Counterintuitions and the well-ordering theorem
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From: George Greene on 20 Nov 2009 13:42 On Nov 20, 9:14 am, WM <mueck...(a)rz.fh-augsburg.de> wrote: > We can write sequences of symbols that allow us to > talk about numbers and to manipulate numbers. Coming from you of all people, that's just silly. You don't know what numbers are. > That's all that exists - > and it's enough to do mathematics. Everything else Who said ANYthing about ANYthing else?? That's ALL *we're* doing IS using sequences of symbols to talk about numbers!! Are you alleging that it was wrong to speak of any set that was not a number??? Are you saying that we may speak of 1 but not of {1} ??????
From: Daryl McCullough on 20 Nov 2009 20:30 George Greene says... > >On Nov 20, 9:18=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough) >wrote: >> >If sets can have powersets, why can't classes have powerclasses? =A0And >> >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! I'm not inclined to keep reading after being yelled at. -- Daryl McCullough Ithaca, NY
From: Nam Nguyen on 21 Nov 2009 00:53 Albrecht wrote: > On 20 Nov., 06:54, Nam Nguyen <namducngu...(a)shaw.ca> wrote: >> Bill Taylor wrote: >>> We have no doubt about >>> individual natural numbers, or even N, because we feel that >>> we know "all about them" in some sense. Though various theorems >>> still surprise us of course, we feel that there can be no further >>> *philosophical surprises* from them, if that phrase strikes a chord. >> The difficulty is what exactly would we mean by "in some sense"? >> There lurks in any concept as strong as the naturals at least one >> formula about the concept that could be impossible for us to know >> its truth value. > > > If we can connect such a undecidable formula with an individual > natural number in a suitable manner, this number would be not fully > determined. But not all formulas are about an individual natural numbers. Some formulas are about infinite numbers of them. And some of these formulas could turn out to be concept-undecidable: we wouldn't be able to know if these formulas are true in the underlying concept. If these formulas exist, they'd trump Godel's results since, e.g., if it's impossible to know the truth value of these formulas they are also unprovable in formal systems adequately formalizing the concept. > If we claim that a set is well defined only in the case > that we are, at least potentially, able to know any element, we had to > conclude that the set of naturals doesn't exist since it would include > at least one such not fully determined element. Again, it's not about any formula that's about any one element. GC for example isn't about any particular even number; it's actually about certain *infinite sets* of prime numbers! > >> In other words, we know as much about the naturals as we know about sets >> in an informal, naive way. >
From: Rupert on 21 Nov 2009 22:52 On Nov 19, 3:02 am, George Greene <gree...(a)email.unc.edu> wrote: > On Nov 12, 6:01 pm, Rupert <rupertmccal...(a)yahoo.com> wrote: > > > Unrestricted comprehension gets you into trouble, right? > > Comprehension IS unrestricted TO BEGIN with in the sense that > even if you do NOT have a SET for every formula, YOU STILL have > a CLASS for every formula (it's just that the class may be proper). > If you go up to a class theory then the "restriction" involves what > you > can quantify over, so it is more unrestricted quantification than > unrestricted > comprehension that is implicated -- you are not supposed to quantify > over a > universe including the thing BEING defined. You have various comprehension axiom schemata in various different second-order set theories, and maybe in the case of Kelly-Morse set theory you would want to call it "unrestricted", but that is quite obviously not what I was talking about. I am obviously talking about the unrestricted comprehension axiom schema for the first-order language of set theory which is of course well-known to be inconsistent.
From: Keith Ramsay on 22 Nov 2009 02:00
On Nov 20, 6:27 am, Herman Jurjus <hjm...(a)hetnet.nl> wrote: |Can you prove "if ZF minus powerset is consistent, then so is ZF |including powerset", without presuming the latter? | |Or even "if ZF minus powerset is consistent, then so is ZF minus |powerset plus (there is a set containing all subsets of N)" ? The hereditarily countable sets form a model of ZF minus the powerset axiom. Each can be represented as a graph on a countable set which can in turn be represented by a real. I can't swear offhand that the latter theory here is strong enough to prove the existence of this model, although I suspect it is. It seems unlikely at least that the theories are equiconsistent the way one might want. Keith Ramsay |