Counterintuitions and the well-ordering theorem
in [Logic]
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From: George Greene on 20 Nov 2009 13:10 On Nov 20, 9:13 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > No, Wikipedia explains that the below is about *finite* games > (or at least games in which, if player 2 wins, then he wins > after a finite number of moves). The POINT is that those are ALL the games that people normally think about playing. Almost any mathematical object you can deal with needs to be finitely specIFIABLE EVEN if it is infinite. Such as an r.e. set, for example (which has a FINITE TM-description). In the case of (e.g.) Chess, (for example), if white does not have a winning strategy THEN BLACK MUST have a drawing one.
From: George Greene on 20 Nov 2009 13:16 On Nov 20, 9:18 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >If sets can have powersets, why can't classes have powerclasses? And > >if they can, why isn't the limit of the power-class process simply a > >hyper-class?? AD NAUSEAM??? > > Exactly right. This IS NOT right! This is RIDICULOUS! It says AD NAUSEAM for A REASON! > If you introduce a class V of all sets, then we can > create a second object that is all subclasses of V, and another object > that is all the subclasses of the second object, etc. Then this whole > *super* universe can be reinterpreted as a theory of sets alone, Are you SERIOUSLY talking about a FIRST-order set theory?? Any such "re-interpretation" as THAT is going to FAIL on cardinality grounds alone! That same theory of sets has both a "whole super-universe" model AND a COUNTABLE model. And in any case, once you have introduced a class V of all sets, the fact that all of ITS subclasses ARE NOT all ELEMENTS of it proves that it WAS NOT the class of all sets! The fact that "re"-interpretation is possible is not even relevant! NO model of set theory contains all of its OWN subclasses as elements! For it to even be claiming about itself that it is ANY sort of concept or model of some collection of "all" sets IS JUST RIDICULOUS! In the context of traditional first-order model theory, the set-class distinction is REAL!
From: George Greene on 20 Nov 2009 13:17 On Nov 20, 9:08 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > Ordered pairs of reals in combination with the axioms of > arithmetic make sense. The axiom of finished infinity does not. The fact that you personally opine that something does not make sense is of no concern to anyone, especially if that something is axiomatic in nature. THE ONLY legitimate objection TO ANYbody's proposed axioms is to DERIVE AN INCONSISTENCY from them. IF you cannot do that then you CAN Sit Down And Shut Up!!!
From: George Greene on 20 Nov 2009 13:23 On Nov 20, 9:18 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > So talk about completed totalities of sets can always be so reinterpreted > without losing anything important. Only if you IN FACT MAKE the set-class distinction. This option is not available if you dismiss the distinction as semantically illegitimate. If the distinction is not available then there simply IS NO class of all classes and talk as though there were IS JUST WRONG AND INCOHERENT.
From: George Greene on 20 Nov 2009 13:40
On Nov 20, 9:13 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > George Greene says... > > > > >Gale, Stewart, and wikipedia are laughing at you. > > No, Wikipedia explains that the below is about *finite* games > (or at least games in which, if player 2 wins, then he wins > after a finite number of moves). I KNOW that. I even SAID that. You know, where *I* said, > >The missing assumption here is basically that the game has a finite > >board and a finite number of pieces and is therefore (like chess or go) > >in some sense finite after all, despite the fact that we were defining > >everything in terms of infinite sequences. You then amazingly go on to opine, > The wikipedia article says the following: "This proof does not actually > require that the game always be over in a finite number of moves, only > that it be over in a finite number of moves whenever II wins." Well, that's just THAT article. That's basically the sub-result (after you translate everything into topology) that the win-set is "closed" or "open" (which is the original Gale- Stewart theorem). THE RESULT IS EXTENDED BY MARTIN in 1975 and 1982 TO A MUCH LARGER class of games: http://en.wikipedia.org/wiki/Borel_determinacy_theorem The original theorem about "closed" or "open" win-sets can be translated back from topology to normal naive gaming concepts as "finite". But what it means for a the game's win-set to have the relevant property of Borel here is FAR from clear. |