{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,4,25]],"date-time":"2025-04-25T04:24:23Z","timestamp":1745555063729,"version":"3.40.4"},"reference-count":15,"publisher":"Springer Science and Business Media LLC","issue":"3-4","license":[{"start":{"date-parts":[[2024,11,10]],"date-time":"2024-11-10T00:00:00Z","timestamp":1731196800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,11,10]],"date-time":"2024-11-10T00:00:00Z","timestamp":1731196800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"HUN-REN Alfr\u00e9d R\u00e9nyi Institute of Mathematics"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Arch. Math. Logic"],"published-print":{"date-parts":[[2025,5]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>We investigate whether classical combinatorial theorems are provable in ZF. Some statements are not provable in ZF, but they are equivalent within ZF. For example, the following statements (i)\u2013(iii) are equivalent: <jats:list list-type=\"order\">\n              <jats:list-item>\n                <jats:p>\n                  <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$$cf({\\omega }_1)={\\omega }_1$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mi>c<\/mml:mi>\n                          <mml:mi>f<\/mml:mi>\n                          <mml:mrow>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\u03c9<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:mo>=<\/mml:mo>\n                          <mml:msub>\n                            <mml:mi>\u03c9<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula>,<\/jats:p>\n              <\/jats:list-item>\n              <jats:list-item>\n                <jats:p>\n                  <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$${\\omega }_1\\rightarrow ({\\omega }_1,{\\omega }+1)^2$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:msub>\n                            <mml:mi>\u03c9<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:mo>\u2192<\/mml:mo>\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mo>(<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>\u03c9<\/mml:mi>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:msub>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>\u03c9<\/mml:mi>\n                              <mml:mo>+<\/mml:mo>\n                              <mml:mn>1<\/mml:mn>\n                              <mml:mo>)<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula>,<\/jats:p>\n              <\/jats:list-item>\n              <jats:list-item>\n                <jats:p>any family <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$$\\mathcal {A}\\subset [{On}]^{&lt;{\\omega }}$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:mo>\u2282<\/mml:mo>\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mo>[<\/mml:mo>\n                              <mml:mrow>\n                                <mml:mi>On<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mo>]<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>&lt;<\/mml:mo>\n                              <mml:mi>\u03c9<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula> of size <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$${\\omega }_1$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:msub>\n                          <mml:mi>\u03c9<\/mml:mi>\n                          <mml:mn>1<\/mml:mn>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula> contains a <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$$\\Delta $$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mi>\u0394<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula>-system of size <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$${\\omega }_1$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:msub>\n                          <mml:mi>\u03c9<\/mml:mi>\n                          <mml:mn>1<\/mml:mn>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula>.<\/jats:p>\n              <\/jats:list-item>\n            <\/jats:list> Some classical results cannot be proven in ZF alone; however, we can establish weaker versions of these statements within the framework of ZF, such as <jats:list list-type=\"order\">\n              <jats:list-item>\n                <jats:p>\n                  <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$${{\\omega }_2}\\rightarrow ({\\omega }_1,{\\omega }+1)$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:msub>\n                            <mml:mi>\u03c9<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:mo>\u2192<\/mml:mo>\n                          <mml:mrow>\n                            <mml:mo>(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>\u03c9<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>\u03c9<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                            <mml:mo>)<\/mml:mo>\n                          <\/mml:mrow>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula>,<\/jats:p>\n              <\/jats:list-item>\n              <jats:list-item>\n                <jats:p>any family <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$$\\mathcal {A}\\subset [{On}]^{&lt;{\\omega }}$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:mo>\u2282<\/mml:mo>\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mo>[<\/mml:mo>\n                              <mml:mrow>\n                                <mml:mi>On<\/mml:mi>\n                              <\/mml:mrow>\n                              <mml:mo>]<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>&lt;<\/mml:mo>\n                              <mml:mi>\u03c9<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula> of size <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$${\\omega }_2$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:msub>\n                          <mml:mi>\u03c9<\/mml:mi>\n                          <mml:mn>2<\/mml:mn>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula> contains a <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$$\\Delta $$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mi>\u0394<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula>-system of size <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$${\\omega }_1$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:msub>\n                          <mml:mi>\u03c9<\/mml:mi>\n                          <mml:mn>1<\/mml:mn>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula>.