An uncountable countable set
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From: Tony Orlow on 23 Aug 2006 11:05 David R Tribble wrote: > Tony Orlow wrote: > David R Tribble 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. > > David R Tribble wrote: >>> 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. > > Tony Orlow wrote: >> Why would you sum them all, when each sequence of bit positions include >> those before it? > > To prove your claim (that countably infinite string of digits produces > uncountably many possible strings) is false. Pay attention. > That's a fact of standard set theory. Are you telling me the countably long bit strings to the right of the digital point don't comprise the uncountably large set of reals in [0,1)??? > > David R Tribble wrote: >>> 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. > > David R Tribble wrote: >> Right, in your theory, which yields an uncountable number of binary >> strings, ala 2^aleph_0. > > I'm proving exactly the opposite. Pay attention. > Then you're proving something contrary to standard set theory. > > David R Tribble wrote: >>> There are a countably infinite number of naturals. Since the sum >>> of a countable number of countable values is itself countable, t is >>> countable. > > Tony Orlow wrote: >> So, what is ceil(log2(aleph_0)+1) again? Is that countably infinite? > > It's meaningless. > Except as the expected number of bits required to list all the naturals. > >> You have missed the point entirely, as usual. Oh well. > > I just proved your claim to be false. I think that's on point. > Great, then you just found a hole in set theory. Go collect your prize.
From: Tony Orlow on 23 Aug 2006 11:10 David R Tribble wrote: > Virgil wrote: >>> So definitions of operations are valid only for their original domains >>> of definition and new definitions are required for different domains of >>> definition. > > Tony Orlow wrote: >>> Bull. The rules for arithmetic manipulation can be applied without >>> problems (for anything but transfinite mathology) to variables of >>> infinite value. > > Virgil wrote: >> Not in mathematics, they can't. > > Indeed. So Tony, what is 1/0 + 2/0? I suppose you will tell us that > it is obviously 3/0. Which is obviously not the same as 4/0. > You see the inconsistencies that you introduce. Lil'un=1/Big'un. Big'un=1/Lil'un. 1/Lil'un+2/Lil'un=3/Lil'un=3*Big'un, and yes that's less than 4*Big'un. Infinitesimals are not absolute 0. > > Virgil wrote: >>> Anyone wishing to extend an operation only defined on one domain to a >>> larger domain must give a clear definition of what that extended >>> definition is to mean. TO has not done this. > > Tony Orlow wrote: >>> The domain is the real line. > > Virgil wrote: >> Which specifically does not include any points at infinity, and at best >> only defines arithmetic on the finite numerical values associated with >> geometrical points, and on nothing else. > > And that's only because we can equate real numbers to points on > a geometric "line" (which actually involves other concepts such as > "distance" and metric spaces). > > Show us where Big'un exists as a point on the real line. > Is it somewhere near 1/0? Or maybe near 3/0? > Big'un is best viewed on one or the other version of the number circle, opposite 0. It's 1/Lil'un. 3*Big'un can be considered to be the size of three infinite lines, perhaps the first three of an infinite number comprising a 2D space.
