How many real numbers exist? New proof moves closer to an answer
Great article. Since it looks like a lot of folks are interested in this article, some extra background.<p>First, what is forcing? The article actually has a great description of ultrapowers (a key part of the construction) but it goes by a little fast, so you might like Tim Chow&#x27;s &quot;A beginner&#x27;s guide to forcing&quot; [1] which does a good job not only laying out the mathematical details at a high level, but also really clearly explaining the philosophy of how you prove something independent of the axioms. I found it very illuminating.<p>Second, the article has some interesting notes on how mathematicians go about what axioms to select and which not to. Penelope Maddy&#x27;s &quot;Believing the Axioms&quot; [2] is the classic on this topic (it has two parts). It is focused on set theory, so it has a nice description of Martin&#x27;s axiom and the arguments for and against. It was nice to read because it is a deeply technical argument (set theory is hard!) but at the same time there is no &quot;right answer&quot;—all of these axioms, after all, are independent of ZFC, and it&#x27;s &quot;ok&quot; to add any of them. The arguments are sometimes aesthetic (&quot;rules&quot; versus &quot;surprises&quot;), sometimes pragmatic (what they can and cannot prove), and sometimes involve deep values of what the universe should be like (should higher cardinalities be like lower ones? weirder? simpler?).<p>It might all seem abstract, but if your day job is programming, imagine an argument over how to architect a large and complex system. Perhaps both architectures are possible, but which one is &quot;right&quot;? What arguments would you deploy? Set theorists are also building a large and complex system (the universe of sets) and are having arguments over how it should be built, which things it should make easy and which hard, which technologies should be supported natively (forcing?) and which should not.<p>[1] <a href="http:&#x2F;&#x2F;timothychow.net&#x2F;forcing.pdf" rel="nofollow">http:&#x2F;&#x2F;timothychow.net&#x2F;forcing.pdf</a>
[2] <a href="https:&#x2F;&#x2F;www.cs.umd.edu&#x2F;~gasarch&#x2F;BLOGPAPERS&#x2F;belaxioms1.pdf" rel="nofollow">https:&#x2F;&#x2F;www.cs.umd.edu&#x2F;~gasarch&#x2F;BLOGPAPERS&#x2F;belaxioms1.pdf</a>