{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,10,29]],"date-time":"2026-10-29T18:32:19Z","timestamp":1793298739283,"version":"4.5.0"},"reference-count":21,"publisher":"National Academy of Sciences","issue":"44","license":[{"start":{"date-parts":[[2026,10,29]],"date-time":"2026-10-29T00:00:00Z","timestamp":1793232000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by-nc-nd\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["10.55776\/I6087"],"award-info":[{"award-number":["10.55776\/I6087"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["10.55776\/Y1498"],"award-info":[{"award-number":["10.55776\/Y1498"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["www.pnas.org"],"crossmark-restriction":true},"short-container-title":["Proc. Natl. Acad. Sci. U.S.A."],"published-print":{"date-parts":[[2026,11,3]]},"abstract":"<jats:p>\n                    If\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mi>\u03b1<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mi>\u03b2<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    are cardinals, then the cardinal exponentiation\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mrow>\n                          <mml:msup>\n                            <mml:mrow\/>\n                            <mml:mi>\u03b1<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:mi>\u03b2<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    is the set of all functions from\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mi>\u03b1<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    into\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mi>\u03b2<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    . The ABCD hypothesis is the statement that for all infinite cardinals\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mrow>\n                          <mml:mi>\u03b1<\/mml:mi>\n                          <mml:mo>\u2264<\/mml:mo>\n                          <mml:mi>\u03b2<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mrow>\n                          <mml:mi>\u03b3<\/mml:mi>\n                          <mml:mo>\u2264<\/mml:mo>\n                          <mml:mi>\u03b4<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    ,\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mrow>\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>\u03b1<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mi>\u03b2<\/mml:mi>\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                              <mml:mo>\u2264<\/mml:mo>\n                              <mml:mo stretchy=\"false\">|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>\u03b3<\/mml:mi>\n                          <\/mml:msup>\n                          <mml:mrow>\n                            <mml:mi>\u03b4<\/mml:mi>\n                            <mml:mo stretchy=\"false\">|<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    if and only if\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mrow>\n                          <mml:mi>\u03b1<\/mml:mi>\n                          <mml:mo>\u2264<\/mml:mo>\n                          <mml:mi>\u03b3<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mrow>\n                          <mml:mi>\u03b2<\/mml:mi>\n                          <mml:mo>\u2264<\/mml:mo>\n                          <mml:mi>\u03b4<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    . The Axiom of Determinacy, AD, states that every two-player infinite game on\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mi>\u03c9<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    has a winning strategy for one of the two players. Let\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"normal\">\u0398<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    be the supremum of the ordinals onto which\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    surjects. This paper will show that if AD holds and\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mrow>\n                          <mml:mi>V<\/mml:mi>\n                          <mml:mo>=<\/mml:mo>\n                          <mml:mi>L<\/mml:mi>\n                          <mml:mo stretchy=\"false\">(<\/mml:mo>\n                          <mml:mi mathvariant=\"double-struck\">R<\/mml:mi>\n                          <mml:mo stretchy=\"false\">)<\/mml:mo>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    , then the ABCD hypothesis holds below\n                    <jats:inline-formula>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" display=\"inline\" overflow=\"scroll\">\n                        <mml:mi mathvariant=\"normal\">\u0398<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:inline-formula>\n                    .\n                  <\/jats:p>","DOI":"10.1073\/pnas.2621223123","type":"journal-article","created":{"date-parts":[[2026,10,29]],"date-time":"2026-10-29T17:37:18Z","timestamp":1793295438000},"update-policy":"https:\/\/doi.org\/10.1073\/pnas.cm10313","source":"Crossref","is-referenced-by-count":0,"title":["The ABCD hypothesis"],"prefix":"10.1073","volume":"123","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0661-1764","authenticated-orcid":false,"given":"William","family":"Chan","sequence":"first","affiliation":[{"id":[{"id":"https:\/\/ror.org\/04d836q62","id-type":"ROR","asserted-by":"publisher"}],"name":"Institut f\u00fcr diskrete Mathematik und Geometrie, Technische Universit\u00e4t Wien","place":["Vienna, Austria"]}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"341","published-online":{"date-parts":[[2026,10,29]]},"reference":[{"key":"e_1_3_4_1_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.24.12.556"},{"key":"e_1_3_4_2_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.25.4.220"},{"key":"e_1_3_4_3_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.50.6.1143"},{"key":"e_1_3_4_4_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.51.1.105"},{"key":"e_1_3_4_5_2","doi-asserted-by":"publisher","DOI":"10.1073\/pnas.85.18.6587"},{"key":"e_1_3_4_6_2","doi-asserted-by":"publisher","DOI":"10.2307\/1971035"},{"key":"e_1_3_4_7_2","first-page":"287","article-title":"Measurable cardinals and analytic games","volume":"66","author":"Martin D. 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