Why should I believe the Singular Cardinal Hypothesis? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T02:24:30Z http://mathoverflow.net/feeds/question/69288 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/69288/why-should-i-believe-the-singular-cardinal-hypothesis Why should I believe the Singular Cardinal Hypothesis? Oliver 2011-07-01T19:19:47Z 2011-07-02T02:45:38Z <p>The Singular Cardinal Hypothesis (SCH) is the statement that $\kappa^{cf(\kappa)} = \kappa^+ \cdot 2^{cf(\kappa)}$ for every singular cardinal $\kappa$ (or various equivalent statements). </p> <p>It is obviously implied by the Generalized Continuum Hypothesis. It is also implied by the Proper Forcing Axiom (under which $2^{\aleph_0} = \aleph_2$). Nonetheless it doesn't seem terribly compelling to me. But I am trying to learn to appreciate it!</p> <p>Why should I believe SCH? (Now when I say "believe," it isn't clear exactly what I mean by that. There are obviously plenty of models of set theory in which SCH holds, and they are certainly worth studying, but somehow they aren't the models that feel most "realistic" in my little head.) <strong>What I want is an answer in the style of Maddy's <em>Believing the axioms</em>,</strong> explaining why I should "like" (or not) this hypothesis.</p> <p>Some thoughts on why SCH isn't so unreasonable: </p> <ul> <li><p>Somehow I find very compelling the result that SCH holds above the first strongly compact cardinal. This is what makes it seem most reasonable to me that SCH should hold everywhere.</p></li> <li><p>SCH is implied by various contradictory axioms. Its negation is equiconsistent with the existence of a fairly large cardinal. </p></li> <li><p>It is obvious that $\kappa^{cf(\kappa)} \geq \kappa^+ \cdot 2^{cf(\kappa)}$, so SCH is just saying that the cardinality on the left shouldn't be any larger than strictly necessary.</p></li> <li><p>It makes cardinal arithmetic much easier! (But maybe this is an argument against SCH too...)</p></li> </ul> <p>Why else should SCH seem reasonable? </p> http://mathoverflow.net/questions/69288/why-should-i-believe-the-singular-cardinal-hypothesis/69308#69308 Answer by Joel David Hamkins for Why should I believe the Singular Cardinal Hypothesis? Joel David Hamkins 2011-07-02T02:10:53Z 2011-07-02T02:45:38Z <p>Solovay proved in ZFC that all instances of the SCH hold above any strongly compact cardinal. Thus, you should believe the <em>eventual</em> SCH to at least the extent that you believe in the very large large cardinals, which I believe are discussed by Maddy in the context of believing in the axioms.</p>