An uncountable countable set
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From: mueckenh on 8 Aug 2006 16:42 Dik T. Winter schrieb: > In article <1154968120.846531.209650(a)75g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > In article <1154722892.397252.6590(a)b28g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > > Dik T. Winter schrieb: > > > ... > > > > > > So the triangle gets infinitely long but not infinitely broad. This > > > > > > means the Diagonal must bend. > > > > > > > > > > Eh? Where do you conclude *that* from? The triangle gets infinitely long > > > > > and infinitely broad. Your conclusion is based on something unstated. > > > > > But this is, again, a distraction. > > > > > > > > 1) Infinitely long means that there are infinitely many natural > > > > numbers. > > > > 2) Infinitely broad means that there are infinite natural numbers. > > > > > > Eh? Why? Can you provide a proof? > > > > You agreed above that the triangle gets infinitely long and infinitely > > broad. > > Yes, I ask for a proof that infinitely broad means infinite natural numbers. The triangle is defined as representing the natural numbers. The third line contains he number 3 represented by three "x". There is no proof because this is a model. > I may note that in *both* cases, infinite has been used losely, meaning that > there is no bound. I have already stated that elsewhere. I think you are > confusing actual infinity (the infinite list exist) with potential infinity > (no member of the list is infinite). At each finite stage, the length and > width of the triangle are equal. And the width of the triangle is the > width of the last element. If you complete the triangle (conforming to > the axiom of infinity) the length of the list is infinite, of course, it has aleph_0 lines. > as is the width, Wrong. There are aleph_0 numbers, but all of tem are finite. > but there is *no* element of the list that has infinite width, because there > is no last element. (If there were a last element, indeed, that element > would have infinite width, but as there is none...) So there are *not* aleph_0 lines now? > > > > Of course both statements are equivalent. > > > > > > There is no "of course" here. Such a statement needs proof. > > > > You want a proof that 3 = 3 or 4 = 4 or oo = oo? > > The last one would be interesting. What does it mean? But I want to see > a proof of "infinitely many natural numbers is equivalent to infinite > natural numbers". That is what you asserted. You see it by your illogic dancing around the list above. > > > You see here the natural numbers in unary representation: > > 0.1 > > 0.11 > > 0.111 > > ... > > > > Can't you see that width and length are the same? > > Yes, see above. Nothing tosee above except illogic dancing. > Still your assertion that "infinitely many natural numbers > is equivalent to infinite natural numbers". Where is the *proof*? > Note that the axiom of infinity asserts otherwise. And still assuming > that you want to prove that that axiom is consistent, you should prove > that assuming that axiom. I did so. I founf aleph_0 lines. But I found that they do not all exist unless there is one line of infinite length. So there are less than aleph_0 lines. > > > > But that is not mathematical thinking. This is the thinking that leads > > > users of (non-symbolic) calculators to complain that when they > > > calculate sin(pi) they do not get 0. Mathematics is not concerned with > > > finite space, time or whatever. But, again, this is a philosophical > > > standpoint, *not* a mathematical one. By the same reasoning sqrt(2) > > > does not exist. > > > > Of course it does not. You cannot distinguish sqrt(2) and the same > > number with the digit of position number 10^100^100 replaced by 5. > > So we are factoring numbers using things that do not exist. Not completely, not as a number. But you have never used more than 10^100 digits of sqrt(2). > > > > That is correct. But in no case the number of paths can be larger than > > > > that of edges (which is countable). > > > > > > No, the number of edges will also be uncountable. > > > > The set of edges is easily enumerated. > > o > > /1 \2 > > o o > > /3\4 /5 \6 > > o o o o > > /7... > > Yes, as long as you remain finite. The lines o Cantor's list can be enmerated. We do not remain in the finite there. The edges of the infinite tree can also be enumerated. See the scheme above. > So the edges leading to final nodes > are countable. What is the natural number that can be mapped to the > final edge in 0.111... ? There is no final edge. Nevertheless all edges can be enumerated. And there are two edges per infinite path. Regards, WM
From: Virgil on 8 Aug 2006 16:46 In article <1155068921.849102.10570(a)h48g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > In article <1154967573.942270.13030(a)p79g2000cwp.