An uncountable countable set
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From: Tony Orlow on 12 Sep 2006 15:34 Virgil wrote: > In article <4506d58e(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> David R Tribble wrote: > >>> You are obviously missing something. >>> >> What I left out was that the power set includes countably infinite >> subsets of N, and such countably infinite bit strings are not considered >> standard naturals. However, they represent finite whole values > > For any standard meaning of finite, TO's claim is FALSE! > > In order for a binary string (function from N to {0,1}) to have a > standard correspondence with any finite natural number value it must > have only finitely many non-zero bits, and thus be equivalent to a > finite string with leading bit 1 or with only one bit. > > Any string with infinitely many non-zero bits cannot correspond to any > finite numerical value. What about an unboundedly large but finite number?
From: imaginatorium on 12 Sep 2006 15:47 Mike Kelly wrote: > Tony Orlow wrote: > > David R Tribble wrote: > > > Mike Kelly wrote: > > >>> Better in what sense? What mathematics are you hoping to be able to do > > >>> with your new foundation that cannot be done with ZFC? > > > > > > Tony Orlow wrote: > > >> It accounts for changes of a single element between infinite sets and > > >> therefore provides for a full spectrum of ordered infinite sets, thus > > >> making the Continuum Hypothesis null and void. It relates the continuum > > >> to the hypernaturals formulaically, and provides for an integration of > > >> sets and measure in the infinite case. It rids us of anomalies like > > >> omega-1=omega. <snippo-bippo> > > > > Use IFR. N maps to S using f(n)=n+1. The inverse of that function is > > g(x)=x-1. So, over the range of 0 to N, |S|=|N|-1. Map N to the Evens E > > using f(n)=2n. The inverse function is g(x)=x/2, so over the range of N, > > the evens have |N|/2 elements. Isn't that intuitively satisfying? > > No. Now what? Well, isn't this "IFR" thing basically Tony's semicomprehension of limiting density, which applies to some infinite subsets of the integers (and to some other sets, but not all). AFAICS, Tony simply ignores everything known about limiting densities, to assume that his version gives a "size" of - presumably - any set. Brian Chandler http://imaginatorium.org
From: Virgil on 12 Sep 2006 15:48 In article <4506e361(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Wolfgang and I see the same problem. Wolfgang and TO HAVE the same problem. They insist that there must be a natural so large that it cannot be increased by one. This in spite of an axiom that says otherwise. You start with 0 and start > enumerating the naturals using increment to generate the next successor. > Increment is defined as the addition of 1. So, the first is 1, the > second is 2, etc, etc. Where you have a set of consecutive naturals > starting at 1, the nth is n, the sum of n increments, and the size of > the set is always equal to the greatest value in the set. If you start with zero and keep adding one to the largest member and appending it, the size of the set is NEVER equal to the largest member: start with 0 = {} and repeat Succ(x) = x union {x}. > So, if you claim the size of the set is aleph_0 We don't. TO is hung up on SIZE only having one meaning, whereas we have many different measures related to "size", not all of them cross-consistent. We have one called cardinality based on injections and bijections between sets, and for a set to have cardinality aleph_0, it must, among other things, allow injections to a proper subsets of itself. As TO's versions fail this test, they are not of cardinality aleph_0. > It is not possible to identify any last step, that's true, but > irrelevant. I am not hinging my argument on any "last natural" as you > are trying to do. For TO's arguments to hold. there must be such a "last natural", but as there is not, TO's arguments fail. Also, TO's definitions of "finite" and "infinite", being in direct conflict with the standard ones, means that what he says about finiteness and infinteness is meaningless in any standard discussion.
From: Virgil on 12 Sep 2006 15:50 In article <4506e412$1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Dik T. Winter wrote: > > In article <45005670$1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> > > writes: > > > Dik T. Winter wrote: > > ... > > > > But you stated "that you can specify which finite number of > > > > iterations" > > > > etc. > > > > That is something different. A number is computable when there is a > > > > Turing > > > > machine such that, when given an arbitrary number n, it will calculate > > > > the > > > > 1-st through n-th digit. There is nothing about a specification of a > > > > finite number of iterations. > > > > > > Well, if you set up a Turing machine such that it will take an infinite > > > number of operations to get to any finite digit, then you have probably > > > not designed it well. > > > > You do not seem to understand. If you set up a Turing machine such that > > given a number n as input calculates the first n digits of some number > > can be very well designed. But can you proof that it will stop? > > Depending on what number you are calculating, you should be able to > prove that it will reach a certain degree of accuracy in a certain > number of steps. I'm not sure how to answer that question in such > general terms. Confessing your ignorance is the first step to wisdom.
From: Virgil on 12 Sep 2006 15:54
In article <4506e465(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Dik T. Winter wrote: > > In article <45004b9b$1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> > > writes: > > > Dik T. Winter wrote: > > ... > > > > > So, while ...111 may not be considered a standard natural, I see no > > > > > reason why it should not be considered, say, an extended natural. > > > > > > > > Why not use the proper name such numbers already have? 2-adics. > > > > > > No reason, except that 2-adics could possibly have infinite bit > > > positions. > > > > They have not. I have yet to come across basic objects in mathematics that > > have infinite posititions in their representation. > > That's why I invented the T-riffic numbers. It's about time. It's about TO's idiocy in trying to invent something for which there is no need, and failing to invent anything self-consistent. |