display: inline-block" data-prod-id="3C9018E702BB8DBA19FAB9E43B1E97BC" data-prod-source="listing" data-prod-type="JOURNAL_ARTICLE" data-file-size="153854" data-has-entitlement="true" data-has-pdf="true" /> </label> </div> <div class="product-listing-with-inputs-content"> <div class="small-10 columns margin-bottom"> <ul class="details"> <li class="title"><h3><a class="part-link" href="/core/journals/bulletin-of-symbolic-logic/article/2023-meeting-of-the-australasian-association-for-logic-university-of-queensland-brisbane-australia-910-november-2023/3C9018E702BB8DBA19FAB9E43B1E97BC"> 2023 MEETING OF THE AUSTRALASIAN ASSOCIATION FOR LOGIC University of Queensland Brisbane, Australia 9-10 November, 2023 </a></h3></li> <li></li> <li class="author"> </li> <li> <dl> <dt class="published dl-inline">Published online by Cambridge University Press: </dt> <dd> <span class="date"> 02 April 2024</span><span class="pages">, pp. 152-165</span> </dd> </dl> </li> <li> <ul class="access links"> <li class="type article">Article</li> <li> <ul class="access-modal"> <li class="status entitled"> <span class="sr-only">You have access </span> <span class="entitled status-text" title="You have access to this product" aria-hidden="true">Access</span> <span class="icon access-icon cursorDefault" title="You have access to this product" aria-hidden="true"> <svg id="access-3C9018E702BB8DBA19FAB9E43B1E97BC" xmlns="http://www.w3.org/2000/svg" viewBox="-701.2 242.7 206.9 328.3"> <circle class="st0" fill="#229619" cx="-597.9" cy="413.8" r="102.5"/> <path class="st1" fill="#fff" d="M-616 472.4l-49.6-49.6 26.5-26.5 23.1 23 59.3-59.3 26.5 26.5z"/> </svg> </span> </li> <li class="status open-access"> </li> </ul> </li> <li> <a href="/core/services/aop-cambridge-core/content/view/3C9018E702BB8DBA19FAB9E43B1E97BC/S1079898624000167a.pdf/2023_meeting_of_the_australasian_association_for_logic_university_of_queensland_brisbane_australia_910_november_2023.pdf" data-pdf-content-id="3C9018E702BB8DBA19FAB9E43B1E97BC" target="_blank" aria-label="Download PDF for 2023 MEETING OF THE AUSTRALASIAN ASSOCIATION FOR LOGIC University of Queensland Brisbane, Australia 9-10 November, 2023 [Opens in a new window]" > <span class="icon icn pdf"> <span class="download-label">PDF </span> </span> </a> </li> <li> <a href="/core/product/3C9018E702BB8DBA19FAB9E43B1E97BC/core-reader" target="_blank" aria-label="View HTML full text for 2023 MEETING OF THE AUSTRALASIAN ASSOCIATION FOR LOGIC University of Queensland Brisbane, Australia 9-10 November, 2023 [Opens in a new window]" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"3C9018E702BB8DBA19FAB9E43B1E97BC","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}'> <span class="icon icn html wide"> <span class="download-label">HTML</span> </span> </a> </li> <li> <script> $(document).ready(function () { /** * Disable tooltip aria label attributes, so the screen reader can correctly read out the core * reader tab navigation. * Set a short delay, so we can be sure the tooltips have rendered, before updating the DOM. */ setTimeout(function () { var $downloadLinks = $('.download-types li a'); 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Our study is not limited to these two areas: indeed, in both cases, the results we obtain are tightly connected to ideas and tools coming from different areas of mathematical logic, including for example <span class='italic'>descriptive set theory</span> and <span class='italic'>reverse mathematics</span>.</p><p>After giving the necessary preliminaries, we first study the uniform computational strength of the <span class='italic'>Cantor</span>–<span class='italic'>Bendixson theorem</span> in the <span class='italic'>Weihrauch lattice</span>. This work falls into the program connecting reverse mathematics and computable analysis via the framework of Weihrauch reducibility. We concentrate on problems related to perfect subsets of Polish spaces, studying the perfect set theorem, the Cantor–Bendixson theorem, and various problems arising from them. As far as we know, this is the first systematic study of problems at the level of <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000118:S1079898624000118_inline1.