An uncountable countable set
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From: Virgil on 18 Aug 2006 03:41 In article <1155885684.223459.232190(a)74g2000cwt.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > > But it is known that this set is countable. Is the set of all subsets of a countably infinite set itself countable?
From: Virgil on 18 Aug 2006 03:52 In article <1155885821.815144.187270(a)m73g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > No infinite list (of independent numbers) is constructible, because no > one can construct infinitely many different real numbers. This argument > shows that the whole Cantor diagonal proof is void. In what axiom system does "Mueckenh" claim that this is the case? Unless we know what he has assumed, we cannot accept his conclusions.
From: Virgil on 18 Aug 2006 03:54 In article <1155886069.568472.204170(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > You are in error. You just proved it to be true. The set of natural > numbers (i.e., finite numbers n, i.e., numbers with finitely many > differences of 1 between 1 and n) does not yield infinitely many > differences of 1. It does in ZF or NBG. What axiom system is a"Mueckenh" assuming?
From: mueckenh on 18 Aug 2006 07:11 Dik T. Winter schrieb: > > Now you may claim, that the mathematics of finity does not hold in > > infinity, but that there everything is different. But that is not > > mathematics in my opinion. May be that it is mathematics in some one > > other's opinion. Nevertheless it is not useful to continue the > > discussion on this topic between us. > > I do not only *claim* that 0.111... can be indexed by the natural numbers; > I can *prove* it, as I have done again and again. You supposed that all digits of 0.111... could be indexed, and than you proved that they can be indexed. > The only things you > are doing is handwaving and circular reasoning. Indeed, the discussion > is useless if you are not able to formulate definitions and proofs. I have formulated the definition according to which my list is constructed. You try to convince us that "there is no n such that 0.111... can be indexed by the numbers 1 to n. This does *not* mean that 0.111... can not be indexed by the complete list." And that is simply nonsense. All finite digit positions are contained in the list, most several times, but every one at least one time. > > > > Each finite segment can be > > > covered, and each finite index can be covered. But the total is not > > > finite, but contains only finite digit positions. The digit positions constitute the number. You cannot build a house of bricks and claim the total house is wooden. > So, in your opinion the axiom of infinity is purest nonsense. No. It leads to nonsense in the framework of mathematics, for instance in contradicts the theorem that 0.111... is not a member of the sequence of sequences of 1's. >You have > a right to opiniate that. But do *not* claim that you have found an > inconsistency with the axiom of infinity, because you have not. Otherwise I had to accept your handwaving argument that all list sequences could index a sequence which does not belong to the list. Why should I? > > > > > Because your assertion that K could be completely indexed and covered > > > > is equivalent to K being in the true list. > > > > > > My assertion is that K can be completely indexed but not completely > > > covered. Again you misquote my assertion. > > > > Indexing n implies covering all digit positions m =< n. Indexing all n > > implies covering all m =< n. > > What is that 'n' in the last equation? It can not be the 'n' in 'all n' > because that is an arbitrary 'n'. n e |N is a whole number. All n e |N are whole numbers. > > > > Do you agree that every natural number which can index a digit position > > is in the list? > > Yes. > > > If a number is not in the list, what does this fact say about its digit > > positions? > > Nothing. If all digit positions are natural numbers, they can all be > indexed by numbers from the list. That is correct. But all numbers with those digit positions are already in the list. All finite digit positions are contained in the list, most several times, but every one at least one time. > But in that case the number of digit > positions is infinite, as there are infinitely many natural numbers. So > the number is not in the list because the list contains only numbers > with finitely many digit positions. > > > > > It is obvious that an infinite > > > > number of differences of 1 leads to an infinite number. > > > > > > Yes. When you can add all those numbers. But the result is not a natural > > > number. > > > > Exactly. Therefore there cannot exist an infinite number of natural > > numbers. > > Why the 'therefore'? Can you provide a proof? Because the sum of infinitely many differences of 1 would make up an infinite number. > > > > > Why should a set which is not finite be actually infinite (i.e. have a > > > > cardinal number)? > > > > > > A set that is not finite is infinite, that is the definition of the word > > > infinite. > > > > No. Every (potentially) infinite set is finite. > > That is quite non-standard terminology. In standard terminology something > that is not finite is infinite and the reverse. However, I have no idea > what a 'potentially infinite set' is. Can you provide a definition? Peano axioms, inductive set. If there is n, then there is n+1. But this does not mean that an infinite set does actually "exist". > > And all extremely false too. > > Elementary mathematics with equivalence classes is false? What is false > about it? The assumption that an infinite set had a cardinal number. The assumption that the limit omega would exist. > > > > > You may deny this, nevertheless it is true. If there is an actual > > > > number of steps, then the width is equal to the lenghts is aleph_0. > > > > > > You keep asserting that. > > > > If you say the length is infinite but the width is not infinite, then > > simply exchange the axes of size of numbers and number of numbers. You > > will easily see your error. > > *My* error? Where did I ever state that? Already before 8 august I have > written an article where I wrote: > > Eh? Where do you conclude *that* from? The triangle gets infinitely long > > and infinitely broad. Your conclusion is based on something unstated. > Please keep the record correct. If it gets infinitely broad, then there must be an infinite number. That is an equivalence, because nothing else than a number could cause the infinite width. As you deny the infinity of any number you deny the infinity of the width of the triangle. > > > > > > > > What I do not understand is how you can believe that someone (except, > > > > perhaps, Virgil) could share your opinon that only in one dimension the > > > > infinity was actually realized, but not in the other. > > > > > > Just because there is no last line. If there were a last line, it would > > > have width aleph-0, but there is no last line. > > > > And the line next to the last line, > > This is uttering nonsense. There is also no line with number 1/2, and no > line with number 3/2, etc. Because the lines are enumerated by positive whole numbers. Regards, WM
From: mueckenh on 18 Aug 2006 07:15
Dik T. Winter schrieb: > In article <1155817978.033024.136310(a)m79g2000cwm.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > > > All digits can be covered, but there is no n that covers all digits. > > > > What is the result of your assertion? > > There is a list number which covers only the first digits of 0.111..., > > while another covers not the first digits but some more behind. > > That is nonsense. I know. I told you several times. > Each finite initial segment can be covered by a natural > number. The complete number can not be covered by a natural number. Hence it cannot be indexed. > > > > You showed it for digit positions only which can be indexed by natural > > > > numbers. > > > > > > But *all* digit positions can only be indexed by natural numbers. > > > There are no digit positions that can not be indexed by natural numbers. > > > > These positions are all in the list - by definition. > > 0.111... is not in the list by mathematical proof. > > Yes. But why should it be in the list? It is not an index position as it > is not a natural number. But you pretend it would consist of index positions which are all in the list. Now find out how that can be: All index positions of 0.111... are in the list, but not in the form of 0.111... . In which form should they exist there? Regards, WM |