Cantor Confusion
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From: mueckenh on 2 Jan 2007 06:59 Virgil schrieb: > > You said: > > It is enough to show that for every finite tree, the given path is not > > in it, in order to show that it is not in the union. > > Since that exactly conforms to the definition of unions, yes. > An object is IN a union if and only if it is in at least one of the sets > of that union, and thus is not in the union unless it is in at least one > of the sets in the union.. The object N = {1,2,3,...} is not in the union of all initial segments {1,2,3,...,n}. > > > > I say: > > Is the union of all finite trees different from the complete infinite > > tree? > > Yes! Which nodes or edges are in one but not the other? I assume you think the union of all finite trees contains less edges or nodes. So which edge or node is in the complete tree but not in the union of all finite trees? > > > It is not what is shared by two sets of paths which separate them but > what is not shared. To separate one path from all others requires an > infinite set of its included edges and/or nodes. For any such infinite > set, any other path will excude infinitely many of those. > > This can be seen by noting that any two infinite paths can have no more > than a finite number of nodes and/or edges in common. So each contains > infinitely many not in the other. Let me know which edges of the complete tree distinguish it form the union of all finite trees (and all the rational numbers from the irrational numbers). Regards, WM
From: Franziska Neugebauer on 2 Jan 2007 09:53 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > >> > Different from what you believe is behind these "numbers". >> >> The contradiction of your answers remains in effect because what "I >> believe" is not part of your contradiction. > > What you believe comes in when we talk about the object which you call > an irrational number. Can't see that. _What_ I clearly see is that _you_ contradicted yourself. See below. >> Do you want to withdraw your answer to question 3.? >> >> >> You have to decide which version of "ideas" shall be valid. >> >> Without that decision you contradict yourself. >> > >> > Common irrational numbers are identical with my "ideas" >> >> Repetition of acknowledgement of your answer to question 2. > > No reason not to repeat it. > >> > But the common interpretation of common irrational numbers is >> > false, in particular the assertion of unlimited facility of >> > approimation. >> >> The contradiction of your answers remains in effect because "common >> interpretations" are not part of your contradiction. > > Where is a contradiction? Read the quote below. Your answers to my questions 2 and 3 constitute a contradiction. Please correct me if I'm wrong: Your answer to my question 2 states x is identical to y <-> there is no difference between x and y (P) Your answer to question 3 states x is different from y (due to some reason) <-> there is a difference between x and y (Q) For P <-> ~Q you said P & ~P which is a contradiction. >> Do you want to withdraw your answer to question 3.? > > Why? Your universe of discourse has become inconsistent. At least if we consider your answers P and Q. ,----[ <4596fc39$0$97262$892e7fe2(a)authen.yellow.readfreenews.net> ] | >>>>>> 2. Is it true that what "we" call an "irrational number" (x1, | >>>>>> x2)) is identical to your "idea"? | >>>>> | >>>>> Yes. | >>>>>> | >>>>>> 3. If so, I cannot see what a meaningful, substantial | >>>>>> difference between your "irrational idea" and the common | >>>>>> "irrational number" could be. What - besides pure terminology | >>>>>> - >> >> >> is that difference? | >>>>> | >>>>> The difference is that an idea has no digits. | >>>> | >>>> 2-3. You contradict yourself answering my question 3. that way. | >>>> As | >>>> you have agreed to in your answer to question 2. common | >>>> "irrational numbers" are identical to your "ideas". There are | >>>> two ways out: | >>>> | >>>> 1) Withdraw your answer to question 2. | >>>> 2) Withdraw your answer to question 3. | >>>> | >>>> Which do you prefer? `---- F. N. -- xyz