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathbf {\Pi }^1_1\mathsf {-CA}_0$</span></span></span></span> in the Weihrauch lattice. We show that the strength of some of the problems we study depends on the topological properties of the Polish space under consideration, while others have the same strength once the space is rich enough.</p><p>We continue considering problems related to <span class='italic'>(induced) subgraphs</span>. We provide results on the <span class='italic'>(effective) Wadge complexity</span> of sets of graphs, that are also used to determine the Weihrauch degree of certain decision problems. The decision problems we consider are defined for a fixed graph <span class='italic'>G</span>, and they take as input a graph <span class='italic'>H</span>, answering whether <span class='italic'>G</span> is an (induced) subgraph of <span class='italic'>H</span>: we also consider the opposite problem (i.e., answering whether <span class='italic'>H</span> is an induced subgraph of <span class='italic'>G</span>). We conclude this part on (induced) subgraphs considering the Weihrauch degree of “search problems.”These problems are defined for a fixed graph <span class='italic'>G</span>, and they take as input a graph <span class='italic'>H</span> such that <span class='italic'>G</span> is an (induced) subgraph <span class='italic'>H</span>: the output is a copy of <span class='italic'>G</span> in <span class='italic'>H</span>. In both cases, we highlight differences and analogies between the subgraph and the induced subgraph relation.</p><p>We then move our attention to <span class='italic'>algorithmic learning theory</span>, and we present the framework we use to study the learnability of families of algebraic structures: here, given a countable family of pairwise nonisomorphic structures <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000118:S1079898624000118_inline2.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathfrak {K}$</span></span></span></span>, a learner receives larger and larger pieces of an arbitrary copy of a structure in <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000118:S1079898624000118_inline3.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathfrak {K}$</span></span></span></span> and, at each stage, is required to output a conjecture about the isomorphism type of such a structure. We say that <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000118:S1079898624000118_inline4.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathfrak {K}$</span></span></span></span> is <span class='italic'>learnable</span> if there exists a learner which eventually stabilizes to a correct guess. The framework was lacking a method for comparing the complexity of nonlearnable families, and so we propose a solution to this problem using tools coming from invariant descriptive set theory. To do so, we first prove that a family of structures is learnable if and only if its learning domain is continuously reducible to the relation <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000118:S1079898624000118_inline5.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$E_0$</span></span></span></span> of eventual agreement on infinite binary sequences and then, replacing <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000118:S1079898624000118_inline6.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$E_0$</span></span></span></span> with Borel equivalence relations of higher complexity, we obtain a new hierarchy of learning problems. This leads to the notion of <span class='italic'>E</span>-<span class='italic'>learnability</span>, where a family of structures <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000118:S1079898624000118_inline7.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathfrak {K}$</span></span></span></span> is <span class='italic'>E</span>-learnable, for a Borel equivalence relation <span class='italic'>E</span>, if there is a continuous reduction from the isomorphism relation associated with <span class='inlineFormula'><span class='alternatives'><img class='inline-graphic mathjax-alternative mathjax-alt-graphic mathjax-off' data-mimesubtype='png' data-type='' src='//static.cambridge.org/binary/version/id/urn:cambridge.org:id:binary:20241108100725073-0860:S1079898624000118:S1079898624000118_inline8.png'><span class='mathjax-tex-wrapper' data-mathjax-type='texmath'><span class='tex-math mathjax-tex-math mathjax-on'>$\mathfrak {K}$</span></span></span></span> to <span class='italic'>E</span>. It is then natural to ask how the notion of <span class='italic'>E</span>-learnability interacts with “classical” learning paradigms.