An uncountable countable set
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From: Mike Kelly on 7 Sep 2006 09:39 Tony Orlow wrote: > Virgil wrote: > > In article <44fe2642(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <44fd9eba(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Dik T. Winter wrote: > >>>>> In article <44ef3da9(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> > >>>>> writes: > >>>>> Your axiom uses things that are not defined. What is the *meaning* of > >>>>> "x<z"? > >>>> Geometrically it means that x is left of z on the number line. > >>> > >>> And for someone standing on the other side of the number line would x be > >>> on the right of z? > >>> > >>> And does the line stay horizontal as one moves around earth? Which way > >>> is larger if the line ever goes vertical. And how does the "larger" work > >>> at antipodes? > >>> > >> Silly questions. > > > > In response to a silly definition. > >>> > >>>> It means > >>>> A y (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y). That says about all it > >>>> needs to, wouldn't you say? > >>> Not hardly. > >>> A y (y = x) v (y = z) v (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y) > >>> is a bit better but still insufficient. > >> True, I should have specified y<>x and y<>z. I guess it's usually done > >> using <= for this reason, eh? > >> > >>>>> > > That is not a definition, because it makes no sense. "The set of > >>>>> > > naturals > >>>>> > > is as large as every natural"? > >>>>> > > >>>>> > It is not larger than all naturals > >>>>> > >>>>> That is something completely different again. > >>>> It's not LARGER than every finite. > >>> Which natural(s) is it "not larger" than", in the sense of not being a > >>> proper superset of that natural or having that natural as a member? > >> ....11111 binary (all bit positions finite) > > > > Unless that string has only finitely many bit positions as well as only > > finite bit positions, it is not a natural at all, as it is then neither > > the first natural nor the successor of any natural, and every natural > > has to be one or the other. > > It is the successor to ....11110. Duh. I've already proven that this is > a finite value, given that all bit positions are finite, and that > therefore no place in that string can achieve an infinite value, and > that any such number has predecessor and successor. The cute thing is > that the successor to ...1111 is 0, and that ...1111 is essentially -1. :) Does it not bother you that nobody else agrees with, or even understands, your proof? -- mike.
From: Tony Orlow on 7 Sep 2006 09:40 Mike Kelly wrote: > Tony Orlow wrote: >> Virgil wrote: >>> In article <44fd9eba(a)news2.lightlink.com>, >>> Tony Orlow <tony(a)lightlink.com> wrote: >>> >>>> Dik T. Winter wrote: >>>>> In article <44ef3da9(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> >>>>> writes: >>>>> Your axiom uses things that are not defined. What is the *meaning* of >>>>> "x<z"? >>>> Geometrically it means that x is left of z on the number line. >>> And for someone standing on the other side of the number line would x be >>> on the right of z? >>> >>> And does the line stay horizontal as one moves around earth? Which way >>> is larger if the line ever goes vertical. And how does the "larger" work >>> at antipodes? >>> >> Silly questions. >> >>> >>>> It means >>>> A y (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y). That says about all it >>>> needs to, wouldn't you say? >>> Not hardly. >>> A y (y = x) v (y = z) v (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y) >>> is a bit better but still insufficient. >> True, I should have specified y<>x and y<>z. I guess it's usually done >> using <= for this reason, eh? >> >>>>> > > That is not a definition, because it makes no sense. "The set of >>>>> > > naturals >>>>> > > is as large as every natural"? >>>>> > >>>>> > It is not larger than all naturals >>>>> >>>>> That is something completely different again. >>>> It's not LARGER than every finite. >>> Which natural(s) is it "not larger" than", in the sense of not being a >>> proper superset of that natural or having that natural as a member? >> ....11111 binary (all bit positions finite) > > That isn't a natural number, Tony. > Are you sure? Pay close attention. For any finite bit position n, it and all predecessors can only sum to a finite bit string value of 2^(n+1)-1. Since there are only finite bit positions in the string, it can never achieve any infinite value at any position in the unending string of bits. Therefore the value must be finite. Furthermore, since any such number does have a predecessor and successor, in this case ....1110 and ...0000, respectively, it fits in the successorship model. The only concept this breaks is that 0 is now a successor as well, creating an infinite ring of successorship. Other than that, it works as a natural, and in fact, this is the way signed integers work in your very computer. So, while ...111 may not be considered a standard natural, I see no reason why it should not be considered, say, an extended natural. Tony
