tag:blogger.com,1999:blog-3722233.post7346788885933681031..comments2024-08-03T11:58:59.111-05:00Comments on Computational Complexity: Revisiting the Continuum HypothesisLance Fortnowhttp://www.blogger.com/profile/06752030912874378610noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-3722233.post-47770796821383531592023-08-12T10:53:01.993-05:002023-08-12T10:53:01.993-05:00It does not. Hamkins is overstating his position w...It does not. Hamkins is overstating his position with the word "impossible" in the above comment. An excerpt from the paper better classifies his argument as a prediction ("the entire episode" refers to his exposition of the reception of Freiling's Axiom):<br /><br />"The entire episode bears out the pattern of response I predict for any attempted use of the dream solution template, namely, a rejection of the new axiom from a perspective of deep mathematical experience with the contrary."<br /><br />To be clear, his prediction might be correct. That seems to be the mood of the times. But that might have more to do with humility and practicality than anything deep; it seems unwise to devote a lot of energy to solving a problem that the greatest geniuses in recent times failed to solve.<br /><br />Hamkins ends his paper with the following (note there is an implied vice-versa after "flawed"): <br /><br />"Before we will be able to accept CH as true, we must come to know that our experience of the ¬CH worlds was somehow flawed; we must come to see our experience in those lands as illusory. It is insufficient to present a beautiful landscape, a shining city on a hill, for we are widely traveled and know that it is not the only one." <br /><br />That is eloquent but misleading. A dream solution need not undermine the reality of nonstandard models. Nor the usefulness or elegance of nonstandard models.<br /><br />A dream solution could even reinforce the usefulness or elegance of nonstandard models; it needn't appear as "a beautiful landscape, a shining city on a hill." It is possible there will be a dream solution proving |ℝ| = Aleph_42 in the cumulative hierarchy (equivalently, in the universe of third-order arithmetic). I would be as shocked as everyone else, but it wouldn't be entirely without precedent; to me, a physics novice, Newtonian mechanics is more of a "beautiful landscape, a shining city on a hill" than general relativity, quantum mechanics, and string theory. Perhaps the universe of iterated powersets is, surprisingly, as weird as physical reality? We are stuck with physical reality, but if the cumulative hierarchy turns out to be ugly and annoying enough to human mathematicians, we could always redefine "the standard model" of set theory, without any intellectual dishonesty.Dustin Wehrhttps://www.blogger.com/profile/06043216009842601898noreply@blogger.comtag:blogger.com,1999:blog-3722233.post-18119900850595506032023-08-12T09:17:14.698-05:002023-08-12T09:17:14.698-05:00You're 100% right. Disputing that CH has a tru...You're 100% right. Disputing that CH has a truth value in the cumulative hierarchy requires disputing that P(P(ℕ)) is well-defined (i.e. claiming that it is undetermined), or something even sillier. It's postmodern mathematics, best suited for (a) contrarians, (b) philosophers, and (c) mathematicians who need to justify to granting agencies their work redoing the foundations of mathematics. I'm sad to say that groups (a) and (c) include a few extremely smart and knowledgeable people. They are few, but they are loud, and their story is titillating for a broad audience, so it persists.Dustin Wehrhttps://www.blogger.com/profile/06043216009842601898noreply@blogger.comtag: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