</p><p>Finally, we study the number of <span class='italic'>mind changes</span> that a learner needs to learn a given family, both from a topological and a combinatorial point of view, and we consider how the complexity of a learner (in terms of <span class='italic'>Turing reducibility</span>) affects the number of mind changes for learning a given family.</p><p>Abstract prepared by Vittorio Cipriani.</p><p><span class='italic'>E-mail</span>: <a href='mailto:vittorio.cipriani17@gmail.com'>vittorio.cipriani17@gmail.com</a>.</p><p><span class='italic'>URL</span>: <a href='https://air.uniud.it/handle/11390/1252226'>https://air.uniud.it/handle/11390/1252226</a>.</p></div> </div> </div> </li> </ul> </div> </div> </div> </div> <h4 class="journal-article-listing-type heading_12 margin-bottom">Meeting Report</h4> <div class="representation overview search" data-prod-id="71FA9B938EBC74EB707E2011A3910EC3" data-mathjax-disabled="false"> <div class="row"> <div class="small-1 columns checklist-col"> <label class="product-listing-with-inputs-input"> <input type="checkbox" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"71FA9B938EBC74EB707E2011A3910EC3","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}' name="productParts" style="display: inline-block" data-prod-id="71FA9B938EBC74EB707E2011A3910EC3" data-prod-source="listing" data-prod-type="JOURNAL_ARTICLE" data-file-size="62244" data-has-entitlement="true" data-has-pdf="true" /> </label> </div> <div class="product-listing-with-inputs-content"> <div class="small-10 columns margin-bottom"> <ul class="details"> <li class="title"><h3><a class="part-link" href="/core/journals/bulletin-of-symbolic-logic/article/2024-spring-meeting-of-the-association-for-symbolic-logic-the-sheraton-times-square-new-york-ny-eastern-apa-meeting-january-1518-2024/71FA9B938EBC74EB707E2011A3910EC3"> 2024 SPRING MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC The Sheraton Times Square, New York, NY Eastern APA Meeting January 15–18, 2024 </a></h3></li> <li></li> <li class="author"> </li> <li> <dl> <dt class="published dl-inline">Published online by Cambridge University Press: </dt> <dd> <span class="date"> 02 April 2024</span><span class="pages">, pp. 172-175</span> </dd> </dl> </li> <li> <ul class="access links"> <li class="type article">Article</li> <li> <ul class="access-modal"> <li class="status entitled"> <span class="sr-only">You have access </span> <span class="entitled status-text" title="You have access to this product" aria-hidden="true">Access</span> <span class="icon access-icon cursorDefault" title="You have access to this product" aria-hidden="true"> <svg id="access-71FA9B938EBC74EB707E2011A3910EC3" xmlns="http://www.w3.org/2000/svg" viewBox="-701.2 242.7 206.9 328.3"> <circle class="st0" fill="#229619" cx="-597.9" cy="413.8" r="102.5"/> <path class="st1" fill="#fff" d="M-616 472.4l-49.6-49.6 26.5-26.5 23.1 23 59.3-59.3 26.5 26.5z"/> </svg> </span> </li> <li class="status open-access"> </li> </ul> </li> <li> <a href="/core/services/aop-cambridge-core/content/view/71FA9B938EBC74EB707E2011A3910EC3/S1079898624000192a.pdf/2024_spring_meeting_of_the_association_for_symbolic_logic_the_sheraton_times_square_new_york_ny_eastern_apa_meeting_january_1518_2024.pdf" data-pdf-content-id="71FA9B938EBC74EB707E2011A3910EC3" target="_blank" aria-label="Download PDF for 2024 SPRING MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC The Sheraton Times Square, New York, NY Eastern APA Meeting January 15–18, 2024 [Opens in a new window]" > <span class="icon icn pdf"> <span class="download-label">PDF </span> </span> </a> </li> <li> <a href="/core/product/71FA9B938EBC74EB707E2011A3910EC3/core-reader" target="_blank" aria-label="View HTML full text for 2024 SPRING MEETING OF THE ASSOCIATION FOR SYMBOLIC LOGIC The Sheraton Times Square, New York, NY Eastern APA Meeting January 15–18, 2024 [Opens in a new