Cantor Confusion
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From: mueckenh on 26 Oct 2006 13:23 Han de Bruijn schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > >>Han de Bruijn wrote: > >> > >>>David Marcus wrote: > >>> > >>>>I don't think so. Bohmian Mechanics is 100% deterministic. All of the > >>>>uncertainty in the results of an experiment is due to uncertainty in > >>>>setting up the initial conditions of the experiment. > >>> > >>>Sure. Back to the dark ages of Laplacian determinism. > >> > >>Don't you mean Newton? > > > > I am sure, Han did not mean Newton. Newton was not a determinist at > > all. He accused Leibniz, who was a bit more deterministic, of being an > > atheist. > > Newtonian mechanics at least raises the _suggestion_ of being completely > deterministic. But I guess it has been Laplace who formulated that idea > explicitly. Laplace was that one who, in conversation with Napoleon, answered the question concerning the role of God: "I don't need that hypothesis". Newton was very religious. He considered his works on theology as by far more important than all his mathematical and physical works. God acts upon the world. The space is his sensory and God has to wind up the great clock. While Leibniz disagreed with the last fact, Samuel Clarke, Newton's deputy in his correspondence with Leibniz, accused Leibniz of being an atheist, a severe accusation at those times, because God was out of work. > Anyway, there is _no_ modern physicist who still believes in > in determinism these days. Due to Quantum Mechanics, the idea has become > entirely outdated. Of course, you are correct. By the way, the first draft of my website on MatheRealism is ready including links to your site. http://www.fh-augsburg.de/~mueckenh/MR.mht Regards, WM
From: mueckenh on 26 Oct 2006 13:27 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. 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? > > and that we > > cannot have any infinity unless we have the axiom of infinity. > > To prove there exists an infinite set requires some axioms. If the > axiom of infinity is not one of them, then, to prove the existence of > an infinite, there has to be some other axiom or axioms aside from the > axioms of Z without the axiom of infinity. > I know that. But some "experts" here are of the opinion that even ZF without INF is an infinite theory. And when I consider the abuse of expressions in set theory, I cannot but believe, that everything is possible there, even without the primordial existence of the set of all sets there: There we have the actual, i.e., finished infinity, the well-order of sets which definitely cannot be defined and, hence, cannot be well ordered, the uncountability of countable sets, ... Why then not the infinity of finite theories and sets? Regards, WM
From: mueckenh on 26 Oct 2006 13:28 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, > > nämlich 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 beschränkterer Mengen durch Aussonderung bzw. > > Auswahl, oder die Bildung von Mengen, die in eng umschriebenem Maß > > 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. Regards, WM
From: mueckenh on 26 Oct 2006 13:33 Sebastian Holzmann schrieb: > mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > > Here I repeat some some education for you (by Fraenkel, that is one of > > those scholars who made ZF): Vor allem ist die Bildung der wichtigsten > > Klasse paradoxer Mengen, nämlich 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 beschränkterer > > Mengen durch Aussonderung bzw. Auswahl, oder die Bildung von Mengen, > > die in eng umschriebenem Maß sozusagen umfassender sind, durch > > Paarung, Vereinigung, Potenzierung usw. > > Please do not quote scholars from some remote time on me. Rest assured > that I try not to sound as foolish as you do. What you regard as foolish is the explanation of the axioms which seem to be your gospel. These axioms and their meaning have not yet changed (as far as I know from modern text books and from the internet page of T. Jech (a leading set theorist of our days)). Regards, WM
From: mueckenh on 26 Oct 2006 13:37
David Marcus schrieb: > > There are many books on binary trees and many others on geometric > > series. I put these topics together. > > Randomly mixing terms from two different fields into a sentence does not > make the sentence meaningful. Yes, I agree. Randomly mixing would probably not be a good approach. > > > That is new and not everybody can > > understand it immediately. You must not be sad on that behalf. > > I'm only sad that you really seem to believe what you say. I am glad to say that I am not the only one. > > > Follow > > just my discussion with those guys who have understood this > > comparatively simple matter. > > Ah, but none of them actually get the same conclusion you do. So, you > and they can't both be right. There are several mathematicians understanding my argument, but the conclusions are no in question here. Some have understood my argument but as it does not fit their expectation and conviction over many years, they try to find a fault. That is a legitimate process of finding the truth by discussion. Even Virgil has understood, although he does not try to find any error but plainly refuses the result because there is a contrary proof by Cantor. And I know from many of my students that they understood me, although they do not study mathematics but various other topics. Regards, WM |