<\/jats:p>\n              <\/jats:list-item>\n            <\/jats:list> Some statements can be proven in ZF using purely combinatorial arguments, such as: <jats:list list-type=\"order\">\n              <jats:list-item>\n                <jats:p>given a set mapping <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$$F:{\\omega }_1\\rightarrow {[{\\omega }_1]}^{&lt;{\\omega }}$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mi>F<\/mml:mi>\n                          <mml:mo>:<\/mml:mo>\n                          <mml:msub>\n                            <mml:mi>\u03c9<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:mo>\u2192<\/mml:mo>\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mo>[<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>\u03c9<\/mml:mi>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:msub>\n                              <mml:mo>]<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>&lt;<\/mml:mo>\n                              <mml:mi>\u03c9<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula>, the set <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$${\\omega }_1$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:msub>\n                          <mml:mi>\u03c9<\/mml:mi>\n                          <mml:mn>1<\/mml:mn>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula> has a partition into <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$${\\omega }$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mi>\u03c9<\/mml:mi>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula>-many <jats:italic>F<\/jats:italic>-free sets.<\/jats:p>\n              <\/jats:list-item>\n            <\/jats:list> Other statements can be proven in ZF by employing certain methods of absoluteness, for example: <jats:list list-type=\"order\">\n              <jats:list-item>\n                <jats:p>given a set mapping <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$$F:{\\omega }_1\\rightarrow {[{\\omega }_1]}^{&lt;{\\omega }}$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mi>F<\/mml:mi>\n                          <mml:mo>:<\/mml:mo>\n                          <mml:msub>\n                            <mml:mi>\u03c9<\/mml:mi>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:msub>\n                          <mml:mo>\u2192<\/mml:mo>\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mo>[<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>\u03c9<\/mml:mi>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:msub>\n                              <mml:mo>]<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>&lt;<\/mml:mo>\n                              <mml:mi>\u03c9<\/mml:mi>\n                            <\/mml:mrow>\n                          <\/mml:msup>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula>, there is an <jats:italic>F<\/jats:italic>-free set of size <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$${\\omega }_1$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:msub>\n                          <mml:mi>\u03c9<\/mml:mi>\n                          <mml:mn>1<\/mml:mn>\n                        <\/mml:msub>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula>,<\/jats:p>\n              <\/jats:list-item>\n              <jats:list-item>\n                <jats:p>for each <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$$n\\in {\\omega }$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mi>n<\/mml:mi>\n                          <mml:mo>\u2208<\/mml:mo>\n                          <mml:mi>\u03c9<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula>, every family <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$$\\mathcal {A}\\subset {[{\\omega }_1]}^{{\\omega }}$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:mo>\u2282<\/mml:mo>\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mo>[<\/mml:mo>\n                              <mml:msub>\n                                <mml:mi>\u03c9<\/mml:mi>\n                                <mml:mn>1<\/mml:mn>\n                              <\/mml:msub>\n                              <mml:mo>]<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mi>\u03c9<\/mml:mi>\n                          <\/mml:msup>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula> with <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$$|A\\cap B|\\le n$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mo>|<\/mml:mo>\n                          <mml:mi>A<\/mml:mi>\n                          <mml:mo>\u2229<\/mml:mo>\n                          <mml:mi>B<\/mml:mi>\n                          <mml:mo>|<\/mml:mo>\n                          <mml:mo>\u2264<\/mml:mo>\n                          <mml:mi>n<\/mml:mi>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula> for <jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$$\\{A,B\\}\\in {[\\mathcal {A}]}^{2}$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mrow>\n                          <mml:mrow>\n                            <mml:mo>{<\/mml:mo>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:mi>B<\/mml:mi>\n                            <mml:mo>}<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:mo>\u2208<\/mml:mo>\n                          <mml:msup>\n                            <mml:mrow>\n                              <mml:mo>[<\/mml:mo>\n                              <mml:mi>A<\/mml:mi>\n                              <mml:mo>]<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msup>\n                        <\/mml:mrow>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula> has property <jats:italic>B<\/jats:italic>.