Counterintuitions and the well-ordering theorem
in [Logic]
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From: George Greene on 18 Nov 2009 11:09 On Nov 17, 5:43 am, Albrecht <albst...(a)gmx.de> wrote: > The trouble involves the > problem with the ostensible conclusion that "for any x" is the same as > "for all x" This IS NOT ANY sort of "ostensible conclusion". NObody "concludes" this! At best, people simply agree to use "all" TO MEAN "any" in SOME contexts. People whose native language is not English should arguably shut up about this unless they have a degree in English.
From: WM on 18 Nov 2009 12:00 On 18 Nov., 16:16, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > I'm not 100% sure, though. Is introducing P(omega) and introducing > square-root(-1) the same sort of mathematical act, or is there some > subtle difference? There is the subtle difference that people can do consistent calculations with sqrt(-1) without blocking inconsistent results whereas people cannot do consistent calculations with P(omega) without blocking the simple result, for instance, that the binary tree cannot have more paths than nodes. There is the obvious difference that nobody questions the usefulness of sqrt(-1) but the majority of scholars doubts the usefulness of any set theoretical result inside and outside mathematics - except the small but loudly shouting minority of mathelogians. Regards, WM
From: Daryl McCullough on 18 Nov 2009 12:17 George Greene says... > >On Nov 18, 10:16=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> I guess I'm confused about the ontological status of abstract >> objects such as sets. > >That's sort of unforgivable. Whatever. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 18 Nov 2009 12:24 George Greene says... >stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: >> What's the argument against the >> possibility that (1) for every strategy for the first player, there >> is a defense for the second player, and (2) for every defense for the >> second player, there is a strategy for the first player that beats it. > >This just irrelevant if true. >The 2nd player will always choose the successfully defending strategy. The second player doesn't *KNOW* what strategy the first player chose. All that he knows is the first *MOVE* in that strategy. -- Daryl McCullough Ithaca, NY
From: Daryl McCullough on 18 Nov 2009 13:02
George Greene says... > >On Nov 17, 6:53=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> But all it takes to be a class is a criterion for membership. A class >> is basically a formula with a single free variable. There is no question >> that the *formula* exists---we can write it down. > >There IS SO TOO a question about what exists, at FIRST order. The issue being discussed is what *axioms* to choose. Should there be, for example, a power set axiom, or the axiom of choice, or whatever. Of course, given a choice axioms, we can certainly ask what existence theorems follow from those axioms. That isn't what is being discussed, however. -- Daryl McCullough Ithaca, NY |