Cantor Confusion
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From: David Marcus on 2 Nov 2006 11:08 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > In article <1162405520.008395.100850(a)e64g2000cwd.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > Where in the above quote is 1/oo defined? > > > > > > It is defined that omega (which Cantor used later instead of oo) is a > > > number larger than any natural number n. Omega is the limit ordinal > > > number. Therefore 1/omega must be a number smaller than every fraction > > > 1/n. > > > > Why? As long as 1/omega is not defined you can not talk about it. You > > simply assume that 1/omega is a number. But that is not the case, it > > is not defined. > > If omega is a number > n, then 1/omega is a number < 1/n. For all n e > N What do you mean by "number"? Normally, "omega" denotes a certain ordinal. Division is not normally defined for ordinals. If you want it to be defined, then you have to define it. So, using a plausible interpretation of your words "number" and "omega", your statement is false because 1/omega is not an ordinal (since it is not defined). -- David Marcus
From: David Marcus on 2 Nov 2006 11:21 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > > > I know of a really good logician who told me that he is unable to > > > > > translate my proof into ZFC. > > > > > > > > Hardly surprising. If he is truly a good logician, then he didn't want > > > > to tell you that you were talking gibberish. > > > > > > No, he (and he is not the only one to do so) does understand the > > > arguing but does not know how to translate it into ZF language. That is > > > all. > > > > If your "really good logician" understands the argument, have him post > > his version of it. > > I have posed for several times a version which can be understood and > has been understood by him and by other mathematicians. Please name these logicians and mathematicians. What do you mean by "understood"? You concede that your logician agrees that the argument is not valid in ZFC. So, what logical system does your logician say it is valid in? > > > > Fine. Then stop claiming that you can prove ZFC is inconsistent. If you > > > > don't like ZFC, that's your concern. But, that is completely different > > > > from saying that you have a proof within ZFC of a contradiction. > > > > > > I have a proof which shows that the real numbers have less elements > > > than a countable set. That is all. ZFC and what there can be formalized > > > does not bother me at all. > > > > If the proof cannot be given in ZFC, then it is irrelevant to whether > > ZFC is inconsistent. > > I cannot construct a horoscope. Nevertheless I can find out whether a > horoscope tells the future correctly, at least when the future has > become present time. I'm afraid I don't follow the metaphor. Mathematics is not physics. What is your conclusion? That ZFC is inconsistent or that ZFC is different from your system? The latter is certainly true and you have failed to show the former. > > So, please desist from making such a spurious claim. > > > > If you can prove that the real numbers are both countable and > > uncountable, then that simply shows that the system that you are using > > for your proofs is inconsistent. It says nothing about the real numbers > > and countability as defined in ZFC. > > The countability of R is not proven in "my system" but in general > mathematics which is valid prior to ZFC and will remain so after ZFC has > gone. Since ZFC is here now, if you wish to publish your proof in a math journal, you will need to wait until ZFC is gone. We'll let you know when that happens. > The proof of uncountability of R therefore shows ZFC is false. The statement "ZFC is false" is meaningless. All you are saying is that your proof cannot be given in ZFC. We all already know that and agree with it. If that is all you want to say, then we can stop now. > > > The maximum of a set of finite numbers which has no maximum is simply > > > not present. It is *not* an infinite number which is larger than any > > > finite number, because a maximum must belong to the set. > > > > If by "maximum" you mean the largest element and by "finite numbers" you > > mean natural numbers, then it is true (in ZFC and standard mathematics) > > that an unbounded set of natural numbers does not have a maximum. > > Correct. > > > > > And a supremum not belonging to the elements of the set does not yield > > > a diagonal digit. > > > > No clue what this is supposed to mean. Feel free to rephrase using > > common mathematical terminology and/or define what a "diagonal digit" > > is. > > A supremum is a least upper bound of a set (like the set of columns of > a list). If there is no maximum, then the supremum is not taken, i.e., > it does not belong to the set. It is true that if there is no maximum, but there is a sup, then the sup is not an element of the set. So what? That doesn't prove anything about the diagonal. -- David Marcus
From: David Marcus on 2 Nov 2006 11:30 mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > Any "diagonal" for this function will have to be of length greater than > every n in N. > > This shows the inconsistency of ZF and NBG. It is amazing that you keep repeating claims after you yourself admit that you have not shown the claim. Is this intentional or don't you realize that you are contradicting yourself? The word "inconsistent" has a technical meaning in logic. The only way to show that ZF is inconsistent is to show a proof within ZF of a statement and its negation. You yourself said that your argument cannot be given in ZF. Hence, you have not shown that ZF is inconsistent. At best, all you can state is that ZF does not agree with the system that you use for your proofs. Everyone already knows this. -- David Marcus
From: David Marcus on 2 Nov 2006 11:35 mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > The diagonal is an infinite sequence. So the diagonal is longer than > > > any of the finite sequences. But the diagonal consists of elements of > > > the finite sequences. So it cannot be longer than the maximum of the > > > finite sequences. If this maximum does not exist, you cannot take the > > > supremum omega for it, because the supremum is not a member of the > > > sequences and does not supply elements of the diagonal. > > > > Let's try a simpler problem. Consider the following list. > > I would hesitate to call it a list, because injectivity is lacking. I've never seen injectivity required for a list. Usually, people use "list" to mean sequence. If your lists require injectivity, you should explicitly state that. People can't read your mind. > But that is not important. > > > > 1 > > 1 > > 1 > > ... > > > > In other words, consider the sequence x where x(n) = 1 for n a natural > > number. How long is this sequence? > > You expect the answer omega. I expect nothing. > No problem. There may be omega elements, the first, the second, and so on. > > The diagonal of that list is very short. It is 1. I didn't say anything about a diagonal. I asked, "How long is this sequence"? I.e., how long is the sequence x? Please try to answer the question I ask, not some other question. -- David Marcus
From: Lester Zick on 2 Nov 2006 12:37
On Thu, 2 Nov 2006 00:28:49 -0500, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: >Dik T. Winter wrote: >> In article <sjnfk256m5tau0bmk5u4vjdg53al1f2sc9(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: >> > Yours are some of the very >> > few that don't. And I've seen posts of other .nl correspondents which >> > come through in color. >> >> It has nothing to do with the domain where you come from. It has everything >> to do with the manner things are quoted and in the way *your* newsreader is >> able to handle that. I know the reason your newsreader does not show colours. But my newsreader does show colors. >> It is because I prepend the '>' sign with a space when quoting. Ok. Mine reacts the same way. But when there is no preceeding space quoted text is shown in a different color. > And I >> have pretty good reasons to do that. (One of the reasons being that this >> article would be rejected because there is more quotation than new text.) I've never experienced that. >My newsreader lets me post articles that have more quotation than new >text. ~v~~ |