<\/jats:p>\n              <\/jats:list-item>\n            <\/jats:list> In contrast to statement (5), we show that the following ZFC theorem of Komj\u00e1th is not provable from ZF + <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$cf({\\omega }_1)={\\omega }_1$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>c<\/mml:mi>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>\u03c9<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>\u03c9<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>: <jats:def-list>\n              <jats:def-item>\n                <jats:term>(6<jats:inline-formula>\n                    <jats:alternatives>\n                      <jats:tex-math>$$ ^*$$<\/jats:tex-math>\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                        <mml:mmultiscripts>\n                          <mml:mrow\/>\n                          <mml:mrow\/>\n                          <mml:mo>\u2217<\/mml:mo>\n                        <\/mml:mmultiscripts>\n                      <\/mml:math>\n                    <\/jats:alternatives>\n                  <\/jats:inline-formula>)<\/jats:term>\n                <jats:def>\n                  <jats:p>every family <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\mathcal {A}\\subset {[{\\omega }_1]}^{{\\omega }}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo>\u2282<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mo>[<\/mml:mo>\n                                <mml:msub>\n                                  <mml:mi>\u03c9<\/mml:mi>\n                                  <mml:mn>1<\/mml:mn>\n                                <\/mml:msub>\n                                <mml:mo>]<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mi>\u03c9<\/mml:mi>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula> with <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$|A\\cap B|\\le 1$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mo>|<\/mml:mo>\n                            <mml:mi>A<\/mml:mi>\n                            <mml:mo>\u2229<\/mml:mo>\n                            <mml:mi>B<\/mml:mi>\n                            <mml:mo>|<\/mml:mo>\n                            <mml:mo>\u2264<\/mml:mo>\n                            <mml:mn>1<\/mml:mn>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula> for <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$$\\{A,B\\}\\in {[\\mathcal {A}]}^{2}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>{<\/mml:mo>\n                              <mml:mi>A<\/mml:mi>\n                              <mml:mo>,<\/mml:mo>\n                              <mml:mi>B<\/mml:mi>\n                              <mml:mo>}<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>\u2208<\/mml:mo>\n                            <mml:msup>\n                              <mml:mrow>\n                                <mml:mo>[<\/mml:mo>\n                                <mml:mi>A<\/mml:mi>\n                                <mml:mo>]<\/mml:mo>\n                              <\/mml:mrow>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msup>\n                          <\/mml:mrow>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula> is <jats:italic>essentially disjoint<\/jats:italic>.<\/jats:p>\n                <\/jats:def>\n              <\/jats:def-item>\n            <\/jats:def-list>\n          <\/jats:p>\n          <jats:p> A function <jats:italic>f<\/jats:italic> is a <jats:italic>uniform denumeration on<\/jats:italic>\n            <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\omega }_1$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>\u03c9<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> iff <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\text {dom}}(f)={\\omega }_1$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mtext>dom<\/mml:mtext>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mi>f<\/mml:mi>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>\u03c9<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, and for every <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$1\\le {\\alpha }&lt;{\\omega }_1$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mn>1<\/mml:mn>\n                    <mml:mo>\u2264<\/mml:mo>\n                    <mml:mi>\u03b1<\/mml:mi>\n                    <mml:mo>&lt;<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>\u03c9<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>, <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$f({\\alpha })$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>(<\/mml:mo>\n                    <mml:mi>\u03b1<\/mml:mi>\n                    <mml:mo>)<\/mml:mo>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> is a function from <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\omega }$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03c9<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> onto <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\alpha }$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mi>\u03b1<\/mml:mi>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. It is easy to see that the existence of a uniform denumeration of <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$${\\omega }_1$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msub>\n                    <mml:mi>\u03c9<\/mml:mi>\n                    <mml:mn>1<\/mml:mn>\n                  <\/mml:msub>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula> implies <jats:inline-formula>\n              <jats:alternatives>\n                <jats:tex-math>$$cf({\\omega }_1)={\\omega }_1$$<\/jats:tex-math>\n                <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>c<\/mml:mi>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:msub>\n                        <mml:mi>\u03c9<\/mml:mi>\n                        <mml:mn>1<\/mml:mn>\n                      <\/mml:msub>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                    <mml:mo>=<\/mml:mo>\n                    <mml:msub>\n                      <mml:mi>\u03c9<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msub>\n                  <\/mml:mrow>\n                <\/mml:math>\n              <\/jats:alternatives>\n            <\/jats:inline-formula>. We prove that the failure of the reverse implication is equiconsistent with the existence of an inaccessible cardinal.<\/jats:p>","DOI":"10.1007\/s00153-024-00946-1","type":"journal-article","created":{"date-parts":[[2024,11,10]],"date-time":"2024-11-10T15:27:24Z","timestamp":1731252444000},"page":"473-491","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Infinite combinatorics revisited in the absence of Axiom of choice"],"prefix":"10.1007","volume":"64","author":[{"given":"Tam\u00e1s","family":"Csern\u00e1k","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Lajos","family":"Soukup","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,11,10]]},"reference":[{"issue":"12","key":"946_CR1","doi-asserted-by":"publisher","first-page":"860","DOI":"10.1090\/S0002-9904-1931-05283-6","volume":"37","author":"B Dushnik","year":"1931","unstructured":"Dushnik, B.: A note on transfinite ordinals. 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