Cantor Confusion
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From: Lester Zick on 30 Oct 2006 12:42 On Sun, 29 Oct 2006 21:02:07 -0700, Virgil <virgil(a)comcast.net> wrote: >In article <MPG.1faf1538d19002279897b9(a)news.rcn.com>, > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >> mueckenh(a)rz.fh-augsburg.de wrote: >> > Dik T. Winter schrieb: >> > >> > > Euclides was apparently smarter than Cantor in this. He used the >> > > parallel axiom (postulate) for something he could not prove from the >> > > other axioms. >> > >> > But which is obviously possible and correct under special >> > circumstances, while Zermelo's AC is obviously false under any >> > circumstances. >> >> What do you mean an axiom is "false"? Are you discussing philosophy or >> mathematics? > >Good point! Axioms cannot be "false" in any real sense. So, Virgil, would you then argue that self contradictions cannot be false in any real sense? ~v~~
From: Lester Zick on 30 Oct 2006 12:52 On 29 Oct 2006 21:18:28 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >mueckenh(a)rz.fh-augsburg.de wrote: >> Every mathematician whose mind has not yet suffered the drill of set >> theory you probably were exposed to knows that my argument is correct. > >How could they know it is correct regarding set theory if they don't >know set theory? "Regarding set theory"? Who suggested that pray tell"? Do you have difficulties construing the language correctly, Moe? >And you know the mind of every mathematician who has not studied set >theory? Such a mind reader you are! Ah the jesuitical approach to modern math set analysis techniques. >> (And there are many mathematicians who never studied set theory but >> every mathematician knows the geometric series.) >> >> However due to the strong pressure of the orthodox fraction it may be >> disadvantageous for a mathematician to see his name printed here. >> Further it is my general principle not to publish names of private >> correspondents. > >Okay, we'll take your word for it then. Better yet, we'll just read >your mind to witness your memories of correspondence you've have. >Better yet, we'll just directly read the minds of every mathematician >who has not studied set theory. Why not? It works for you! Seems to work pretty well for you too, Moe. ~v~~
From: Lester Zick on 30 Oct 2006 12:53 On 29 Oct 2006 21:40:05 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: [. . .] >> Therefore, the real numbers show ZFC is consistent, because it has a >> countable model. > >The real numbers do no such thing. You have no idea what you're talking >about. It's offensive, actually, the way you throw jargon of set theory >and mathematical logic around though you have no idea what it means. It is enough that it is jargon. ~v~~
From: MoeBlee on 30 Oct 2006 12:53 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > > 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". > > What is the set N? Where are all the natural numbers? They are nowhere > and they will not come along anywhere. You write N or, equivalently, > the axiom INF. That is all. You name these things like infinite sets > and then you think that they existed and that you had it. But you > don't. As I said, post-Frege, we do not prove the existence of a set having a certain property just by referring to "the set that has property P". That is, we do not have a general scheme that permits us to assert the existence of a set having a certain property just by referring to "the set that has property P". And N is not even a special case of such method. > > > 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 am not a native English speaker. Perhaps this is not an original > English word. I used it for "prior to the ordinary theory". If you > can't understand it, then ask. If you understand it, then do just > understand it. The word is fine. The problem is your application of it to a subject of which you know nothing. > > > 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). > > A theory is a systems of ideas. The axioms belong to it, not > necessarily sentences which may or may not haven been derived. That's not what I mean by 'theory', and it is not what anyone who studies set theory in even a modest amount of depth means by 'theory'. You may use the word in your sense, but you should understand then that you are talking about something different from what discussants in set theory mean. Usually, there are two different definitions of 'theory': (1) A theory is a set of sentences (cf., e.g., Chang & Keisler). (2) A theory is a set of sentences closed under entailment (cf., e.g., Enderton). That difference is unfortunate, so we just have to be clear which definition we're using. I use (2). > > 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. > > One expert wrote: > I wouldn't necessarily call ZFC - AoI "finite set theory". > > And I replied: > It is a theory without the actual infinite, a theory without omega. One > > may execute any operation. The finite domain will never be left. Which > of the remaining axioms should yield infinity? Probably what he is trying to get across to you is that dropping the axiom of infinity does not entail that all sets are finite. Dropping the axiom of infinity only entails that it is undecided whether there are infinite sets. You can't infer from Z without the axiom of infinity that there are no infinite sets. To prove that there are no infinite sets in a Z set theory requires adopting an axiom that there are no infinite sets not just dropping the axiom of infinity. > Please adhere to what I wrote, if you are so bad in interpreting. I regret if I mischaracterized your intent. However, you should recognize that you use terminology of set theory and mathematical logic without understanding the meanings of those terms, so it is indeed very likely that you will be misunderstood. MoeBlee
From: Lester Zick on 30 Oct 2006 12:55
On 29 Oct 2006 21:13:01 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >Lester Zick wrote: >> On 27 Oct 2006 16:56:30 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >> >> >Lester Zick wrote: >> >> >> You mean that mathematical definitions can't have different "domains >> >> >> of discourse" and mathematical definitions in different domains of >> >> >> discourse can't borrow from one another? >> >> > >> >> >No, that's not what I said. >> >> >> >> Then why exactly are you complaining about what I said? Frankly, Moe, >> >> you don't seem to have said much of anything that I can make out. If >> >> mathematical definitions can have different domains of discourse then >> >> what I wrote should be perfectly acceptable according to your own >> >> definition of mathematical definitions and domains of discourse.. >> > >> >Since I never said anything that can be paraphrased as the jumble of >> >nonsense you just mentioned, nothing I did write entails that the >> >jumble of nonsense you wrote needs to be acceptable to me. >> >> What kind of jumble of nonsense do you prefer then, Moe? >> >> >> >No, you posted utter nonsense ("Cardinality(x)=least ordinal(y) with >> >> >equinumerosity(z)") as if it is something that I had said. >> >> >> >> I never said you had said that. >> > >> >You're absurd. You quoted me asking you what I said that justified a >> >certain statement you made. >> >> Gee it's sure too bad the relevant citations appears to have gone with >> the wind. No doubt my fault as well. > >Gone in your one post memory span. But the post in which you posted >"Cardinality(x)=least ordinal(y) with equinumerosity(z)", in reply to >my question, hasn't been swept by any winds. You mean the question where you demand I research and justify your opinions for you, Moe? ~v~~ |