tag:blogger.com,1999:blog-6839602.post2078216538647453280..comments2019-11-24T09:28:13.037-08:00Comments on Everyone Else is Crazy: Constructability, Uncountability, and ω-HaskellJim Applehttp://www.blogger.com/profile/11080395413026172939noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-6839602.post-7789358493198712532011-12-15T22:02:07.639-08:002011-12-15T22:02:07.639-08:00'"From the viewpoint of ZFC, there are an...'"From the viewpoint of ZFC, there are an uncountable number of ZFC functions from the natural numbers to the booleans.'<br /><br />Isn't that only if we accept the continuum hypothesis?"<br /><br />No. CH is about the relative sizes of aleph_1 and c. My statement is only that c > aleph_0.<br /><br />"'From the viewpoint of Haskell, there are an uncountable number of Haskell functions [from the natural numbers to the booleans.]'<br />--- emphasis added<br /><br />Why? Is this provable or a conjecture?"<br /><br />It's sloppy, and possibly meaningless.<br /><br />"'But, from the world of ZFC, the number of Haskell functions from the natural numbers to the booleans is countable: we can simply order them by their definitions lexicographically.'<br /><br />Can't you implement this lexicographic ordering in Haskell?"<br /><br />I think in ZFC as you can enumerate all constructible ZFC functions from the natural numbers to the booleans, but c is still greater than aleph_0 because ZFC by itself can't show that the enumeration is an enumeration of all ZFC functions from the natural numbers to the booleans.<br /><br />I think I was trying to say that that is true of Haskell, but if you step outside, you get some extra power.<br /><br />My wording was imprecise, to say the least.Jim Applehttps://www.blogger.com/profile/11080395413026172939noreply@blogger.comtag:blogger.com,1999:blog-6839602.post-31816811046459569802011-12-12T13:29:23.325-08:002011-12-12T13:29:23.325-08:00Forgive me for giving you a hard time, but I (a ra...Forgive me for giving you a hard time, but I (a random mathematician) have some questions with some of your statements...<br /><br />You said:<br /><br />"From the viewpoint of ZFC, there are an uncountable number of ZFC functions from the natural numbers to the booleans."<br /><br />Isn't that only if we accept the continuum hypothesis?<br /><br />The continuum hypothesis is independent of the axioms for ZF(C)...<br /><br />"From the viewpoint of Haskell, <b>there are an uncountable number of Haskell functions</b> [from the natural numbers to the booleans.]" <br />--- emphasis added<br /><br />Why? Is this provable or a conjecture?<br /><br />"But, from the world of ZFC, the number of Haskell functions from the natural numbers to the booleans is countable: we can simply order them by their definitions lexicographically." <br /><br />Can't you implement this lexicographic ordering in Haskell?<br /><br />Just curious...pqnelsonhttps://www.blogger.com/profile/12779680952736168655noreply@blogger.comtag:blogger.com,1999:blog-6839602.post-22927886241403889512007-06-30T18:42:00.000-07:002007-06-30T18:42:00.000-07:00evilness, thats what he's getting at.Pure, Functio...evilness, thats what he's getting at.<BR/>Pure, Functional, Unadulterated Evilness. <BR/><BR/><BR/>~~jfredett (Joe)Joe Fredettehttps://www.blogger.com/profile/07170469679136904832noreply@blogger.comtag:blogger.com,1999:blog-6839602.post-59107206269498002142007-06-30T15:37:00.000-07:002007-06-30T15:37:00.000-07:00I don'T think I understand what you're getting at....I don'T think I understand what you're getting at.Jim Applehttps://www.blogger.com/profile/11080395413026172939noreply@blogger.comtag:blogger.com,1999:blog-6839602.post-34866139053098334612007-06-30T12:08:00.000-07:002007-06-30T12:08:00.000-07:00We can also imagine having an operation of typeget...We can also imagine having an operation of type<BR/><BR/>getNonComputableOrdinal :: IO (Ordinal A)<BR/><BR/>where the ordinal is determined, e.g., by a random process.augustsshttps://www.blogger.com/profile/07327620522294658036noreply@blogger.com