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > Example: > > > The third 1 of 0.111... can index the third 1 of 3 = 0.111 as well as > > > the third one of 5 = 0.11111. > > > > > > Since the third 1 and the fourth 1 are identical, absent their > > predecessors, neither can index anything by itself. > > > > If "mueckenh" wants the string of the first three 1's as index for the > > third 1, that is possible. > > > > > > > > > > > > If 1/9 is not in the list > > > > > but > > > > > can be indexed completely by list numbers > > > > Each digit in the decimal expansion can be indexed by a natural number > > but the 'completed' expansion can only be "indexed" by the 'completed' > > set of all naturals. Which is perfectly symmetric. > > > > > > , this symmetry is broken, > > > > > because: What means "not in the list"? It means that 1/9 has more > > > > > 1's > > > > > than every list number. > > > > It does not mean that 1/9 has a decimal expansion with "more" 1's than > > there are naturals in the set of all naturals. > > All numbers with as many 1's as are indexible by naturals are in the > true list. 0.111... is not in the true list. That is proof enough. We can take 0,111... and insert a"2" after each "1" and still have enough naturals to index every digit and the list of digit too. Given any endless list indexed by the set of naturals we can, a la the Hilbert Hotel method, accommodate as many more without running out of indices. http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel Hilbert's paradox of the Grand Hotel From Wikipedia, the free encyclopedia In mathematics, the German mathematician David Hilbert (1862 1943) presented the following paradox about infinity: In a hotel with a finite number of rooms, once it is full, no more guests can be accommodated. Now imagine a hotel with an infinite number of rooms. You might assume that the same problem will arise when all the rooms are taken. However, there is a way to solve this: if you move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3, etc., you can fit the newcomer into room 1. Note that such a movement of guests would constitute a supertask. It would seem to be possible to make place for an infinite (countable) number of new clients: just move the person occupying room 1 to room 2, occupying room 2 to room 4, occupying room 3 to room 6, etc., and all the odd-numbered new rooms will be free for the new guests. However, this is where the paradox lies. Even in the previous statement, if an infinite number of people fill the odd numbered rooms, then what amount is added to the infinite that was already there? Can one double an infinite? Also, for example, say the infinite number of new guests do come and fill all of the odd numbered rooms, and then the infinite number of guests in the even rooms leave. An infinite has just been subtracted from a still existing infinite, yet an infinite still exists.
From: Virgil on 8 Aug 2006 17:10 In article <1155069769.640195.127530(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > > > You agreed above that the triangle gets infinitely long and infinitely > > > broad. > > > > Yes, I ask for a proof that infinitely broad means infinite natural > > numbers. > > The triangle is defined as representing the natural numbers. The third > line contains he number 3 represented by three "x". There is no proof > because this is a model. It is "Mueckenh" who needs proof that his finite model translates to an infinite model in the way he insists it does. Since the row number and the last x position number are finite for every row in every finite model if the WM triangle, where is the proof that either ever becomes infinite? > > of course, it has aleph_0 lines. But no line in numbered aleph_0. > > > as is the width, > > Wrong. There are aleph_0 numbers, but all of tem are finite. There is no x in any line that is numbered as the aleph_0_th 'x' in that line. > > I did so. I founf aleph_0 lines. But I found that they do not all exist > unless there is one line of infinite length. Unless you provide us with a system of axioms and definitions in which such peculiar results can be shown to hold, we will continue to reject your "findings". >So there are less than > aleph_0 lines. Each line is less that an aleph_0 line but the set of lines contains aleph_0 lines. "Mueckenh" is trying to sell the idea that an ordered set must contain a largest element. This is provably false in ZF or NBG, and "Mueckenh" cannot invalidate either without making his own set of assumptions about what is "true". > > There is no final edge. Nevertheless all edges can be enumerated. And > there are two edges per infinite path. There has been constructed here a bijection between the set of edges and the set of naturals. There has also been constructed here a bijection between the set of infinite paths and the power set of the naturals. Neither construction has been challenged. Thus "mueckenh" is claiming to be able to inject the power set of the naturals into the set of naturals. We should be interested in seeing his attempts to perform this impossibility.
From: Virgil on 8 Aug 2006 17:13 In article <1155069769.640195.127530(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > So the edges leading to final nodes > > are countable. What is the natural number that can be mapped to the > > final edge in 0.111... ? > > There is no final edge. Nevertheless all edges can be enumerated. And > there are two edges per infinite path. So we have infinitely many finite triangles, but without that "final edge", no infinite triangle. And "Mueckenh" loses again.
From: David R Tribble on 8 Aug 2006 18:49
Dik T. Winter schrieb: >> The last line of the triangle will have equal width as the length of the >> triangle. If we get at an infinitely long triangle, there is no last line. > Mueckenh wrote: > And there is no definite number aleph_0 of lines. > > But if there were actually infinitely many, namely aleph_0 lines, then > already the first 10 % of lines were infinitely many. And 90 % of the > lines were infinitely long. What is 10% of Aleph_0? If you start counting your lines, at what point do you know that you've counted the first 10% of them? |