From: Tony Orlow on 7 Sep 2006 09:52 Mike Kelly wrote: > Tony Orlow wrote: >> Dik T. Winter wrote: >>> In article <44ef3b88(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: >>> > Dik T. Winter wrote: >>> ... >>> > > > > There is no such specific natural number. It is when we have them >>> > > > > all, but as there is no largest number, this can not be achieved by >>> > > > > taking them one by one. >>> > > > >>> > > > The set of all naturals numbers consists of only natural numbers. There >>> > > > is NO natural number where the count becomes infinite. So there is no >>> > > > point in the set, even if you COULD get to the "last" one, where any >>> > > > infinite set has been achieved. >>> > > >>> > > And there is no point in the set where you have the complete set. Yes. >>> > > Indeed. So what? >>> > >>> > So, if there is no point in the set which can even remotely be >>> > considered infinitely far from the beginning, what makes it actually >>> > infinite? >>> >>> What the set of all finite numbers makes infinite? The axiom of infinitys >>> states that the set of all finite (natural) numbers does exist. From that >>> is is easy to prove that that set is not finite, hence infinite. >> Yes, according to the Dedekind definition of an infinite set, but that >> definition leads to some conclusions that offend some people's >> sensibilities. >> >>> > If no element of the set can be an infinite number of steps >>> > from the start, you may not be able to find an end. >>> >>> And indeed, when you go step by step you will not get at the end. >> Right, you will always have gone a finite number of steps and have >> arrived at a finite natural. >> >>> > But does that mean >>> > it's "greater than" every finite, or only "greater than or equal"? >>> >>> What is the difference? Assuming you mean "aleph-0" when you write "it", >>> it is easily proven that: >>> aleph-0 is greater than or equal to each natural >>> gives the theorem: >>> aleph-0 is greater than each natural. >>> >>> Because: suppose aleph-0 >= each n in N. Now suppose in addition that it >>> is equal to some particular n. Well, n + 1 is in N, and so aleph-0 >>> should also be larger or equal to n + 1. Hence it can not be equal to n. >>> So it is not equal to any n at all. And so aleph-0 > each n in N. >> Only assuming that you have identified some largest natural. This is the >> form of most proofs in this theory: assume a largest finite, find a >> contradiction, and blame it something else. Of course you cannot find a >> largest natural such that this is the size of the set. You also cannot >> take an infinite number of increments without achieving an infinite value. >> >>> > > Indeed. The set of all natural numbers is just sufficient. >>> > >>> > No, it is far too great. If you have a countably infinite number of bit >>> > positions, then you have an uncountably infinite set of strings. Where >>> > bit positions are indexed by the naturals, the naturals are the power >>> > set of the number of bit positions, >>> >>> Wrong. This is plain nonsense. Suppose there are three bit positions. >>> The set of naturals {1, 2, 3} is sufficent to index them. In what way >>> is {1, 2, 3} the power set of the number of bit positions (3)? >> Let me explain, using your example. We have the first three bit >> positions 0, 1 and 2, representing 1, 2 and 4, the fist three powers of >> 2. With those bit positions we can produce eight unique binary strings, >> as follows: >> >> 0 1 2 >> >> 0 0 0 0 >> 1 1 0 0 >> 2 0 1 0 >> 3 1 1 0 >> 4 0 0 1 >> 5 1 0 1 >> 6 0 1 1 >> 7 1 1 1 >> >> The set of binary naturals which can be represented by this set of bit >> positions is the power set of the bit positions, each being a unique >> combination, in the form of a sum, of those powers of 2. Given n bit >> positions, we can produce 2^n unique strings. Given n unique powers of >> 2, we can produce 2^n unique sums. Thus the set of binary strings is the >> power set of the number of bit positions. >>> > > No one is obligated to accept the theory at all. Whether it is proven >>> > > to be "correct" or not, as I have no idea what "correct" in this context >>> > > means. Is Euclidean geometry "correct"? Is hyperbolic geometry "correct"? >>> > > Is elliptic geometry "correct"? >>> > >>> > Ah, now you bring up a prime example. Euclid set down laws for flat 2D >>> > geometry, and questioning those axioms led to new shapes for space. >>> > Accrdingly, the axioms of set theory might work together to describe a >>> > system, but it is not impossible that entirely other systems might arise >>> > from different starting assumptions. >>> >>> And indeed, I never did state the opposite. But if you want to get at a >>> new system, provide axioms, definitions, and whatever. Tell us what >>> axioms to retain and what axioms to reject. And if there are axioms to >>> be rejected, come up with alternatives. >> Well, that's what I'm trying to do, to provide an alternative >> perspective and system. I certainly reject the Dedekind definition of an >> infinite set as being inconsistent with notions of infinite measure when >> it comes to the finite naturals. By viewing the situation in terms of >> density and range, the conclusion can be quite different. Since no two >> naturals are infinitely distant, and since each natural occupies a unit >> of measure on the real line, there cannot be an infinite number of them, >> because you cannot fit an infinite number of unit segments in a finite >> range. >> >>> > > But what is the case is that if you accept the axioms, you also have >>> > > to accept what follows from the axioms. >>> > >>> > Yes, I understand that, and much to the consternation of some, I don't. >>> >>> Yes, much consternation, I can understand that. So apparently you are >>> accepting the axioms, but not what follows from the axioms. What kind >>> of logic are you using? >> I am NOT accepting the axioms. The axioms are artificial statements >> which can be made to work together, but which do not follow from >> elementary logic the way they should. Every axiom should be justifiable >> based on fundamental concepts. I don't see that here. > > Which axiom of ZFC set theory do you find objectionable? I object to the claim that the Axiom of i
From: Tony Orlow on 7 Sep 2006 09:53 Mike Kelly wrote: > Tony Orlow wrote: >> MoeBlee wrote: >>> Tony Orlow wrote: >>>> And yet, you are saying it's not technically correct. So, 1 is a natural >>>> but not a rational or real? If rigorous formulations come to that >>>> conclusion, then rigor does not ensure correctness. >>> You're hopeless. You didn't understand a thing I said, which may be my >>> fault for not providing adequate explanation in the context of your >>> ignorance of the subject; but it is no my fault that you won't even >>> look at a book on mathematics, such as even a introductory text in real >>> analysis. >>> >>> MoeBlee >>> >> Your whole point here is ludicrous. You are arguing that the naturals >> are not a subset of the rationals, which are not a subset of the reals. >> While each superset may require a more complex construction than the >> subset, all elements of the subset are covered by the superconstruction. >> As points on the real line, they are all real numbers. What you're >> saying amounts to what I said above, which is indeed absurd. So, no, I >> don't understand a thing you're saying, when you say naturals aren't >> reals. Sorry. > > All natural numbers are finite ordinals. All real numbers are infinite > sets of rationals. Obviously a set which contains only finite ordinals > is not a subset of a set that contains only infinite sets of rationals. > Both denote a point on the real line. Those set constructions are a means to define the point. Different means of specification may define the same point.
From: Tony Orlow on 7 Sep 2006 09:58
Mike Kelly wrote: > Tony Orlow wrote: >> Dik T. Winter wrote: >>> In article <44fdcaf1(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> writes: >>> > Dik T. Winter wrote: >>> > > In article <1157367096.604428.36330(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: >>> ... >>> > > > A number has only one of the following properties: It is larger than or >>> > > > smaller than or equal to any natural number. >>> > > >>> > > So omega and aleph-0 are numbers. They satisfy the definition. Thanks. >>> > >>> > So, they are numbers which are larger than any finite number? Why then >>> > do we not consider an inductive proof of the form E y e N A x>y P(x) not >>> > to prove P(aleph_0) or P(omega)? >>> >>> That would be a new axiom, and you may consider it, but it leads to >>> contradictions when you retain the current definitions of aleph_0 and >>> omega. >> Yes, I am well aware of that. That's why I have chosen to reject those >> concepts in favor of finding something better, based on this >> infinite-case induction. Noting personal. >> >> :) > > Better in what sense? What mathematics are you hoping to be able to do > with your new foundation that cannot be done with ZFC? > 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. If you remove an element, the proper subset should ALWAYS be smaller by 1. That is the case for me. For a theory to claim a proper subset is the same "size" as the proper superset is an immediate deal-breaker for me. |