window]" data-cup-event-payload='{"eventCode":"DL-AR","eventContext":"/core/journals/bulletin-of-symbolic-logic/volume/9EBD0A609ABA5800D0BC9A55948ECAE8","productId":"71FA9B938EBC74EB707E2011A3910EC3","referringURL":"","authenticationMethods":[],"usedIdentities":["0"],"searchScope":"PRODUCT"}'> <span class="icon icn html wide"> <span class="download-label">HTML</span> </span> </a> </li> <li> <script> $(document).ready(function () { /** * Disable tooltip aria label attributes, so the screen reader can correctly read out the core * reader tab navigation. * Set a short delay, so we can be sure the tooltips have rendered, before updating the DOM. */ setTimeout(function () { var $downloadLinks = $('.download-types li a'); 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Most of these systems crucially rely on the use of intensional methods, treating sets with potentially high quantifier complexity in the defining matrix via characteristic functions and axioms that describe only their properties and do not completely characterize the elements of the sets.</p><p>The applicability of all of these metatheorems is then substantiated by a range of case studies for the respective areas which in particular also highlight the naturalness of the use of intensional methods in the design of the corresponding systems.</p><p>The first new area covered thereby is the theory of nonlinear semigroups induced by corresponding evolution equations for accretive operators. In that context, we present (besides an initial foray into the area from 2015) essentially the first applications of proof mining to the theory of partial differential equations. Concretely, we provide quantitative versions of four central results on the asymptotic behavior of solutions to such equations.</p><p>The second new area unlocked in this thesis is that of the continuous dual of a Banach space and its norm (which are also approached via intensional methods). This in particular relies on a proof-theoretically tame treatment of suprema over (certain) bounded sets in this intensional context which is further exploited later on. These systems, which give access to this until now untreated fundamental notion from functional analysis, are then used to provide further substantial extensions to treat various notions from convex analysis like the Fréchet derivative of a convex function, Fenchel conjugates, Bregman distances, and monotone operators on Banach spaces in the sense of Browder.</p><p>These systems are then utilized to provide applications in the context of Picard- and Halpern-style iterations of so-called Bregman strongly nonexpansive mappings where we provide both new quantitative and qualitative results.</p><p>Lastly, we discuss the key notion of extensionality of a set-valued operator and its relation to set-theoretic maximality principles in more depth (which was already singled out—to some degree—in previous work). We thereby exhibit an issue arising with treating full extensionality in the context of these intensional approaches to set-valued operators and present useful fragments of the full extensionality statement where these issues are avoided.</p><p>Corresponding to these fragments, we discuss a range of uniform continuity statements for set-valued operators beyond the usual notion involving the Hausdorff-metric. In particular, in that context, we utilize the previous tame treatment of suprema over bounded sets to also provide the first proof-theoretic treatment of that Hausdorff-metric in the context of systems for proof mining.</p><p>The applicability of this treatment of the Hausdorff-metric is then in particular substantiated by a last case study where we provide quantitative information for a Mann-type iteration of set-valued mappings which are nonexpansive w.r.t. the Hausdorff-metric.</p><p>The abstract was taken directly from the thesis.</p><p>Abstract prepared by Nicholas Pischke, Technische Universität Darmstadt, Darmstadt, Germany</p><p><span class='italic'>E-mail</span>: <a href='mailto:pischke@mathematik.tu-darmstadt.de'>pischke@mathematik.tu-darmstadt.de</a>.</p><p><span class='italic'>URL</span>: <a href='https://doi.org/10.26083/tuprints-00026584'>https://doi.org/10.26083/tuprints-00026584</a>.</p></div> </div> </div> </li> </ul> </div> </div> </div> </div> <h4 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Bulletin of Symbolic Logic: Volume 30 - | Cambridge Core