Cantor Confusion
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From: Virgil on 25 Oct 2006 21:25 In article <1161806304.579580.247890(a)m73g2000cwd.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: > > > 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 the fact that it does not exist and that we > > > cannot have any infinity unless we have the axiom of infinity. Yes, > > > some very naive (formalized?) opinion. > > > > We need something to talk about before we can talk about it. If you > > But we need certainly nothing infinite, because all our talk is finite. > > > > want, you can call the "background sets" not "sets" but "gleeb" to avoid > > misunderstandings. > > Of course these entities are no sets in the sense of set theory. > > > And, please, do educate yourself on what you are > > talking about. > > Do you in fact consider this the best way of hiding your "no-ledge". > > 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, 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. > > Regards, WM "Mueckenh" is apparently one who cannot think for himself, but relies solely on the authority of others.
From: Virgil on 25 Oct 2006 21:26 In article <1161806391.469976.252900(a)m73g2000cwd.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: > > > 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! > > One cannot get an infinite set without the axiom of infinity. That is > fact! One cannot refute the possibility of infinite sets without an "axiom" explicitly denying them.
From: MoeBlee on 25 Oct 2006 21:29 Lester Zick wrote: > So then maybe you can give us a mathematically exhaustive definition > of "the" and "with"? I can put it all in symbols drawn from these: One symbol that we read as 'for all'. One symbol that we read as 'not both'. One symbol that we read as 'is a member of'. And denumerably many variables, though only a finite number of them are needed at any given point. The words 'for all', 'not both', 'is a member of' are nicknames and, though they are very strongly connected with notions about mathematics, but they are not associated on the specific and exact venue of syntax. Then there is a formal semantics also. > After all the original objective as I recall was to > define cardinality and not cardinal(x). No, I don't offer a definition of 'cardinality' by itself, nor does 'card(x)' stand for cardinal(x). Rather, 'card(x) stands for 'cardinality(x)', which is to say, 'the cardinality of x'. card(x) is the cardinality of x. So the definition is: The cardinality of x = the least ordinal equinumerous with x. And in the actual formal language, the English nickname 'card' would not appear, but rather a single 1-place operation symbol (say, 'C') would be there (also the parenthesis would not be required, and since '=' is a 2-place predicate symbol, officially, it goes to the front of the formula). And, of course, the English 'the least ordinal equinumerous with' would be replaced by strict formal symbols too. > I mean the only reason I can > think of to specify the particular x Again, we do not specify a particular x. The variable 'x' ranges over the entire domain of discourse. The definition is saying in effect, "Whatever x is, its cardinality is the least ordinal that is equinumerous with it." That is, for any object, we want to know what we are saying when we refer to the cardinality of that object. So x stands for any object, and we are saying that the cardinality of that object is the least ordinal that is equinumerous with that object. This is not arcane or exotic as a definitional form. It's no more arcane or exotic than Algebra 1 in freshman high school where we give such definitions as 'the square root of x = the number that when multiplied by itself equals x". MoeBlee
From: Virgil on 25 Oct 2006 21:30 In article <1161806870.567125.312870(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > There are many books on binary trees and many others on geometric > series. I put these topics together. That is new and not everybody can > understand it immediately. You must not be sad on that behalf. Follow > just my discussion with those guys who have understood this > comparatively simple matter. At the end, you may get it too. May God protect me from such evil. > > > > Or, he could continue to do as Humpty Dumpty did and use his own > > > meanings for words without telling people what they are. > > You must not think that everything you cannot understand is > ununderstandable. It is just some new idea. Otherwise it would not be > so interesting. You prefer to learn old knowledge from your old books > with their old trodden paths? My paths are new! And go nowhere.
From: Virgil on 25 Oct 2006 21:37
In article <1161807131.665622.111610(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1161683454.085728.22020(a)m73g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > > > > Therefore we write the limit. The limit of all n is omega. > > > > By what definition of "limit"? > > > > > But the *facts* are based on the number of transactions. The trick with > > > the vase is simply to force the infinite number of transaction into a > > > finite time interval. > > > > The only relevant question is "According to the rules set up in the > > vase problem, is each ball which is inserted into the vase before noon > > also removed from the vase before noon?" > > > > An affirmative answer confirms that the vase is empty at noon. > > A negative answer directly violates the conditions of the problem. > > > > How does Mueckenheim answer? > > You see my name written above. My first name is Wolfgang, my short sign > is WM. Unless you switch to one of these alternatives, this is my last > answer to you - be sure. > > All the balls have been removed before noon. > But more balls are in the vase. Where do those ephemeral balls materialize from? > Conclusion: It is nonsense to talk about the complete set N and about > any infinite set, no matter, whether it is done in English or in any > formal language. It is only WM's version of nonsense. To everyone else it is more sensible that WM's convoluted nonsense like " All the balls have been removed before noon. But more balls are in the vase." > > I understand that many of those mathematicians who spent years to > study set theory, are upset because of the wasted time, and tend to bad > behaviour. Nowhere nearly as bad as WM's claiming " All the balls have been removed before noon. But more balls are in the vase." > > > > Come on, real numbers do exist. > > > > >Where? > > > >The same place that naturals, ordinals and rationals exist, in the mind. > > >And nowhere else. > > Correct. And therefore no such thing can exist unless it exists in the > mind. But we know that there is no well order of the reals. in any > mind, because it is proven non-definable. It is perfectly definable, and perfecty defined, it is merely incapable of being instanciated. > Nevertheless some cranks > insist, it would exist somewhere. Do you know where? In minds less handicapped that WM's, |