tag:blogger.com,1999:blog-25562705.post219098248767529765..comments2023-10-20T12:34:59.317-04:00Comments on Adventures in Computation: Are the Reals real?Aaronhttp://www.blogger.com/profile/09952936358739421126noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-25562705.post-15673103850829218902007-12-14T21:13:00.000-05:002007-12-14T21:13:00.000-05:00This is maybe alluded to above, but I want to be m...This is maybe alluded to above, but I want to be more explicit: A single choice from an uncountable set (or any set) does not require the axiom of choice. If you want to avoid the ability to choose a single element from an uncountable set, you need to avoid allowing uncountable sets to exist... for that, I guess you need to get rid of the power set axiom or the axiom of infinity. (Avoid having infinite sets at all, or avoid having power sets.)Davidhttps://www.blogger.com/profile/10565910956857563935noreply@blogger.comtag:blogger.com,1999:blog-25562705.post-21299483907592256412007-12-13T23:43:00.000-05:002007-12-13T23:43:00.000-05:00"If quantities in the universe such as position, m..."If quantities in the universe such as position, mass, velocity, temperature, etc. are represented by (continuous) real values, then they can't be perfectly simulated."<BR/><BR/>You've convinced me that the Reals cannot be perfectly simulated. But you haven't convinced me that physical quantities can be perfectly simulated. So why should I question whether or not pi is physically real?<BR/><BR/>Anyway, imperfect simulations are quite valuable!Anonymousnoreply@blogger.comtag:blogger.com,1999:blog-25562705.post-66342077934636108722007-10-07T14:47:00.000-04:002007-10-07T14:47:00.000-04:00The version of the two-envelope paradox I describe...The version of the two-envelope paradox I describe actually doesn't rely on the ability to pick two elements from a countable set with equal probability -- I had always thought that the non-existence of a uniform distribution over the integers was the "solution" to the two-envelope paradox, but apprantly it is not.Aaronhttps://www.blogger.com/profile/09952936358739421126noreply@blogger.comtag:blogger.com,1999:blog-25562705.post-58206009237278429572007-10-07T08:42:00.000-04:002007-10-07T08:42:00.000-04:00Woah, I just noticed that you covered the "can't p...Woah, I just noticed that you covered the "can't pick a natural number at random with equal probability" in the two envelope paradox. There you go. :-)Pseudonymhttps://www.blogger.com/profile/04272326070593532463noreply@blogger.comtag:blogger.com,1999:blog-25562705.post-84423843094737800552007-10-07T08:39:00.000-04:002007-10-07T08:39:00.000-04:00Can we really choose an element from an uncountabl...<I>Can we really choose an element from an uncountable set?</I><BR/><BR/>I don't think that the Banach-Tarski paradox is a problem, because any physical object is made of a countable set of "things" (virtual particles notwithstanding).<BR/><BR/>But here's a related problem: In probability theory, we can't choose an element from a countably infinite set in such a way that each element has the same chance of being chosen.<BR/><BR/>(To see why, suppose we pick two naturaly at random, say, x and y. There are a finite number of numbers less than x, and an infinite number greater. If each number has the same chance of being chosen, then with probability 1, y is greater than x. But by the same argument, with probability 1, x is greater than y.)<BR/><BR/>However, we <I>can</I> (and routinely do, at least theoretically) choose an element from an uncountably infinite set such as [0,1), such that each number has equal chance of being chosen.<BR/><BR/>How does <I>that</I> work?Pseudonymhttps://www.blogger.com/profile/04272326070593532463noreply@blogger.comtag:blogger.com,1999:blog-25562705.post-87592635642598497622007-09-25T08:20:00.000-04:002007-09-25T08:20:00.000-04:00Good point. Fortunately the integers can be placed...Good point. Fortunately the integers can be placed in 1-1 correspondence with the naturals. :-) Actually, I suspect that the entire postscript is nonsensical.Aaronhttps://www.blogger.com/profile/09952936358739421126noreply@blogger.comtag:blogger.com,1999:blog-25562705.post-78494856440469554562007-09-25T01:00:00.000-04:002007-09-25T01:00:00.000-04:00"We can define a choice function by first putting ..."We can define a choice function by first putting the elements of our countable set in one to one correspondence with the integers." - Aren't you going to need a correspondence to the natural numbers in order to be able to get a smallest element?Robhttps://www.blogger.com/profile/05106663398227635415noreply@blogger.com