From: mueckenh on 2 Jan 2007 12:37 Franziska Neugebauer schrieb: > > Where is a contradiction? > > Read the quote below. Your answers to my questions 2 and 3 constitute a > contradiction. Please correct me if I'm wrong: # You are wrong. > > Your answer to my question 2 states > > x is identical to y > <-> there is no difference between x and y (P) Correct. x is an idea. It is identical to the entity which you call an irrational number, i.e., an idea y. > > Your answer to question 3 states > > x is different from y (due to some reason) > <-> there is a difference between x and y (Q) You simply fail to distinguish sharply enough between y and y'. x is different from y', i.e., from what you think an irrational number is., i.e., from an idea which can be approximated to any given precision by rational numbers. > > For P <-> ~Q you said > > P & ~P > > which is a contradiction. > > >> Do you want to withdraw your answer to question 3.? > > > > Why? > > Your universe of discourse has become inconsistent. At least if we > consider your answers P and Q. P = "What I call the idea of an irrational number is what you call an irrational number (although you don't know the characteristic of an irrational number)." Q = "The idea of an irrational number cannot be approximated up to an arbitrarily small positive epsilon (as you erroneously seem to believe)." Regards, WM
From: Franziska Neugebauer on 2 Jan 2007 14:27 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> > Where is a contradiction? >> >> Read the quote below. Your answers to my questions 2 and 3 constitute >> a contradiction. Please correct me if I'm wrong: # > > You are wrong. Just to remind you what we were originally discussing: ,----[ <1167256258.863156.191060(a)73g2000cwn.googlegroups.com> ] | > 1. So you agree that there exist two entities x1, x2 e R which solve | > the equation x^2 = 2? | | Yes. [<- this is WM's answer] | > | > 2. Is it true that what "we" call an "irrational number" (x1, x2)) | > is identical to your "idea"? | | Yes. [<- this is WM's answer] `---- This answer to question 2 is (P) ,----[ continued ] | > 3. If so, I cannot see what a meaningful, substantial difference | > between your "irrational idea" and the common "irrational number" | > could be. What - besides pure terminology - is that difference? | | The difference is that an idea has no digits. [<- this is WM's | answer] `---- This part of your answer to question 3 is (Q). Irrefutibly it states that there _is_ a difference, hence Q <-> ~P. ,----[ continued ] | And the numerical approximation of an idea like sqrt(2), written in a | list, has not enough digits to determine whether it is different from | infinitely many other numbers or numerical approximations of ideas | contained in that list. [<- this is WM's answer continued] `---- This part of your answer to question 4 we may name (Q'). Your three answers together state P & Q & Q' According to the rules of logic which still _are_ in effect a & b & c -> a & b regardless of the truth of c. Hence from stating P & Q & Q' it follows that you also state P & Q Since P <-> ~Q you are stating a contradiction: P & ~P >> Your answer to my question 2 states >> >> x is identical to y >> <-> there is no difference between x and y (P) > > Correct. x is an idea. It is identical to the entity which you call an > irrational number, i.e., an idea y. > >> Your answer to question 3 states >> >> x is different from y (due to some reason) >> <-> there is a difference between x and y (Q) > > You simply fail to distinguish sharply enough between y and y'. > x is different from y', i.e., from what you think an irrational number > is., i.e., from an idea which can be approximated to any given > precision by rational numbers. Everything that you uttered wrt approximations is already "absorbed" in Q' (see above). Truth of Q' is irrelevant for the observation that you stated the aforementioned contradiction. >> For P <-> ~Q you said >> >> P & ~P >> >> which is a contradiction. >> >> >> Do you want to withdraw your answer to question 3.? >> > >> > Why? >> >> Your universe of discourse has become inconsistent. At least if we >> consider your answers P and Q. > > P = "What I call the idea of an irrational number is what you call an > irrational number (although you don't know the characteristic of an > irrational number)." The real Q is missing: ,----[ from obve ] | The difference is that an idea has no digits. [<- this is WM's | answer] `---- > Q = "The idea of an irrational number cannot be approximated up to an > arbitrarily small positive epsilon (as you erroneously seem to > believe)." Name that part Q'. To restore the consistency*) of your universe of discourse you must refrain from P or Q. A discussion of Q' is no remedy. *) only wrt to the current contradiction. F. N. -- xyz
From: Virgil on 2 Jan 2007 16:25
In article <1167737700.201830.142800(a)n51g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: > > > > > Different from what you believe is behind these "numbers". > > > > The contradiction of your answers remains in effect because what "I > > believe" is not part of your contradiction. > > What you believe comes in when we talk about the object which you call > an irrational number. What WM believes in comes in all the time. All those claims he makes but cannot prove or even justify. |