Counterintuitions and the well-ordering theorem
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From: Albrecht on 19 Nov 2009 01:50 On 18 Nov., 17:09, George Greene <gree...(a)email.unc.edu> wrote: > 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. It is a logical conclusion to say if F holds for any x <=> F holds for all x This conclusion, or equivalence, is surely correct for any finite case. People who don't understand this simple fact should arguably shut up about this. Best regards Albrecht
From: George Greene on 19 Nov 2009 08:02 On Nov 18, 12:00 pm, WM <mueck...(a)rz.fh-augsburg.de> wrote: > There is the subtle difference that people can do consistent > calculations with sqrt(-1) All the calculations that everybody does (correctly) in ZFC are consistent, TOO, as far as YOU know. > without blocking inconsistent results Native language problem again. In English, that phrase is somewhere between ambiguous and ungrammatical. Who or what is supposed to be blocking inconsistent results? Or are you alleging that the inconsistent results are in fact occurring, and that they are blocking something? > 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. We do not BLOCK that result! That result DOES NOT EXIST! Every time you claim that THAT is a result, YOU ARE JUST LYING! DISPROVING your alleged result (which we do) does NOT constitute "blocking" anything! > 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. One of the main pieces of usefulness of i is in the context of e^(i*pi) +1=0, i.e., of using complex numbers to deal with rotation and angular or polar-co- ordinate representations (of things like alternating current, for example). That relates to combinations of sinusoidal functions, which relates to Fourier series, which relates to limit points of infinite sums of these functions, WHICH IS WHAT MOTIVATED Cantorian ordinals in the first place! The connections were there FROM BIRTH! This is NOT some disconnected realm!
From: George Greene on 19 Nov 2009 08:05 On Nov 18, 1:02 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > The issue being discussed is what *axioms* to choose. Should There CANnot be any "should" around this. All consistent axiom-sets are equally legitimate. More to the point, it is very obvious for a whole host of reasons that NO FIRST-order axiom-set (or no r.e. 1st-order axiom-set anyway) is going to be adequate. If you really are trying to investigate collections then you might as well go to 2nd-order ANYhow. At 1st-order you get this weird confusion where adding existence assumptions, trying to make more things exist, can actually make your universe SMALLER. For example, in order to get a countable model of ZFC, you have to throw some things OUT of the domain, DESPITE the fact that they "really" exist. In the case of the conflict between AC and AD, it is far from clear which class of models is supposed to be more inclusive.
From: George Greene on 19 Nov 2009 08:11 On Nov 18, 12:24 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > The second player doesn't *KNOW* what strategy the first player > chose. All that he knows is the first *MOVE* in that strategy. Well, your use of the word "defense" was confusing. What you actually MEANT by "defense" was "strategy". A defense is a strategy that actually successfully defends. And in any case, you are still inconsistent. You said, >> (1) for every strategy for the first player, there >> is a defense for the second player, This means a successfully defending strategy. >> and (2) for every defense for the second player, This is incoherent; there is nothing (yet) for this strategy TO be defending AGAINST. This(2) IS JUST a strategy, and IF >> there is a strategy for the first player that beats it. Then (2) IS NOT a defense.
From: George Greene on 19 Nov 2009 12:44
On Nov 12, 1:23 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote: > People skeptical of the power set usually balk at the very first > application that gets you something really new: P(omega), or the > reals. The other time to balk is at the very end, because there in fact is no end. This is a place&case where people could legitimately talk about potential infinity. There is no last ordinal and there is no ultimate universe of all powerclasses. |