tag:blogger.com,1999:blog-6642011.post582927013960932209..comments2022-05-17T13:54:19.476-04:00Comments on Philosophy, et cetera: Infinity ain't everythingRichard Y Chappellhttp://www.blogger.com/profile/16725218276285291235noreply@blogger.comBlogger7125tag:blogger.com,1999:blog-6642011.post-29648912184776302282007-05-21T20:02:00.000-04:002007-05-21T20:02:00.000-04:00There is a very precise way in which infinite sets...There is a very precise way in which infinite sets can be of “different sizes”. Notably, the set of real numbers is bigger than the set of counting numbers. What this means is that if you try and create a one to one correspondence between counting numbers and real numbers, it is always possible to show that you have missed some real numbers even though your list of counting numbers is infinitely long. Look at Cantor’s diagonal argument” if you want to see how this is done—it is a clever proof.<BR/><BR/>http://en.wikipedia.org/wiki/Cantor_diagonal_argument<BR/><BR/><BR/>So yes, it is possible to have infinite sets that do not contain all the elements of larger infinite sets. I guess the question then is would infinite space/time be such a set? I know physicists have considered the question, but I don’t know if current theory points to much of an answer.driftwoodhttps://www.blogger.com/profile/06307983826591633035noreply@blogger.comtag:blogger.com,1999:blog-6642011.post-25425539400237480782007-05-20T18:28:00.000-04:002007-05-20T18:28:00.000-04:00One has to ask what one means by an actual univers...One has to ask what one means by an actual universe being infinite. Are we speaking temporally, spatially or something like the many-worlds interpretation of quantum mechanics?<BR/><BR/>After all if we're just talking about temporal then we'd have very limited possibilities if the universe just reaches heat death. That is still infinite but hardly encompasses all possibilities.<BR/><BR/>The spatially infinite universe also would seem to limit possibilities. Certainly not everything would happen.<BR/><BR/>So it seems by "infinite" one is talking in a more Lewis sense. In a Lewis sense what the person says is true but almost no one is talking about Lewis' sort of realism towards possibilities when they talk about infinite. Certainly the MWI is more narrow.<BR/><BR/>I <I>think</I> Tipler goes in the direction of dealing with finite patterns. But as you note that is false. (I'm not sure what exactly Tipler's position is, I should note)Clark Goblehttps://www.blogger.com/profile/03876620613578404474noreply@blogger.comtag:blogger.com,1999:blog-6642011.post-81761104810625889112007-05-19T14:43:00.000-04:002007-05-19T14:43:00.000-04:00Hi Richard. I think your first point is right tha...Hi Richard. I think your first point is right that one can restrict the class of possibilities in the infinite set. In the context of a world, there still may be limits of this sort, I guess. For instance if laws of nature are fixed and not variable. This is certainly a difficult topic. <BR/> - Steve EsserStevehttps://www.blogger.com/profile/14851240963321295307noreply@blogger.comtag:blogger.com,1999:blog-6642011.post-2029221260221357632007-05-19T09:02:00.000-04:002007-05-19T09:02:00.000-04:00Ha, yeah, fair point!Ha, yeah, fair point!Richard Y Chappellhttps://www.blogger.com/profile/16725218276285291235noreply@blogger.comtag:blogger.com,1999:blog-6642011.post-23837285300270742072007-05-19T06:58:00.000-04:002007-05-19T06:58:00.000-04:00> But is it guaranteed to occur, at some point in ...> But is it guaranteed to occur, at some point in the infinite sequence? Well, no.<BR/><BR/>In your example is it not as or more guaranteed than anything else you have ever guaranteed in your life? I also presume the x/infinity probably isn't large enough to make Steve feel any better about it.<BR/><BR/>GNZAnonymousnoreply@blogger.comtag:blogger.com,1999:blog-6642011.post-33067428560679528652007-05-19T06:27:00.000-04:002007-05-19T06:27:00.000-04:00I'm not sure. But note that the "all heads" outcom...I'm not sure. But note that the "all heads" outcome is not any less likely than any other particular (fully specified) outcome. And some or other particular outcome must result. So it can't be impossible. (It may be "probability zero" in some weird mathematical sense that doesn't entail actual impossibility.)<BR/><BR/>Here's another example: a random sequence of natural numbers isn't guaranteed to "eventually" contain the number 7. For there's some (again, infinitesimal) chance that the sequence will just happen to mimic the sequence of even numbers. And in that case, wait as long as you like, the number 7 will never, ever, show up.Richard Y Chappellhttps://www.blogger.com/profile/16725218276285291235noreply@blogger.comtag:blogger.com,1999:blog-6642011.post-7079309973683445862007-05-19T05:42:00.000-04:002007-05-19T05:42:00.000-04:00Are you sure you can generalize the coin-flip reas...Are you sure you can generalize the coin-flip reasoning to an actual infinity? I don't have a full grasp of the mathematics, but the impression I have is that infinitesimals are only good when you're approaching infinity, not at actual infinity.<BR/><BR/>So in this case, the probability of at least one coin being tails approaches one as the number of coin flips approaches infinity, retaining the infinitesimal possibility of all the flips being heads. However, for an infinite number of flips, the probability of at least one coin being tails is one, and there is a zero probability of all the flips being heads.<BR/><BR/>Intuition says that there is always the chance of getting yet another heads, but intuition tends to break down with true infinity.Anonymousnoreply@blogger.com