tag:blogger.com,1999:blog-3722233.post7346788885933681031..comments2022-12-05T10:50:26.666-06:00Comments on Computational Complexity: Revisiting the Continuum HypothesisLance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger3125tag:blogger.com,1999:blog-3722233.post-52024989096951741132020-10-08T13:38:24.580-05:002020-10-08T13:38:24.580-05:00Does your argument rule out the possibility of fin...Does your argument rule out the possibility of finding a new, natural axiom (presumably based on new phenomenological insights) which is specifically applicable to the cumulative hierarchy (rather than ALL set-theoretic worlds)?antianticamperhttps://www.blogger.com/profile/11156250444026317037noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-56162290450765300362020-10-08T12:23:28.912-05:002020-10-08T12:23:28.912-05:00Regarding your proposal for how to settle CH by fi...Regarding your proposal for how to settle CH by finding a new axiom that is intuitively appealing and yet still settles CH, I argue in my paper "Is the dream solution of the continuum hypothesis attainable?" (http://jdh.hamkins.org/dream-solution-of-ch/) that this is impossible. The essence of my argument is that for us to learn that an axiom candidate decides CH will automatically undermine any attempt to take it as natural or intuitive, because of our prior extensive familiarity (via forcing and so on) with set-theoretic worlds having the opposite outcome.Joel David Hamkinshttps://www.blogger.com/profile/03016500743689022122noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-5204030507164587892020-10-08T10:15:42.656-05:002020-10-08T10:15:42.656-05:00Quite rightly, bullet point #4 above illustrates t...Quite rightly, bullet point #4 above illustrates that the parallel postulate is only true or false in a specified model. So shouldn't it be true that CH is either true or false in the specified model called "the cumulative hierarchy?" Or is the cumulative hierarchy "underdetermined" in some way? Model theory makes no allowances for semantic underdetermination because every sentence is either true or false in a given model. Is this somehow philosophically wrong or naive? If so, how? And this touches on bullet point #1. Are you guys not Platonist regarding the cumulative hierarchy? I grant you that ZFC is not quite as "intuitively" elementary as PA but it feels close, right? (I am aware, of course, that mathematically ZFC and PA are NOT close in power.) Is it not the case that both PA and ZFC both "indicate" canonical models in spite of the existence of non-standard models, i.e. the natural numbers and the cumulative hierarchy, and that every sentence should indeed either be true or false in the canonical model? <br /><br />As an aside, if "Platonism" makes you uncomfortable at cocktail parties you may wish to switch to a less metaphysical mathematical realism. One such can be found described quite well in Tragesser's book "Husserl and Realism in Logic and Mathematics."antianticamperhttps://www.blogger.com/profile/11156250444026317037noreply@blogger.com