An uncountable countable set
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From: imaginatorium on 20 Aug 2006 12:09 Tony Orlow wrote: > Dik T. Winter wrote: > > In article <ec8h1q$prm$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: > > ... > > > Yes, that's a shame, isn't it? It would seem to me that any good theory > > > of infinite sets could be applied to infinite sets of points, such as > > > the reals in (0,1] or those in (0,2], and be able to draw conclusions > > > such as that there is twice as many points and twice as much space in > > > the second as in the first, rather than coming to to useless conclusion > > > that the infinite sets are equal in size. > > > > You are looking for measure theory. > > I looked at measure theory a little. Lebesgue measure doesn't come close > to what I'm concocting. I am trying to integrate set theory with > combinatorics, set density, Lebesgue measure, topology, etc, and all it > takes is a housecleaning and a few new simple rules. Fine. When's the book coming out, then? > > > Indeed, they don't form > > > a field. Tsk tsk. I think this is you - I wonder what you mean by a "field"? > > There is not something inherently wrong with a collection of objects > > not forming a field or a division ring, or whatever. However, if that > > is the case you must be prepared to find that division is not generally > > possible. Take for instance the Sedenions, an extension of the Cayley > > numbers (or Octonians). In the Sedenians general division is not > > possible because there are zero divisors. > > That is an area I don't know too much about except that it's an > extension of the complex numbers. Complex numbers are really not single > quantities, at least in terms of linear order, and so I wouldn't expect > them to form a ring, but perhaps more of a 2-D ring, or toroidal > surface, in that sense. In what sense, exactly? Do you have the faintest clue what Dik means by 'ring' in this context? (Hint: you've demonstrated, perhaps even stated, that you know nothing about (abstract) algebra. If so you would make less of a fool of yourself if you avoided mimicing words like this.) <snip even more...> Yeah, my dog speaks chocolate. Does yours? Brian Chandler http://imaginatorium.org
From: Franziska Neugebauer on 20 Aug 2006 12:16 mueckenh(a)rz.fh-augsburg.de wrote: > Franziska Neugebauer schrieb: >> >> So you claim my proof is wrong. Where exactly? > > The result. You are free to /dislike/ the result P ~ N. That neither renders the proof false nor wrong. >> > There are not infinitely many differences, if we consider the >> > finity of each number. >> >> Proof? > > Consider a staircase: Set theory is not about staircases. > Without a stair surpassing the height H the > staircase cannot surpass height H. Without a stair surpassing every > finite height the staircase cannot surpass every finite height, i.e., > cannot have infinitely many steps of height 1. (surpassing every > finite = infinite). But the length surpasses every finite length > (according to Cantor). I don't discuss Cantorisms. >> > There are infinitely many differences, if we consider infinitely >> > many numbers. >> >> There _are_ infinitely many numbers n e omega and therefore there are >> "infinitely many differences of 1" (as proven). > > If you climb 10 meters, then you arrive at height 10 m. > But you climb infinitely many meters without arriving at an infinite > height? I don't discuss physicalisms. Set theory does not claim to rule physics therefore "physical" arguments are unapt to explore set theory. >> > Therefore one of the assumptions is wrong: Either there are not >> > infinitely many numbers or there is at least one number which is >> > infinite by size. >> >> There are infinitely many numbers n e omega _and_ there is not a >> single number which is "infinite by size". This is fact > > Is infinitely not a number in your opinion? "Infinitely" is an adverbial modifier and not a number. > Then infinity cannot be surpassed. I don't discuss *the* infinity. > Then omega + 1 is purest nonsense. You already uttered your dislike. > Under this condition I agree with you. > You are back to the pre-Cantorism. > >> >> Do you agree that P ~ omega? >> > >> > P has the same cardinality as |N. >> >> So P ~ omega. >> >> > But either there are not infinitely many numbers or there is at >> > least one number which is infinite by size. >> >> Why (please prove!)? Do you want to ignore facts? > > You must not think that facts are facts because you call them facts. You mean I shall not call a fact "a fact", even if you have previously agreed to it? > Fact is: if omega is a number, then omega differences of height 1 > result in the total size omega, because > > omega * 1 = omega. > > Isn't it a fact? What are you talking about? F. N. -- xyz
