Counterintuitions and the well-ordering theorem
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From: WM on 20 Nov 2009 09:08 On 20 Nov., 13:12, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Bill Taylor says... > > > > >stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >> Does it really make sense to doubt the truth of the > >> claim that there is a number i such that i*i = -1? > > >That's totally different and completely unobjectionable. > >Complex numbers can be defined trivially as ordered pairs with > >the appropriate operations, and all the usual results obtained. > >It has no logical or philosophical problematicity at all - > >i.e none beyond whatever the reals that make them up, already have. > > I don't think it is different at all. The fact that you can > interpret complex numbers as ordered pairs is just a simple > case of coming up with a *model* of complex numbers, which > exists if the theory complex numbers is consistent. The same > is true of the power set. We define the power set by axioms. > If those axioms are consistent, then there exists a model > of those axioms. The fact that it is more complicated than > ordered pairs doesn't seem particularly important to me. > > >> We essentially *define* i into existence by that claim. > >> It seems to me that the > >> power set of omega is similarly defined into existence. > > >It is a totally different question in every respect. > > No, it isn't. It is. Ordered pairs of reals in combination with the axioms of arithmetic make sense. The axiom of finished infinity does not. Regards, WM
From: Daryl McCullough on 20 Nov 2009 09:13 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). >wikipedia rebuts: >> The proof that such games are determined is rather simple: >> Player I simply plays not to lose; that is, he plays to make >> sure that player II does not have a winning strategy after I's move. >> If player I cannot do this, then it means player II had a winning >> strategy from the beginning. >> On the other hand, if player I can play in this way, then he must >> win, because the game will be over after some finite number of moves, >> and he can't have lost at that point. >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. 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." -- Daryl McCullough Ithaca, NY
From: WM on 20 Nov 2009 09:14 On 20 Nov., 13:37, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > Bill Taylor says... > > > > >stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > >You admit doubts about the O.S. of sets; I presume > >(maybe wrongly?) you have no, or at least much lesser, > >doubts about the O.S. of natural numbers. > > I don't see any big difference between the two. The set > of naturals and the set of reals are both abstractions. > I don't understand in what sense either exists, other > than exists as a coherent topic of study. None of them does exist other than as a name and a wrong, i.e., self- contradictory idea. We can write sequences of symbols that allow us to talk about numbers and to manipulate numbers. That's all that exists - and it's enough to do mathematics. Everything else is a useless object for useless Fools Of Matheology. Regards, WM
From: Daryl McCullough on 20 Nov 2009 09:18 George Greene says... >If there can be a class of all sets but not a set of all sets, then is >the set/class distinction real or fake? >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. 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, where V is reinterpreted not as *all* sets, but all sets of hereditary cardinality less than some inaccessible cardinal alpha. Under this reinterpretation, there *is* no object containing all of the (new) sets. So talk about completed totalities of sets can always be so reinterpreted without losing anything important. -- Daryl McCullough Ithaca, NY
From: Marshall on 20 Nov 2009 10:43
On Nov 20, 6:14 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > > None of them does exist other than as a name and a wrong, i.e., self- > contradictory idea. Snooze. Marshall |