An uncountable countable set
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From: Randy Poe on 24 Jul 2006 12:29 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > Just that is in question. More precisely: Are there actually or only > > > potentially infinitely many numbers. > > > > With respect to what axiom system are you asking this? > > > > Or do you maintain that there is some sort of absolute truth in > > mathematics that does not require any assumptions at all? > > Of course. Real space is Euclidean or non-Euclidian irrespective of the > axioms which mathematicians prefer. Aithmetics is finite or infinite > irrespective of the axioms which mathematicians prefer. You mean there's something called "arithmetic" which exists outside mathematics? > Do you really think reality would care if you set up axioms? I think "arithmetic" is the application of abstract operations to abstract symbols, whose rules are defined by axioms. What "arithmetic reality" are you talking about which is outside mathematics? - Randy
From: mueckenh on 24 Jul 2006 13:54 Franziska Neugebauer schrieb: > Dear Wolfgang! > > mueckenh(a)rz.fh-augsburg.de wrote: > > > Franziska Neugebauer schrieb: > > > >> mueckenh(a)rz.fh-augsburg.de wrote: > >> > >> > > >> > Dik T. Winter schrieb: > >> > > >> >> With the axiom of infinity there are infinitely many finite > >> >> numbers. And I still do not know the exact difference between > >> >> actual infinite > >> >> and potential infinite. But that is (in my opinion) something > >> >> that is best left to philosphy. > >> > > >> > In set theory "infinity" means "actual infinity". Potential > >> > infinity can neither be counted nor be surpassed. > >> > >> _In_ set theory "actual" or "potential" have no meaning. You produce > >> meaningless verbiage. > > > > Obviously you have not the foggiest idea about set theory. > > I'm not keen on ancient set theory, that's true. > > > Here are few quots from many: Fraenkel, Abraham A., Levy, Azriel: > > "Abstract Set Theory" (1976) > > p. 6 the set of all integers is infinite (infinitely comprehensive) in > > a sense which is "actual" (proper) and not "potential". > > 1. Page 6 (six)! What's the name of the Chapter? "Preface"? > 2. "in a sence". You know what that means? An eidetic elucidation. > 3. What this sentence states is: omega exists. > 4. Where do Fraenkel/Levy use (!) the aforementioned terms > _mathematically_? In set theory. It is full of the actual infinite. Set theory is theory of the transfinite. Transfinite is actually beyond the finite. Potentially infinite means always finite but without border. > > > One may doubt whether this example really illustrates the abyss > > between finiteness and actual infinity. > > One may also doubt that you understood my objection. _In_ (not in the > elucidation of) set theory there is no meaning of "actual" or > "potential". So you have no clue of set theory. I suspected that. Consider at the begining for instance the axiom of infinity or the cardinal number of the reals which is larger than the infinite cardinal numbe of the naturals. > > > p. 240 Thus the conquest of actual infinity may be considered an > > expansion of our scientific horizon no less revolutionary than the > > Copernican system or than the theory of relativity, or even of quantum > > and nuclear physics. > > > > Potential infinity existed already long before set theory. > > Where is the mathematical use of the aforementioned terms? In set theory! Regards, WM
From: mueckenh on 24 Jul 2006 14:01 Randy Poe schrieb: > mueckenh(a)rz.fh-augsburg.de wrote: > > Virgil schrieb: > > > > > > > > Just that is in question. More precisely: Are there actually or only > > > > potentially infinitely many numbers. > > > > > > With respect to what axiom system are you asking this? > > > > > > Or do you maintain that there is some sort of absolute truth in > > > mathematics that does not require any assumptions at all? > > > > Of course. Real space is Euclidean or non-Euclidian irrespective of the > > axioms which mathematicians prefer. Aithmetics is finite or infinite > > irrespective of the axioms which mathematicians prefer. > > You mean there's something called "arithmetic" which > exists outside mathematics? It is the core of mathematics. > > > Do you really think reality would care if you set up axioms? > > I think "arithmetic" is the application of abstract operations > to abstract symbols, whose rules are defined by axioms. There are no "abstract" symbols. Every symbol has a representation, at least in your mind. Otherwise it does not exist. And if you add two and two symbols, you get four, at least in that arithmetics which handles concrete representations. > > What "arithmetic reality" are you talking about which is > outside mathematics? > Mathematics does not exist other than in reality. Therefore it *is* reality (like everything which really *is*). There are only some persons who think that they need no reality for thinking. But that is thoughtless thinking. Regards, WM
From: Virgil on 24 Jul 2006 15:09 In article <1153748034.871369.133580(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > > Just that is in question. More precisely: Are there actually or only > > > potentially infinitely many numbers. > > > > With respect to what axiom system are you asking this? > > > > Or do you maintain that there is some sort of absolute truth in > > mathematics that does not require any assumptions at all? > > Of course. Real space is Euclidean or non-Euclidian irrespective of the > axioms which mathematicians prefer. It may be that our perception of "real" space is compelled to be , at least locally, Euclidean, but perception is not reality. > Aithmetics is finite or infinite > irrespective of the axioms which mathematicians prefer. Do you really > think reality would care if you set up axioms? Does "meuckenh" think that that part of our imaginings we call artithmetic has any reality outside our heads? We impose patterns on reality which only exist in our own imaginations, and are no part of reality. Those patterns which we see" area a part of us, not a part of what is outside us. > > > > > > The proof is given by the list: > > > > > > 1 0.1 > > > 2 0.11 > > > 3 0.111 > > > ... ... > > > w 0.111... > > > > That is not a list as presented. > > The upper part is a list. The last line is the LUB. All together ths > may be called an "extended list" It may equally well be called a non-list or a hulaballoo. > To spell it out clearly: 1/9 is the supremum of many sets, including > the list above, the set {1/9} and the set {1/10, 1/9}. But it is not > expected to count the elements of the sets to which it is supremum. But "muecknh"'s rule for "*+" has valid interpretation as the LUB of a set of ordinals, when he interprets his "list" as a list of ordinals, and then the LUB of the list is omega = "0.111..." > > omega, however, is, according to Cantor, the number of elements of N. > It can be put in order with other ordinals like the naturals and omega > + n, to name only few. This is contradicted because we see from the > above extended list, that the number of elements of N is *not* yet > omega. The ordinality of N is called omega despite any idiotic refusal by "Mueckenh" to acknowledge that truth. > It is less than omega unless omega appeared twice in the order > of ordinals. Then is 1 not 1 unless it appears trice in the list of ordinals? Makes as much sense. > > > > Again you forgot the special from of unary representations. Is the > > > special property of unary representation so difficult to see, so hard > > > to remember? > > > > Properties of sets of numbers are not dependent on any representational > > scheme. > > But their recognition depends on the representation. When I realized > that the set of naturals cannot be counted by omega, It ain't what "mueckenh" doesn't know that cripples him, it's what he "knows" that ain't so. With apologies to Twain. > I did not yet have > this lucid and plausible scheme. > > Regards, WM We frequently note your lack of lucidity and plausibility.
From: Virgil on 24 Jul 2006 15:11
In article <1153750179.424604.45850(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > With the axiom of infinity there are infinitely many finite numbers. > > And I still do not know the exact difference between actual infinite > > and potential infinite. But that is (in my opinion) something that is > > best left to philosphy. > > In set theory "infinity" means "actual infinity". Potential infinity > can neither be counted nor be surpassed. And need not be considered at all, at least not in set theory, nor any other part of mathematics. |