An uncountable countable set
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From: Virgil on 13 Jul 2006 13:57 In article <1152785777.506058.199660(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > David Hartley schrieb: > > > The transpositions are relevant in so far as WM claims that Cantor wrote > > that a sequence of transpositions can not change the order-type. Either > > Cantor was wrong (or using a different idea of the limit), or WM is > > misinterpreting him, my German is not good enough to tell. > > Daraus folgt, da? solche Umformungen einer wohlgeordneten Menge die > Anzahl derselben unge?ndert lassen, welche sich auf eine endliche oder > unendliche Folge von Transpositionen von je zwei Elementen > zur?ckf?hren lassen, ... > > It follows that only such transformations of a well-ordered set leave > its (ordinal) number unchanged, which can be derived from a finite or > infinite sequence of transpositions of each two elements, ... From which, "mueckenh" concludes that such a sequence can convert a well ordered set into a set which is simultaneously well ordered and not well ordered. > > > > In any case, > > it's a proposition WM needs to prove to complete his argument, and he > > hasn't done so. > > What should that be good for? > > In the binary tree everyone can see that no path can split without two > additional edges. So it is by no means possible that there were more > paths than edges. As long as paths all end. > Nevertheless the set theorists go through eyes wide > shut and do not care. Set theorists see that the set of infinite binary paths can be bijected with the power set of the set of edges of those paths. That this result differs from that for finite trees may seem anomalous, but it is in no way contradictory. Infinite is different than finite in many surprising ways. > > Who would accept logic arguments if set theory was concerned? > > Regards, WM
From: Virgil on 13 Jul 2006 14:12 In article <1152787358.727132.194480(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > In article <1152735445.836339.213400(a)i42g2000cwa.googlegroups.com> > > mueckenh(a)rz.fh-augsburg.de writes: > > ... > > > The first well-order taken from Cantor's. The others remain well-orders > > > because: wenn in einer wohlgeordneten Menge irgend zwei Elemente m und > > > m' ihre Pl?tze in der gesamten Rangordnung wechseln, so wird dadurch > > > der Typus nicht ver?ndert, also auch nicht die "Anzahl" oder > > > "Ordnungszahl". Daraus folgt, da? solche Umformungen einer > > > wohlgeordneten Menge die Anzahl derselben unge?ndert lassen, welche > > > sich auf eine endliche oder unendliche Folge von Transpositionen von je > > > zwei Elementen zur?ckf?hren lassen, d. h. alle solche ?nderungen, > > > welche durch Permutation der Elemente entstehen. > > > > I translate (not needed for Franziska, but needed for others): > > When in a well-ordered set any two elements m and m' change place in > > the common order, the order-type will not change, so also the ordinal > > number will not change. From that it follows that those transformations > > of a well-ordered set leave the ordinal number unchanged, that can be > > written as a finite or infinite sequence of transpositions of two > > elements, that is, all such changes that emerge through permutations > > of elements. > > > > I see a problem here: > > > > It is not clear what the meaning is of the words "transformations" > > ("Umformungen") and "permutations" ("Permutation"). I think, it is clear > > that it does not include *any* re-ordering, although many of them can be > > written as an infinite sequence of transpositions, while some those *do* > > change the ordinal number. > > These words are uninteresting. Transpositions or exchanges of two > elements are clearly defined. Infinitely many are admitted by Cantor. > > > > The reordering of the naturals (0, 1, 2, ...) to (1, 2, 3, ..., 0) can > > be written as an infinite sequence of transpositions, but it changes the > > ordinal number. But is it a permutation? > > Unimportant for Cantor's statement. That must mean that "meuckenh" does not see any sequence of permutataions that can produce that result. but even if one finds such a sequence of transpostions, the result is still well ordered, though of a different order type, that of the successor of N. > There is no transposition which > could change the ordinal number. Only the infinite process does what > cannot be done by any finite set of transpositions. This shows that > infinite processes and infinity is doubtful. My example with |Q shows > that infinity is impossible. To those who believe it is impossible, perhaps it is, but to those who believe otherwise, it is not. > > I think the interpretation by WM is wrong. > > I know your interpretation is wrong. It ain't what you don't know that harms you, it's what you "know" that ain't so. Mark Twain.
