Cantor Confusion
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From: Dik T. Winter on 20 Oct 2006 21:23 In article <1161378001.475899.279610(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > Dik T. Winter schrieb: > > In article <1161276322.150252.120060(a)i42g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > > > > > Again replying to more than one article at once without giving proper > > > > references. That makes it very difficult to follow threads. > > > > > > I am sorry, I have only 15 shots per several hours, but far more opponents. > > > > Why do you think that is the case? > > I have not the slightest idea. I have some ideas about it. You *never* provide definitions for the terms you use, but the terms you use are not those from standard mathematics, so there is a huge miscommunication. > > I do not understand you. You cannot count all naturals and at some time > > saying that you are finished. > > (1) The set of natural numbers exist, by the axiom of infinity. > > (2) You will never terminate when counting through the natural numbers. > > If we could not get to omega, we would not need it. If you do not need it, so be it. Do not use current set-theory and base your analysis on whatever is provided with a finitistic view. There is *no* reason to attack people for using it, except that it is possibly against your ethical views. > And one would not > be able to count omega + 1 and so on. Cantor could have refrained from > introducing it. But he did not because he thought it had its uses, and I think so too. When doing differentiation (Newton and Leibniz) there was no proper (mathematical) founding of the concepts they were using. Leibniz even stated that he did not know how to define his 'infinitesimals' (and that is the reason that he did not give any proof when presenting his first paper). It was only *much* later that through set theory a proper mathematical foundation could be given, and that you could provide *proofs* of some of the properties. Later Robinson provided alternative ways to provide proofs, and again later Anders Kock did the same in a different way. > In order to count all finite sets you need omega. That > is the essence of Cantor's theory. Yes, that was Cantor's theory, but that was an error, and it was not the essence. See my discussion with Dave Seaman about just that point. I do not think he still had that position when he wrote his 'Contributions to set theory' some 11 years after his 'About infinite linear point-sets', page 213 in the 'Gesammelte Abhandlungen'. The essence of his theory was that there were sets that could not be put in bijection with the set of natural numbers. > > Yes to the first, no to the second. f(oo) does not belong to set theory, > > it belongs to analysis. Also note that Cantor at some time refrained from > > using oo and started to use w, because there was too much confusion. > > The oo from pre-Cantorian mathematics is different from the omega of > > Cantorian mathematics. > > According to Cantor oo denotes potential infinity, omega denotes actual > infinity. If all natural numbers exist, then the infinite oo of > analysis in our vase problem is exactly the omega of set theory. No. The oo in standard analysis is still potential. That is, it can not be attained. So the limit lim{n = 1 --> oo} 1/n = 0, does indicate a potential infinity, not an actual infinity. But you are dishonest in transforming the vase problem (where the answer was asked at t = 0) to another problem (where the answer was asked at t = oo). The letter makes no sense. The former *might* make sense when you properly define the problem. > > > > Yes. You can construct a number from any list, but you can not > > > > construct a constructable number from a list that is itself not > > > > constructable. > > > > > > Every diagonal number is constructed and, therefore, is constructible. > > > > When constructed from a constructable list. > > What is an unconstructable list? Do you call any undefinite mess a > list? > Any list is construtced. Any diagonal number is constructed. Some confusion, I think (and I am myself also guilty). A constructible number is a number that can be represented by a finite number of additions, subtractions, multiplications and finite square root extractions. So you can not prove that a list of constructable numbers gives as the diagonal a constructable number. The very reason is that taking the diagonal is not a construction according to the definition. On the other hand, I think you are meaning computable (which would make more sense). But, a computable number is *by definition* a number such that a Turing machine is given and a digit n, the n-th digit of that number is printed after a certain (finite) amount of time. There is a list of Turing machines (by the very definition of Turing machines their number is countable). But that gives *not* a list of computable numbers. It is known that the computable numbers are a "subset" of the Turing machines, namely that subset that are known to stop after a certain time. So a proper list of computable numbers would be equivalent to a list of Turing machines that do stop (excluding those that do not stop). But whether a Turing machine stops or not is not computable, so a complete list of stopping Turing machines is not computable. > > > p. 52 explains it: "Lemma. Given any denumerable subset C0 of C, there > > > exist members of C which are not contained in C0; that is to say, C0 is > > > a proper subset of C." So C is the continuum R and C0 is the set of > > > list numbers > > > > Yes, any diagonal number is constructed from the list. That does *not* > > make any diagonal number constructible. > > Why then should it be important to stress that it is constructed? > However, the set of constructed numbers is countable. Because the term 'constructed' when going from the list to the diagonal has a meaning different from 'constructed' in constructable number. > > Obviously if the list is not > > constructable, the diagonal number is constructed from the list, but is > > itself not constructable. You apparently understand 'constructable' as > > being able to be constructed from something, whatever that is. But > > that term has a
