An uncountable countable set
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From: Virgil on 24 Jul 2006 15:21 In article <1153750477.050796.204330(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1153674914.649461.71180(a)m73g2000cwd.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > In article <1153478871.303029.5360(a)i3g2000cwc.googlegroups.com> > > > > mueckenh(a)rz.fh-augsburg.de writes: > > > > > But it is obvious for *all* *finite* *unary* number. And only > > > > > finite > > > > > unary numbers are involved in the second list below: > > > > > > > > > > The *- sum of > > > > > > > > > > 0.1 > > > > > 0.11 > > > > > 0.111 > > > > > ... > > > > > 0.111... > > > > > > > > 0.111... is not a finite unary number. Rather, I would say it is not > > > > a > > > > unary number at all. > > > > > > 0.111.. is not a terminating rational number. > > > > Eh, can we please keep to the subject. You state you have a list > > that only finite numbers are involved in the "list" above. I state > > that 0.111... is not a finite unary number. Pray show that > > 0.111... is a finite unary number. (I neither have stated that it > > is a terminating rational number, because it is not.) > > The pure list stretches only from top to the three ... In which case your "*+" operation produces only 0.111, not 0.111... . The last > element does not beong to it. My reply concerned your statement that > 0.111... was not a unary number at all and the difference between > decimal and unary interpretation you could perhaps try to derive from > your observation. Since "mueckenh" uses the same notation for both unary natural/omega and decimal fractions, and seems to switch meanings at his own whim, we reserve the right to do the same. So that when WE look at the members of your sequence or its limit, they means only what WE intend them to mean, no more and no less, and any other interpretation "mueckenh" tries to put on our meanings is wrong. We can play his Humpty-Dumpty game too.
From: Virgil on 24 Jul 2006 15:28 In article <1153751354.788807.293880(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Couldn't it be that those who > designed modern set theory erred as much as those who studied his > original papers without noting this error? Given the present choice between believing many profound and very clever mathematicians are all making the same mathematical error and believing one shallow and not very clever non-mathematician is making a mathematical error, most of us will wisely side with the many.
From: Virgil on 24 Jul 2006 15:54 In article <1153753292.535535.35240(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > > Not necessary. I confirm: It makes no sense because omega makes no > > > sense. > > > > In what way makes the definition above no sense? I asked you before and > > now you finally reply that that is because omega makes no sense. While > > I do not understand that at all, this would mean that you think that all > > limits of the form lim{n -> oo} 1/n = 0 make no sense. > > I am convinced that infinity makes sense only in the form of potential > infinity. There are many who are convinced that, at least until some actual contradiction is found in ZF or NBG, they are safe to use. I am one of them. And I do not choose to limit my options merely to assuage the prejudices of "Mueckenh". > I think that my "list" has proved that (among other proofs). And everyone else thinks otherwise. In particular that your alleged "proofs" prove nothing. > You must insist that a simple law that is valid for all finite numbers > *suddenly* becomes invalid, if the infinite case is encountered. I will > not follow you into this realm. The simple "law" that prohibits a set from being injectable into a proper subset holds for all finite sets but *suddenly* becomes invalid if the infinite case is encountered. If there were no differences between finiteness and infiniteness of sets, there could be no "laws" which hold for one type but not the other. But as soon as there is any difference, there MUST be "laws" which hold for one but not the other. > Do you believe that there are actually infinitely many finite numbers? If "number" means cardinal or ordinal within some standard axiom system, then "yes". > Why do you not assert there were finitely many infinite numbers? The > logic would be the same. There are at least finitely many infinite ordinals in ZF or NBG. But absent some axiom system in which to define them, I do not know what you mean by "number". > > Mathematics cannot be done without matter. For thinking, speaking, > writing matter is required. Much matter allows much mathematics, less > matter allows less mathematics, no mater allows no mathematics. That is where we differ. To me axiomatic mathematics is entirely mental. While the physical world has certainly suggested the directions in which mathematics has developed, those developments are themselves not a part of the physical world, but only of the mental world. The patterns humans see in nature are imposed by the humans seeing them and are not inherent in nature itself. The observer is a part of what is observed. > It is easy to show that the number of infinite paths is countable too, > by he fact that each path carries two edges as its load. If "mueckenh" claims it is so easy to show, I challenge "mueckenh" to establish a bijection, or even a surjection, from the set of naturals (or finite ordinals) to the set of all maximal paths in an infinite binary tree. Since I and others have already produced several times bijections between the power set of the naturals and the set of all such paths, "mueckenh" would in effect have to produce a bijection between a set and its power set.
From: Virgil on 24 Jul 2006 15:58 In article <1153753753.367949.81300(a)75g2000cwc.googlegroups.com>, 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. > > 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". > One may doubt whether this example really illustrates the abyss between > finiteness and actual infinity. > 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. But actual infinity exists now, at least according to the quotes above. So that "mueckenh"'s own citations contradict his own thesis. Like an engineer hoist with his own petard!
From: Virgil on 24 Jul 2006 16:00
In article <1153763693.433966.220360(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > 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 Ther are none so blind as "mueckenh" who will not see that even his own citations give him the lie. |