Cantor Confusion
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From: Sebastian Holzmann on 24 Oct 2006 07:33 mueckenh(a)rz.fh-augsburg.de <mueckenh(a)rz.fh-augsburg.de> wrote: > One can prove the nonexistence, because there are no sets possible > which cannot be constructed from the empty set. Otherwise you could > also assert that the set of all sets was in your theory. Just in oder > to avoid that, the axioms were made. [...] > So we cannot be sure in ZFC that the set of all sets does not exist? Or > what is the difference? The existence set of all sets contradicts with the other axioms. One cannot have a consistent theory that combines ZF(C) with "\exists x \forall y y\in x". One can combine the axiom of infinity with the other axioms of ZF(C) without getting a contradiction (provided there wasn't one inside ZF - AoI). That is the difference!
From: Han de Bruijn on 24 Oct 2006 08:04 David Marcus wrote: > 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. > > Are you saying that we should reject a physical theory because of our > philosophy? Is it a philosophy that there is _no_ decisive empirical evidence for determinism as "present" in Newtonian mechanics? Han de Bruijn
From: Han de Bruijn on 24 Oct 2006 08:11 David Marcus wrote: > You mean all the cranks think they understand it. If they really > understood it (and it made sense--a rather big assumption that), they > could explain it in standard language. It isn't even that much fun to > banter with Mueckenheim because so little of what he says is even > understandable. Haha! What you say about Bohmian mechanics is understandable for anyone who has studied Newtonian mechanics. But it's nevertheless outdated. As one of your "cranks", I am at least up to date as Quantum Mechanics and Relativity are concerned. But you can not say such a thing. So who is a "crank" in this debate? Han de Bruijn
From: Han de Bruijn on 24 Oct 2006 08:21 David Marcus wrote: > Han de Bruijn wrote: > >>And no, binary trees will not be found in Halmos' "Naive Set Theory". >>Because it's too naive, I suppose .. > > I don't see how you would know it is not there since you said that while > you own the book, you've never read it. Funny how books aren't much use > if you don't actually read them. > > Anyway, I didn't say binary trees were in the book. I said that if > Muckenheim wants to talk using the mathematical concepts of sets, > functions, and relations, he should use standard terminology, where > "standard" means the same as in some mathematics book. It is like saying > that if he wants to speak in English he should use words according to > the meanings as given in an English dictionary. Certainly. In very much the same way as you think you can talk about phenomena in the atomic domain by using the standard terminology from Newtonian mechanics. While Quantum Mechanics instead uses a different terminology that you seem not to be able to comprehend. > Or, he could continue to do as Humpty Dumpty did and use his own > meanings for words without telling people what they are. Han de Bruijn
From: georgie on 24 Oct 2006 08:30
David Marcus wrote: > georgie wrote: > > > > David Marcus wrote: > > > georgie wrote: > > > > > > The OP said that a definition was invalid because it was self- > > > referential. However, there is no rule against self-reference in modern > > > mathematics, so the OP's objection is not valid. > > > > So the set of sets containing themselves is ok by you. > > I don't see where I said that. I said there is no rule against self- > reference. The rules are specified by the axioms of ZFC. So it's valid or it isn't and it's only valid when you like it. Is that your position. From your reactions it appears that way. If you don't explain why you think self-reference is ok in some cases and not others, you lose credibility. > > > Almost a century ago, Russell and Whitehead attempted to develop such > > > rules as a way of avoiding the paradoxes, but their approach was too > > > cumbersome. So, ZFC avoids the paradoxes in a different way. The > > > diagonal argument follows the rules of ZFC. If you want more details on > > > what the rules are, there are quite a few good books on the subject. > > > > This is a bunch of BS. There are no references to ZFC in Cantor's > > proof. > > All the steps of Cantor's argument can be justified within ZFC. People > don't normally trace their arguments all the way back to the axioms. The > whole point of Mathematics is that we build definitions and theorems and > then use those to build more. > > Which books on set theory have you read? All the ones you've read plus some more. |