From: Tony Orlow on 20 Aug 2006 12:20 imaginatorium(a)despammed.com wrote: > Tony Orlow wrote: >> Dik T. Winter wrote: >>> In article <ec8h1q$prm$1(a)ruby.cit.cornell.edu> aeo6(a)cornnell.edu writes: >>> ... >>> > Yes, that's a shame, isn't it? It would seem to me that any good theory >>> > of infinite sets could be applied to infinite sets of points, such as >>> > the reals in (0,1] or those in (0,2], and be able to draw conclusions >>> > such as that there is twice as many points and twice as much space in >>> > the second as in the first, rather than coming to to useless conclusion >>> > that the infinite sets are equal in size. >>> >>> You are looking for measure theory. >> I looked at measure theory a little. Lebesgue measure doesn't come close >> to what I'm concocting. I am trying to integrate set theory with >> combinatorics, set density, Lebesgue measure, topology, etc, and all it >> takes is a housecleaning and a few new simple rules. > > Fine. When's the book coming out, then? I am working on an outline right now. When the boys are back in school, I'll have time and peace to actually start putting it all together. Thanks for asking. PS - you still owe me an actual answer to my question, or at least an explanation as to why you don't accept my answer to yours. I thought we had a deal, that didn't include pink elephants. > >>> > Indeed, they don't form >>> > a field. Tsk tsk. > > I think this is you - I wonder what you mean by a "field"? Dik said transfinite numbers didn't form a field. Look it up. > >>> There is not something inherently wrong with a collection of objects >>> not forming a field or a division ring, or whatever. However, if that >>> is the case you must be prepared to find that division is not generally >>> possible. Take for instance the Sedenions, an extension of the Cayley >>> numbers (or Octonians). In the Sedenians general division is not >>> possible because there are zero divisors. >> That is an area I don't know too much about except that it's an >> extension of the complex numbers. Complex numbers are really not single >> quantities, at least in terms of linear order, and so I wouldn't expect >> them to form a ring, but perhaps more of a 2-D ring, or toroidal >> surface, in that sense. > > In what sense, exactly? Do you have the faintest clue what Dik means by > 'ring' in this context? (Hint: you've demonstrated, perhaps even > stated, that you know nothing about (abstract) algebra. If so you would > make less of a fool of yourself if you avoided mimicing words like > this.) You answer my question and I'll answer yours. You condescending attitude has earned you that much. > > <snip even more...> > Yeah, my dog speaks chocolate. Does yours? Your dog will die of liver failure. > > Brian Chandler > http://imaginatorium.org >
From: David R Tribble on 20 Aug 2006 12:22 Dik T. Winter wrote: >> Let L be the list of all natural numbers >> Let K be the sequence of digits such that for each p in L, K[p] = 1. >> What is *wrong* with that definition? > Tony Orlow wrote: >> Sorry to interject, but could you tell me how long a sequence K is? If >> it's finite, then you only have a finite list L, and if it's countably >> infinite, then you have an uncountable list of naturals? > Virgil wrote: >> How does having a countable list of naturals require an uncountable list >> of naturals? > Tony Orlow wrote: > Is K a list of natural numbers? Pay attention. K is the sequence of > digits required to represent the naturals in binary. The set of binary > strings of length n has size 2^n. If n is aleph_0, then your countably > infinite string of digits produces uncountably many possible strings. > We've been through this. You cannot cull bits or strings in any way to > reduce this uncountably infinite set to produce the countably infinite > set of naturals you claim exists. In other words, there is no TYPE of > number which could represent the size of K. Yes, we've been though all this before and you still won't listen to reason. It takes ceil(log2(k)+1) binary digits to encode each natural k, which we'll call D(k). So: D(1) = 1, D(2) = 2, D(3) = 2, D(4) = 3, etc. Now add up all those D(k) for all natural k (all k in N), and call it t, the total number of binary digits for all naturals: t = sum D(k), for all k in N. Obviously, since each k is a finite natural, then each D(k) is a finite natural as well, i.e., each natural k requires a finite number of binary digits to represent it. Adding each finite D(k) to the finite sum D(1)+D(2)+...+D(k-1) up to that point yields a countable finite total number of digits for each k. There are a countably infinite number of naturals. Since the sum of a countable number of countable values is itself countable, t is countable.
From: Tony Orlow on 20 Aug 2006 12:23
David R Tribble wrote: > Tony Orlow wrote: >>> It would seem to me that any good theory >>> of infinite sets could be applied to infinite sets of points, such as >>> the reals in (0,1] or those in (0,2], and be able to draw conclusions >>> such as that there is twice as many points and twice as much space in >>> the second as in the first > > Once again, you are confusing 'cardinality' with 'volume' . The first > deals with denumeration of sets, while the second deals with geometric > distance. > I'm not confusing them, I'm relating them. Do you think someone describing the acceleration of a body is confusing meters with seconds, and then confusing grams with those two when they talk about energy or force? Do you know what a dimension is? Sheesh! I'm not the confused one here. With a declared specific infinite density of reals on the real line, the two are easily integrated. :) TO |