From: Virgil on 13 Jul 2006 14:16 In article <1152787731.201543.289190(a)35g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > In "usual" mathematics we have 10 * 0.999... = 9.999... and from that > we get easily 0.999... = 1. > > I argue that 0.111... does not exist. When I will have succeded, also > 0.999... will be abolished. But that has not yet been generally > accepted. OK? As "mueckenh" will never succeed in convincing the majority of mathematicians to believe what they do not believe, 0.111... and 0.999... will both "exist" forever.
From: Virgil on 13 Jul 2006 14:39 In article <1152788625.612288.116290(a)i42g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > The axiom says "there is an infinite set". It does not say that > > > 0.111... does belong to that set, in particular because all numbers > > > which in fact do belong to the set are different from 0.111... . > > > > You are using the word set with two different meanings in the same > > sentence. > > The set of naturals shold be the same set as the set of indices of > 0.111... . But it cannot. Maybe not in "mueckenh"'s world but it works fine in mine. if "mueckenh" chooses to exile himself in such a way from the mainstream, he should not be surprised when the mainstream elects not to join him in his self-exile. > > There is a specific definition that the set of all naturals has a > cardinal number larger than any element of the set. This definiion is > wrong. Not in Zf or ZFC or NBG. "Mueckkenh" must make some assumptions or he cannot conclude anything. But absent a statement of anything of his assumptions except for his distaste for infinite sets, we have no way to test the validity of his conclusions. > I show this by the fact that 0.111... is not in the set of unary > reprsetations of all natural numbers. Neither is 2. So what? > Aleph_0, the number of 1's in > 0.111... is *not* the cardinal number of |N. Really? I seem to see a trivial bijection between the numbers of 1's preceding any of the 1's in 0.111... and the members of N, which bijextion disproves "mueckenh"'s claim. > Te assumption that all digits of 0.111... can be inexed by natural > numbers while 0.111... has the property to to be outside of the list. But none of 0.111,,,'s digits need go unindexed. One can, in fact, use the members of the endless well ordered set {0.1, 0.11, 0.111, ...} to index the digits of 0.111... . > > K cannot have infinitely many digits if all can be indexed by finite > numbers. Assumptions made without proof, such as the above, must either be axioms or are of no consquence. > > Whatever p you choose, there is always a place in 0.111... which cannot > be indexed. Therefore there is a place which cannot be indexed in > principle. Whatever finite natural one picks, there is always a larger one, but how does that prohibit existence of a totality of all of them? Only if one ->assumes<- some axiom or set of axioms justifying such a prohibition. So that "mueckenh"'s views are quite as artificial as ours, in that he makes unjustifiable asssumptions about what one may imagine and then claims they are justifiable. When we make assumptions, we merely call them axioms and, until someone deduces a conflict between axioms, we go on from there. > > You know that one list number is not capable of indexing 0.111... , bu > you assume that 0.111... can be indexed completely. Our axiom systems provide adequate mechanisms. "Mueckenh"'s objections are only valid in a different axiom system which has no more ultimate justification than our own.
From: Virgil on 13 Jul 2006 15:16
In article <1152790284.160766.50890(a)s13g2000cwa.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > > >That the transpositions are > > > conditional does not disturb the proof. > > > > It does mean that they must be performed in order though. > > > > And that does distrub the proof. > > Why should it? Do you dislike order? I dislike "meuckenh"'s sort of disorder. > Cantor's list is ordered, the set N is ordered. Do they not exist for > this sake? is there any hint why order should destroy infinity? Is there any hint of how that question is relevant to anything? "Mueckenh" is trying vainly to link his necessarily sequential process of applying a sequence conditional transpositions to a list to the application of the Cantor rule simultaneoulsy to all members of a list. > > > > > Cantor's diagonal argument is > > > also conditional. > > > > But there are no conditions on any digit which depend on any other digit > > having been calculated previously, i.e., no sequential conditions. Thus > > one can have a general rule that does each digit without reference to any > > others. so is valid for all in one rule and one step. > > In order to identify any digit, you must find it. That requires > counting. That requires order. If one already has a completed N, as we have, we can prove that the Cantor rule creates a number which is simultaneously different from each member of any given list. "Mueckenh" cannot show that there is any "Limit" order after his process has proceeded forever that is anything other than well ordered. > > > > > That the conditions have to be executed one after the > > > other does not disturb the proof. > > > > It means that the"mueckenh" process, as described, can never end. > > > This is a set and it exists as well as any other infinite set. Not more > and not less. What set? "Mueckenh" has a whole sequence of differently ordered sequences with none of them entirely ordered by magnitude. |