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Cardinal and Ordinal Numbers
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<h2 id="topics">Topics</h2> <p>After five introductory chapters on <a href="/facts/Naive_set_theory/hYIC5xPf">naive set theory</a> and set-theoretic notation, and a sixth chapter on the <a href="/facts/Axiom_of_choice/1Ura2HTh">axiom of choice</a>, the book has four chapters on <a href="/facts/Cardinal_number/ccoKwU4R">cardinal numbers</a>, their arithmetic, and series and products of cardinal numbers, comprising approximately 50 pages. Following this, four longer chapters (totalling roughly 180 pages) cover orderings of sets, <a href="/facts/Order_type/koXjnax6">order types</a>, <a href="/facts/Well-order/GrWEBWlW">well-orders</a>, <a href="/facts/Ordinal_number/GelFLjJt">ordinal numbers</a>, ordinal arithmetic, and the <a href="/facts/Burali-Forti_paradox/NQusk9Ik">Burali-Forti paradox</a> according to which the collection of all ordinal numbers cannot be a set. Three final chapters concern <a href="/facts/Aleph_number/kwe9I4v5">aleph numbers</a> and the <a href="/facts/Continuum_hypothesis/R0yGqHND">continuum hypothesis</a>, statements equivalent to the axiom of choice, and consequences of the axiom of choice.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a><a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p><p>The second edition makes only minor changes to the first except for adding footnotes concerning two later developments in the area: the proof by <a href="/facts/Paul_Cohen/wMLaE3Vz">Paul Cohen</a> of the independence of the continuum hypothesis, and the construction by <a href="/facts/Robert_M._Solovay/75zxRfT4">Robert M. Solovay</a> of the <a href="/facts/Solovay_model/chwwCgwt">Solovay model</a> in which all sets of <a href="/facts/Real_number/R02gw5Pb">real numbers</a> are <a href="/facts/Lebesgue_measurable/8us9cGoW">Lebesgue measurable</a>.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h2 id="audience-and-reception">Audience and reception</h2> <p>Sierpiński was known for his significant contributions to the theory of transfinite numbers;<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>, reviewer <a href="/facts/Reuben_Goodstein/7sSAxMPt">Reuben Goodstein</a> calls his book "a goldmine of results",<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> and similarly <a href="/facts/Leonard_Gillman/Xp25hwTv">Leonard Gillman</a> writes that it is highly valuable "as a compendium of interesting mathematical information, presented with care and clarity". Both Gillman and <a href="/facts/John_C._Oxtoby/6Dz6GjAC">John C. Oxtoby</a> call the writing style "leisurely" and "unhurried",<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> and although Gillman criticizes the translation from an earlier Polish-language manuscript into English as unpolished, and points to several errors in the bibliography, he does not find the writing in the text of the book to be problematic.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p><p>In the 1970 text <i>General Topology</i> by Stephen Willard, Willard lists this book as one of five "standard references" on <a href="/facts/Set_theory/1ptfsvUP">set theory</a>.<a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p> <h2 id="further-reading">Further reading</h2> <ul><li><a href="/facts/Lida_Barrett/VIceMCgN">Barrett, L. K.</a> (1960), "Review of <i>Cardinal and Ordinal Numbers</i> (1st ed.)", <i><a href="/facts/Scripta_Mathematica/0aHBgzdC">Scripta Mathematica</a></i>, 25: 247</li> <li>Neumer, W., "Review of <i>Cardinal and Ordinal Numbers</i> (1st ed.)", <i><a href="/facts/ZbMATH/P6rFxKKx">zbMATH</a></i> (in German), <a href="/facts/Zbl_(identifier)/P6rFxKKx">Zbl</a> <a href="https://zbmath.org/?format=complete&q=an:0083.26803">0083.26803</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Oxtoby, J. C. (1959), "Review of Cardinal and Ordinal Numbers (1st ed.)", Bulletin of the American Mathematical Society, 65 (1): 