Cantor Confusion
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From: Virgil on 3 Nov 2006 19:39 In article <1162555814.977259.219580(a)f16g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > mueck...(a)rz.fh-augsburg.de wrote: > > > William Hughes schrieb: > > > > > > > Each diagonal digit is formed by a line. > > > > > > and by a column! > > > > Yes, and the number of columns is equal to the number of lines. > > The fact that no single line contains all the columns is not > > important. > > > The fact that *no* set of lines contains all the line is not important > too. > > I think we have obtained agreement, that we disagree in this point and > that further discussion will not lead to a result acceptable for both > parties. The final state is: > > 1) The number of columns is equal to the number of lines. Not necessarily. The number of "columns" can be finite if there is a uniform bound on the length of lines even when the numer of "lines" is not finite. Similarly with a finite number of lines, unless every line is of less width that the number of lines, the width of the list can be grater than its umber of lines. > > 2) Both are infinite, according to set theory this means omega or > aleph_0. > > But you believe: omega or aleph_0 is the maximum of the set of lines Wrong. It is the supremum of the numbers of the lines, each line having a separate natural number. > and the supremum of the set of columns, which, however, is not taken by > any column. You believe the different meaning is not important. Not only that, we believe it is nonexistent. > > I believe that a taken maximum and a not taken supremum are so > different (although the difference is very tiny) that set theory, > necessarily assuming this, is wrong. When one mislabels a LUB as a max, as WM does here, the one doing the mislabeling is the one who is wrong. > > The difference between maximum und supremum is, by the way, the reason > why I insist that an actually infinite set of natural numbers must > contain a non-natural number. And this is why a set of rational people does not ever contain a WM. > If we do not intermingle supremum and maximum, it does so in fact. Not in ZF or NBG. Only in WM's world.
From: Virgil on 3 Nov 2006 19:52 In article <1162559082.298879.296060(a)e3g2000cwe.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1162470378.676172.129570(a)h48g2000cwc.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > In article <1162405520.008395.100850(a)e64g2000cwd.googlegroups.com> > > > > muecke= > > > nh(a)rz.fh-augsburg.de writes: > > ... > > > > > > > > And in mathematics 1/oo is *not* defined. > > > > > > > > > > > > > > Not in mathematics. But in a theory which assumes omega to be > > > > > > > a whole number. > > > > > > > > > > > > Which theory? > > > > > > > > > > Set theory. > > > > > Cantor invented omega and defined omega as a whole number. > > > > > Who changed this standard meaning? > > > > > Why do you think this meaning was changed? > > > > > When do you think the contrary meaning became standard? > > > > > What is the contrary meaning? > > > > > Do you agree that A n: n < omega is incorrect? > > > > > If not, why do you complain about on-standard meaning on Cantor's > > > > > definition? > > > > > > > > A nice rant. > > > > > > Why don't you answer? > > > > Because it is unrelated with the question where 1/oo was defined. > > But to answer, I do not know who and when the definition of ordinal > > number was introduced. I know *why* it might have been introduced: > > to remove possible confusion. I have no idea what you mean with > > "contrary meaning". As your statement A n: n < omega does not > > indicate where n is coming from, I can not state whether it is > > correct or not. But if the natural numbers are meant, than it is > > certainly correct (if we assume the natural comparison between ordinal > > numbers and natural numbers). > > n e N was implied. In such matters, ZF does not allow quantified membership by unstated implication, but requires an explicit and pre-existing set to be for every quantification. http://en.wikipedia.org/wiki/ZFC#The_axioms 3) Axiom scheme of separation (also called the Axiom scheme of comprehension): If z is a set and is any property which may be possessed by elements x of z, then there is a subset y of z containing those x in z which possess the property. > > Again, depends on the model. > > Of course. But the usual set theory without yet specifying any models > starts from this point. > Everything is a set is true in theories like ZF, with restricted > comprehension. Unrestricted comprehension requires a universal set. And "unrestricted comprehension " also allows contradictions like Russell's paradox, which is why sensible mathematicians avoid it.
From: Virgil on 3 Nov 2006 19:58 In article <1162559784.572072.34420(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > 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). > > By "number" I understand those mathematical objects which can be > compared by size or magnitude, i.e., which observe trichotomy. I know > that division is not normally defined for transfinite ordinals, but > with omega > n we can define 1/omega < 1/n for n e N as some kind of > abbreviation. What does Wm allege it abbreviates? > > Don't forget, most things come into being by intuition, not by > formalization. But those things which cannot be formalized tend to go as fast as they come. > > It is by logic we prove, it is by intuition that we invent. (Henry > Poincar?) But most of what gets invented ends up, as it deserves to, on the scrap heap. So it what intuition invents cannot be proved by logic, the scrap heap is its logical resting place. > > When Cantor introduced omega, there was no formal definition. But logic, in that case, provided one.
From: Virgil on 3 Nov 2006 20:06 In article <1162561055.651700.229760(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > 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. > > Read this thread. > > > > What do you mean by "understood"? You concede that your logician agrees > > that the argument is not valid in ZFC. > > He does not know how to translate it. > > >So, what logical system does your > > logician say it is valid in? > > Do you really believe that ZFC is the only and ultimate system of > thinking? When someone claims that an argument is, or is not, valid in ZF, no other system is relevant in establishing or invalidating that claim. When WM claims to have found a contradiction in ZF, no other system is relevant in establishing or invalidating that claim. > > > 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. > > I would not rely on your message, I am afraid you will not notice that > ZFC is gone. As we are still using it, it is not gone yet. > > > > > 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. > > The proof has not yet been given. That is a bit different from cannot > be given. Then lets see you give it, WM. A claim here unaccompanied by a proof isn't worth the electrons used to post it.
From: Virgil on 3 Nov 2006 20:15
In article <1162561552.371410.43580(a)m7g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > 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? > > It is amazing that you don't understand the meaning of my proof. We understand the meaning, but we deny that what WM calims to be a proof is a proof. So what has been claimed is of no consequence while the alleged proof remains dubious. > I have shown that the set of paths of the binary tree has a cardinality > not less than R. Others preceded you in this proof. > I have shown that the set of edges has the cardinality of N. Others preceded you in this proof, too. > And finally I have shown that the cardinality of the set of paths is > not larger than the cardinality of the set of edges. No you have not, at least not in any system which you are willing to identify. Such identification requires a listing of all the statements which are asssumed to be true without proofs. > > In ZFC Cantor's proof is valid, according to which the cardinality of R > is larger than that of N. > > Therefore there is a false result obtained from ZFC. Or a false result obtained by WM. In the ZFC system, what is being assumed is clearly stated. E.g., at http://en.wikipedia.org/wiki/ZFC In the WM system, what is being assumed is carefully hidden. Given this sort of choice, I, for one, opt for ZFC, where one knows precisely what is going on. |