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Jack Silver
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<h2 id="contributions">Contributions</h2> <p>In his 1975 paper "On the Singular Cardinals Problem", Silver <a href="/facts/Mathematical_proof/7ECDrU80">proved</a> that if a <a href="/facts/Cardinal_number/ccoKwU4R">cardinal</a> <i>κ</i> is <a href="/facts/Singular_cardinal/wrGvPz2c">singular</a> with <a href="/facts/Uncountable/zl2WqyJQ">uncountable</a> <a href="/facts/Cofinality/cC8dl3oW">cofinality</a> and 2<i>λ</i> = <i>λ</i>+ for all infinite cardinals <i>λ</i> < <i>κ</i>, then 2<i>κ</i> = <i>κ</i>+. Prior to Silver's proof, many mathematicians believed that a <a href="/facts/Forcing_(mathematics)/3vMqqKcq">forcing</a> argument would yield that the negation of the theorem is <a href="/facts/Consistency/IHkdnA9M">consistent</a> with <a href="/facts/ZFC/UYIsPbXw">ZFC</a>. He introduced the notion of a <i>master condition</i>, which became an important tool in forcing proofs involving large cardinals.<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> </p><p>Silver proved the consistency of <a href="/facts/Chang%27s_conjecture/ewfwgkVC">Chang's conjecture</a> using the Silver collapse (which is a variation of the <a href="/facts/List_of_forcing_notions/WKNzD87w">Levy collapse</a>). He proved that, assuming the consistency of a <a href="/facts/Supercompact_cardinal/I9FN4T8T">supercompact cardinal</a>, it is possible to construct a model where 2<i>κ</i> = <i>κ</i>++ holds for some <a href="/facts/Measurable_cardinal/2Er8f5zL">measurable cardinal</a> <i>κ</i>. With the introduction of the so-called <a href="/facts/Silver_machine/E81Tb5ed">Silver machines</a> he was able to give a fine structure free proof of <a href="/facts/Ronald_Jensen/Q5PxW1Mr">Jensen's</a> <a href="/facts/Covering_lemma/JVCh3MLe">covering lemma</a>. He is also credited with discovering <a href="/facts/Silver_indiscernible/u0xrkn6l">Silver indiscernibles</a> and generalizing the notion of a <a href="/facts/Kurepa_tree/N5NIcfTx">Kurepa tree</a> (called Silver's Principle). He discovered <a href="/facts/Zero_sharp/u0xrkn6l">0#</a> ("zero sharp") in his 1966 Ph.D. thesis, discussed in the graduate textbook <i>Set Theory: An Introduction to Large Cardinals</i> by Frank R. Drake.<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p><p>Silver's original work involving large cardinals was perhaps motivated by the goal of showing the inconsistency of an uncountable measurable cardinal; instead he was led to discover indiscernibles in <i>L</i> assuming a measurable cardinal exists. </p> <h2 id="selected-publications">Selected publications</h2> <ul><li>Silver, Jack H. (1971). "Some applications of model theory in set theory". <i>Annals of Mathematical Logic</i> 3(1), pp. 45–110.</li> <li>Silver, Jack H. (1973). "The bearing of large cardinals on constructibility". In <i>Studies in Model Theory</i>, MAA Studies in Mathematics 8, pp. 158–182.</li> <li>Silver, Jack H. (1974). "Indecomposable ultrafilters and 0#". In <i>Proceedings of the Tarski Symposium</i>, Proceedings of Symposia in Pure Mathematics XXV, pp. 357–363.</li> <li>Silver, Jack (1975). "On the singular cardinals problem". In <i>Proceedings of the International Congress of Mathematicians</i> 1, pp. 265–268.</li> <li>Silver, Jack H. (1980). "Counting the number of equivalence classes of Borel and coanalytic equivalence relations". <i>Annals of Mathematical Logic</i> 18(1), pp. 1–28.</li></ul> <h2 id="external-links">External links</h2> <ul><li><a href="https://math.berkeley.edu/people/faculty/jack-h-silver">Jack Silver</a> at Berkeley</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>Group in Logic and the Methodology of Science, "Jack Howard Silver", University of California–Berkeley <a href="http://logic.berkeley.edu/news.html" target="_blank">http://logic.berkeley.edu/news.html</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Jack Silver at the Mathematics Genealogy Project <a href="https://mathgenealogy.org/id.php?id=22309" target="_blank">https://mathgenealogy.org/id.php?id=22309</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Cummings, James (2009). "Iterated Forcing and Elementary Embeddings". In Handbook of Set Theory, Springer, pp. 775–883, esp. pp. 814ff. <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Drake, F. R. (1974). "Set Theory: An Introduction to Large Cardinals". Studies in Logic and the Foundations of Mathematics 76, Elsevier. ISBN 0-444-10535-2 <a href="/wiki/ISBN_(identifier)" target="_blank">/wiki/ISBN_(identifier)</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> </ol>