21–23, doi:10.1090/S0002-9904-1959-10264-0, MR 1565962 <a href="/wiki/John_C._Oxtoby" target="_blank">/wiki/John_C._Oxtoby</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Gillman, L., "Review of Cardinal and Ordinal Numbers", Mathematical Reviews, MR 0095787; briefly updated for 2nd ed., MR0194339 <a href="/wiki/Leonard_Gillman" target="_blank">/wiki/Leonard_Gillman</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Oxtoby, J. C. (1959), "Review of Cardinal and Ordinal Numbers (1st ed.)", Bulletin of the American Mathematical Society, 65 (1): 21–23, doi:10.1090/S0002-9904-1959-10264-0, MR 1565962 <a href="/wiki/John_C._Oxtoby" target="_blank">/wiki/John_C._Oxtoby</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Gillman, L., "Review of Cardinal and Ordinal Numbers", Mathematical Reviews, MR 0095787; briefly updated for 2nd ed., MR0194339 <a href="/wiki/Leonard_Gillman" target="_blank">/wiki/Leonard_Gillman</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Goodstein, R. L. (December 1966), "Review of Cardinal and Ordinal Numbers (2nd ed.)", The Mathematical Gazette, 50 (374): 437, doi:10.2307/3613997, JSTOR 3613997 <a href="/wiki/Reuben_Goodstein" target="_blank">/wiki/Reuben_Goodstein</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Oxtoby, J. C. (1959), "Review of Cardinal and Ordinal Numbers (1st ed.)", Bulletin of the American Mathematical Society, 65 (1): 21–23, doi:10.1090/S0002-9904-1959-10264-0, MR 1565962 <a href="/wiki/John_C._Oxtoby" target="_blank">/wiki/John_C._Oxtoby</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Gillman, L., "Review of Cardinal and Ordinal Numbers", Mathematical Reviews, MR 0095787; briefly updated for 2nd ed., MR0194339 <a href="/wiki/Leonard_Gillman" target="_blank">/wiki/Leonard_Gillman</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Gillman, L., "Review of Cardinal and Ordinal Numbers", Mathematical Reviews, MR 0095787; briefly updated for 2nd ed., MR0194339 <a href="/wiki/Leonard_Gillman" target="_blank">/wiki/Leonard_Gillman</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Oxtoby, J. C. (1959), "Review of Cardinal and Ordinal Numbers (1st ed.)", Bulletin of the American Mathematical Society, 65 (1): 21–23, doi:10.1090/S0002-9904-1959-10264-0, MR 1565962 <a href="/wiki/John_C._Oxtoby" target="_blank">/wiki/John_C._Oxtoby</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Goodstein, R. L. (December 1966), "Review of Cardinal and Ordinal Numbers (2nd ed.)", The Mathematical Gazette, 50 (374): 437, doi:10.2307/3613997, JSTOR 3613997 <a href="/wiki/Reuben_Goodstein" target="_blank">/wiki/Reuben_Goodstein</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Goodstein, R. L. (December 1966), "Review of Cardinal and Ordinal Numbers (2nd ed.)", The Mathematical Gazette, 50 (374): 437, doi:10.2307/3613997, JSTOR 3613997 <a href="/wiki/Reuben_Goodstein" target="_blank">/wiki/Reuben_Goodstein</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Oxtoby, J. C. (1959), "Review of Cardinal and Ordinal Numbers (1st ed.)", Bulletin of the American Mathematical Society, 65 (1): 21–23, doi:10.1090/S0002-9904-1959-10264-0, MR 1565962 <a href="/wiki/John_C._Oxtoby" target="_blank">/wiki/John_C._Oxtoby</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Gillman, L., "Review of Cardinal and Ordinal Numbers", Mathematical Reviews, MR 0095787; briefly updated for 2nd ed., MR0194339 <a href="/wiki/Leonard_Gillman" target="_blank">/wiki/Leonard_Gillman</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Gillman, L., "Review of Cardinal and Ordinal Numbers", Mathematical Reviews, MR 0095787; briefly updated for 2nd ed., MR0194339 <a href="/wiki/Leonard_Gillman" target="_blank">/wiki/Leonard_Gillman</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Willard, Stephen (1970), General topology, Reading, Massachusetts: Addison-Wesley Publishing Co., p. 297, ISBN 9780486434797, MR 0264581 <a href="9780486434797" target="_blank">9780486434797</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> </ol>