<story><title>Mathematicians Measure Infinities, Find They’re Equal</title><url>https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/</url></story><parent_chain></parent_chain><comment><author>abnry</author><text>I am having trouble grasping the result.&lt;p&gt;As far as I understand it, any proof of any result has to be with respect to some axiom system. The continuum hypothesis result is that you cannot prove a infinity exists between the naturals and the real numbers under the usual axiom system A.&lt;p&gt;How that was proved, I don&amp;#x27;t know, and I also don&amp;#x27;t know what axiom system was used in that proof. If the axiom system was also A, then it seems to me that it follows we cannot have p&amp;lt;t, and hence p=t, because otherwise by staying in the same axiom system A you violate the unprovability of the continuum hypothesis in A. This supplies a proof in A that p=t. But this seems trivial.&lt;p&gt;So the axiom system used to prove that &amp;quot;the continuum hypothesis is unsolvable in A&amp;quot; must be a larger system A+.&lt;p&gt;I think in A+ we again have p=t. The question this paper answers is whether we can prove &amp;quot;you cannot prove p=t using A&amp;quot; or &amp;quot;you can prove p=t using A&amp;quot;. The latter means there is a straight up proof.&lt;p&gt;Any people who really know this stuff can you comment if I got the gist right?</text></comment>
<story><title>Mathematicians Measure Infinities, Find They’re Equal</title><url>https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/</url></story><parent_chain></parent_chain><comment><author>arstin</author><text>I only had one one set theory course so can&amp;#x27;t vouch for the details, but this seemed like a well done article on a subject that could easily have been more eyeglazingly superficial or obscure.</text></comment>