From: imaginatorium on 23 Aug 2006 13:11 Tony Orlow wrote: > David R Tribble wrote: > > Tony Orlow wrote: <snip> > >> Please > >> show where I am being inconsistent, not with set theory, but within my > >> own assumptions. Remember, I don't claim to believe in transfinite set > >> theory, and don't intend to be consistent with it. > > > > The problem is that your own assumptions are not consistent with > > each other - they don't form a coherent theory. Many of these > > inconsistencies have been pointed out to you before, but you choose > > to not believe them or simply to ignore them. > > > > It's easy to say that without mentioning any specific inconsistencies. Well, here are a few statements from you, culled from nearby posts (at least in the google "chronological order")... (Quoted in post by Moeblee above) Tony Orlow wrote: > I have never said I think there is a largest natural. I have said that > some of your assumptions lead to that conclusion. So you think there is no largest natural? (Quoted in post by you, three down the list) David R Tribble wrote: > Tony Orlow wrote: >> But Monsieur, what about the injection from P(N) into N, via the bit >> strings which denote set membership, each of which also corresponds to a >> binary natural? Tsk, tsk. Mustn't forget that one! Remember, the only >> set which doesn't map is the entire set, and that maps to the largest >> natural, that is, ...1111 with all bits in finite positions. But you think that "...1111" _is_ the largest natural. No inconsistency so far, I suppose, in your view. I wonder if you see any inconsistency in the Finlayson thingies: where between every pair of rationals is another rational, yet each rational has an adjacent rational to the right of it? Does this thing that is the largest natural, that doesn't exist, i.e. "...1111", does it have a left end? Perhaps it goes on for ever to the left, and never stops, and the place where it stops is called Infinity. No, no inconsistency here, eh? Brian Chandler http://imaginatorium.org
From: Virgil on 23 Aug 2006 14:04 In article <echoo8$f2k$1(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > David R Tribble wrote: > > Tony Orlow wrote: > >> I think the responsibility lies with you to point out an inconsistency > >> that arises from my assumptions. > > > > You're kidding, right? > > No. You complain that IFR, N=S^L and infinite-case induction lead to > some kind of contradiction, but I don't see it. The problems with TO's IFR and N+S^L interpretations have been carefully and adequately pointed out by several people. Anyone with his vision not obscured by such bias as TO has regularly exhibited would easily have seen the conflicts with standard theories that TO's assumptions give rise to. > If set theory were > "proven" true Goedel showed that any set theory which could serve as a basis for standard arithmetic AND could prove itself consistent must be inconsistent. So we who are aware of that are quite thankful that our set theories cannot prove themselves consistent. > >> Please > >> show where I am being inconsistent, not with set theory, but within my > >> own assumptions. Remember, I don't claim to believe in transfinite set > >> theory, and don't intend to be consistent with it. > > > > The problem is that your own assumptions are not consistent with > > each other - they don't form a coherent theory. Many of these > > inconsistencies have been pointed out to you before, but you choose > > to not believe them or simply to ignore them. > > > > It's easy to say that without mentioning any specific inconsistencies. AS we do not have from TO in any one place any complete collection of his undefined terms, basic definitions and his set of axiomatic assumptions about them, TO does not have a system at all. > > > In fact, mapping the naturals in [1,Big'un] to the reals in [Lil'un,1] > using the mapping function f(x)=x/Big'un yields Ross' Finlayson numbers Appeals to Ross' creations do not inspire confidence it the reliability of the appealer.
From: Virgil on 23 Aug 2006 14:35
In article <echosu$f2k$2(a)ruby.cit.cornell.edu>, Tony Orlow <aeo6(a)cornell.edu> wrote: > David R Tribble wrote: > > Tony Orlow wrote: > >> But Monsieur, what about the injection from P(N) into N, via the bit > >> strings which denote set membership, each of which also corresponds to a > >> binary natural? Tsk, tsk. Mustn't forget that one! Remember, the only > >> set which doesn't map is the entire set, and that maps to the largest > >> natural, that is, ...1111 with all bits in finite positions. > > > > ... as well as all the infinite subsets of N. You keep forgetting > > about those, don't you? > > > > You must be forgetting that, given that all bit positions are finite, > even your countably infinite bit strings only can represent finite > values In binary fraction notation, infinite bit strings can represent uncountably many finite real values. And merely as strings, there is one for each member of P(N). > and since they also all have distinct successors and > predecessors What string of only 0's and 1's is the successor to an infinite string of 1's? > your claim that they are not natural numbers is rather > unjustified, wouldn't you say? Not at all. One can prove at necessity that one does not need any infinite strings in order to represent all naturals. And that even if one should choose to use infinite strings, one does not need to use any string with infinitely many 1's in it in order to have represented every natural. > You notice that I called ....1111 the > largest natural, don't you? Sillyness noted! If it has infinitely many 1's, it does not represent a natural. Representing infinite binary strings with their least significant digits to the right and increasing significance to the left, ....000, the string of all zeros is smallest. ....0001, the string with only one 1 and that in the least significant place is next smallest. If the string for any natural has any 0's to the right of any 1's, then its successor string (corresponding to the successor of the natural) has a left most 1, and it is in the same position. If the string for any natural has no 0's to the right of any 1's, then its successor string has a left most 1, and it is one position further to the left . By simple induction on the two rules above, it is obvious that the string for any natural has a leftmost 1. Thus TO's ...111, by not having a left most 1, is NOT the string for any natural number. This holds regardless of whether one assumes ZF, ZFC and NBG. TO cannot, or at does has not, give us an axiom system in which his infinite string of 1's can ever represent any natural number. |