A question about a question and answer. - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-22T03:55:50Z http://mathoverflow.net/feeds/question/79881 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/79881/a-question-about-a-question-and-answer A question about a question and answer. Erin K Carmody 2011-11-03T00:39:43Z 2012-03-10T03:03:04Z <p>That is wrong or right about this question and answer?</p> <p>Question: Is there a cardinality which is greater than the continuum?</p> <p>Answer: Yes and No. If there is a Universe where a given cardinal kappa is greater than the size of the continuum, then there is a Generic-Extension of this Universe where the size of the continuum is greater than kappa.</p> <p>Edit: This question is basically about the size of the continuum, which has been discussed several times on mathoverflow. It is also about the philosophical position of whether there is the reality of the multiverse.</p> http://mathoverflow.net/questions/79881/a-question-about-a-question-and-answer/79882#79882 Answer by Andreas Blass for A question about a question and answer. Andreas Blass 2011-11-03T01:02:09Z 2011-11-03T01:02:09Z <p>In ordinary set theory, "Yes" is right and "No" is wrong. Even after you generically extend the universe to make the cardinal of the continuum bigger than a given $\kappa$, there are plenty of other cardinals that are even bigger than your new continuum. As M Turgeon says, to avoid cardinals larger than the continuum, you'd have to revoke the axiom of power set. </p> <p>Before he became a science fiction writer, Rudy Rucker did some work on the set theory that you get by revoking power set and adding Martin's axiom for all cardinals. That makes the continuum a proper class. </p> <p>There has also been a little work on a set theory that allows only countably infinite sets, like what you'd get by generically collapsing all cardinals. But all these ways of getting a "No" answer to your question are far from the usual set theory and are probably best understood as being about some notion other than "set".</p>