Cantor Confusion
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From: Virgil on 26 Oct 2006 15:32 In article <1161883732.413718.244570(a)k70g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Sebastian Holzmann schrieb: > > > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > > I am afraid, it will be useless for you to consult any books. > > > Nevertheless, here is my attempt to teach you some real logic: > > > Vor allem ist die Bildung der wichtigsten Klasse paradoxer Mengen, > > > n?mlich der allzu umfassenden Mengen (Antinomien von Burali-Forti, > > > Russell usw.) durch unsere Axiome ausgeschlossen. Denn diese gestatten, > > > eine oder mehrere gegebene Mengen als Ausgangspunkt nehmend, nur > > > entweder die Bildung beschr?nkterer Mengen durch Aussonderung bzw. > > > Auswahl, oder die Bildung von Mengen, die in eng umschriebenem Ma? > > > sozusagen umfassender sind, durch Paarung, Vereinigung, Potenzierung > > > usw. > > > > Mathematics has advanced beyond the stages of the 19th century. > > Fraenkel wrote that in 1923. That is 20th century. > > What you propose, namely the infinity of ZF without the axiom INF would > not be an advance. But meanwhile you may have recognized that your > assertion (ZF even without INF is not finite) is false. It is, however, quite true that ZF without INF need not be finite.
From: MoeBlee on 26 Oct 2006 15:43 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > > > Cantor laid the foundations of sets theory. If modern set theory would > > > not accept his definition, then it should not call itself set theory. > > > > It's called Z set theory. If you don't want it to be called 'set > > theory' just because it has refinements not in Cantor's work, then I > > guess we could call it 'zet zery' or whatever. But that's missing the > > point. Upon Skolem's refinement of Zermelo's set theory, we have a > > formal set theory. > > Why did Skolem not like it? Skolem wrote a very strong crtique of set theory. It's in the Van Heijenoort anthology 'From Frege To Godel'. But Skolem made an important suggestion. In the axiom schema of separation, Zermelo had used a notion of a 'definite property', which is not rigorous (what in heck is a definite property?). So Skolem suggested dumping 'definite property' and instead using well formed formulas (in a certain way). And that alteration makes it possible for us to fully formalize Zermelo set theory. > I proved that there are not more real numbers than a countable set has > elements. I proved it in a manner which everybody with moderate > mathematical knowledge can understand. And, what is important, in a > manner completely independent of your special language. Then yours is a proof in an informal context of your own mathematical understanding (and what CLAIM to be a context shared by anyone with moderate mathematical knowledge). And, so, as I've said already about a half a dozen times by now, that is not a proof that set theory is inconsistent. To prove set theory is inconsistent, you must provide prove, IN set theory, a sentence P and a sentence ~P that are sentences in the language of set theory. > It is clear and proven *from outside* that their results > are wrong and, therefore, uninteresting for me. It matters little to me that set theory is uninteresting to you on account of your having convinced yourself that it conflicts with certain ideas of yours that are outside set theory. I view set theory (in any of a number of different formulations) as one among many possible formal axiomatizations to provide the usual theorems of real analysis. If you have a proof that set theory is inconsistent, then a lot of people, including me, would like to see the proof. But what you've given is just an argument as to why you think set theory is wrong. MoeBlee
From: William Hughes on 26 Oct 2006 15:52 mueckenh(a)rz.fh-augsburg.de wrote: > jpalecek(a)web.de schrieb: > > > > The model is that simple that any student in the first semester could > > > understand it. Every paths which branches into two paths necessarily > > > needs two additional edges for this sake. It is only your formalistic > > > attitude that blocks your understanding. But you must not think that > > > anybody is blocked like you. > > > > This looks like a proof by induction. Indeed, you can prove your > > formula > > by induction for FINITE paths. > > Thanks for admitting your understanding so far. > > > But for infinite ones, you must do one > > more transfinite step. > > No. Any real number has only finite digit positions And once again WM deliberately confuses, "all positions are finite", with "there are a finite number of positions". - William Hughes
From: MoeBlee on 26 Oct 2006 15:53 mueckenh(a)rz.fh-augsburg.de wrote: > You are so much caught inside your theory that you are unable to look > at it from outside. No, I'm not. I think about philosophical problems with set theory frequently. And I keep in mind my agenda to learn about other foundational proposals as I try to get at least some of the outline or even just the flavor (if that's all I'm capable of) of such alternative proposals while I study to put myself in a position to rigorously understand them. > In the formal theory, there is the only predicate > infinite used for 'is actually infinite' and there is no use of 'is > potentially infinite'. But, alas, if there appear contradictions, then > the interpretation of the predicate infinite is quickly adjusted. There's no such adjustment in the formal theory. Please stop spouting misinformation. MoeBlee
From: MoeBlee on 26 Oct 2006 16:08
mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Sebastian Holzmann schrieb: > > > > > > > > > > Oh, I think I begin to see your problem here. But before we can speak of > > > > ZFC as a theory, we must first have some sort of "background set theory" > > > > available. And if we do not allow that background theory to "have" > > > > infinite sets (in some naive way), we cannot even formulate Z, because > > > > it consist of infinitely many sentences... > > > > > > So in principle we need the set of all sets in order to talk about > > > every thing including > > > > Who said anytying about talking about "every thing"? > > > > > the fact that it does not exist > > > > No, we don't need to hold that there exists a set of all sets to prove, > > in a given theory, that there does not exist a set of all sets. > > According to set theorists, naming it is having it. Please stop spouting blatant misinformation. Since Russell wrote that letter to Frege, there is no such thing in set theory as proving the existence of a set having a certain property just by referring to "the set that has property P". > At least the > natural numbers and other infinite sets are present by naming N etc. > How else should an infinite set come to existence in any primordial > model without an axiom of infinity securing its existence? Oh, please, just read a damn book on the subject already! "Primordial model". Oy vey. > I know that. But some "experts" here are of the opinion that even ZF > without INF is an infinite theory. You don't mean an 'infinite theory' (a theory is a set of sentences closed under entailment). You mean a theory that has a theorem that there exists an infinite set. And I have no idea who the "experts" are that think there is a theorem that there exists an infinite set without using the axiom of infinity to prove the theorem. I have no interest in talking about such "experts". MoeBlee |