From: Dik T. Winter on 20 Oct 2006 21:50 In article <1161378187.155995.290420(a)f16g2000cwb.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > In article <1161276574.792436.186750(a)i3g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: .... > > Yes, but it is infinitely many times not continuous in each neighbourhood > > of the limit point. > > Let t be the ordinal number of transactions. > > It *is* continuous like a staircase. > X(t) = 9 for t = 1 until t = 2 where X(t) switches to 18. That is > enough to excflude X(omega) = 0. How do you *define* X(omega). As far as I know X is only defined for real numbers, and omega is not one of them. And I see no reason to exclude X(omega) = 0, = 1, = -1 at all from this reasoning. > > That is still not a mathematical formulation. More is required. You > > need to actually state what you mean with 'number of balls in the vase > > at noon' or 'natural numbers in the set at noon'. > > The number of transactions t is then t = omega at noon. Is *that* a mathematical definition? Pray provide a real mathematical definition. > > The limit can in mathematics *always* be determined by the terms (if there > > is a limit). But the limit in no case defines the function value at the > > limit point. > > Then the irrational numbers as limit points are undefined. You think so. The irrational numbers are defined to be the limits of some particular sequences (or rather as equivalence classes of sequences). I think you have no idea how numbers (yes, I use that term while you think it is disgusting) are defined. I will repeat: (1) start with the natural numbers as defined by Peano (you may start with 0, 1 or 2, but starting with 0 makes everything a bit easier). (2) define arithemetic with those numbers, using the axioms. (3) define negative numbers as pairs of two elements, the first is a single bit (the sign), the second is a natural number. (4) provide arithemetic with the negative numbers. Show that the natural numbers from (2) can be embedded in this, Call this the integers. (5) assume pairs of integers, and provide equivalence classes amongst these pairs (i.e. (a, b) ~ (p, q) iff a.q = b.p). (6) provide arithmetic with such pair. Show that the integers can be embedded in this. Call these the rationals. (7) assume sequences of rationals. Create equivalence classes amongst those sequences (a_n ~ b_n if |a_n - b_n| goes to 0; but this is losely speaking and quite a few other methods are known, all equivalent). (8) provide arithmetic for those sequences. Show that the rationals can be embedded in this. Call these the reals. So, at what stage in this process is the limit of a function used to define the irrationals? > > The irrational numbers are by definition the limits of particular > > sequencens. And that is not set theory. > > Everything is a set. Will you object that irrational numbers are > defined on the basis of set theory? No, not in principle, because also the natural numbers are defined on the basis of set theory. Unless you can give a proper mathematical definition that goes beyond handwaving like: III = apple+orange+pear = 3. -- dik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/
From: Virgil on 20 Oct 2006 22:42 In article <J7GqFL.9AE(a)cwi.nl>, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: > In article <1161378187.155995.290420(a)f16g2000cwb.googlegroups.com> > mueckenh(a)rz.fh-augsburg.de writes: > > > In article <1161276574.792436.186750(a)i3g2000cwc.googlegroups.com> > > > mueckenh(a)rz.fh-augsburg.de writes: > ... > > > Yes, but it is infinitely many times not continuous in each > > > neighbourhood > > > of the limit point. > > > > Let t be the ordinal number of transactions. > > > > It *is* continuous like a staircase. > > X(t) = 9 for t = 1 until t = 2 where X(t) switches to 18. That is > > enough to excflude X(omega) = 0. > > How do you *define* X(omega). As far as I know X is only defined for real > numbers, and omega is not one of them. And I see no reason to exclude > X(omega) = 0, = 1, = -1 at all from this reasoning. > > > > That is still not a mathematical formulation. More is required. You > > > need to actually state what you mean with 'number of balls in the vase > > > at noon' or 'natural numbers in the set at noon'. > > > > The number of transactions t is then t = omega at noon. > > Is *that* a mathematical definition? Pray provide a real mathematical > definition. > > > > The limit can in mathematics *always* be determined by the terms (if > > > there > > > is a limit). But the limit in no case defines the function value at the > > > limit point. > > > > Then the irrational numbers as limit points are undefined. > > You think so. The irrational numbers are defined to be the limits of some > particular sequences (or rather as equivalence classes of sequences). I > think you have no idea how numbers (yes, I use that term while you think > it is disgusting) are defined. I will repeat: > (1) start with the natural numbers as defined by Peano (you may start with > 0, 1 or 2, but starting with 0 makes everything a bit easier). > (2) define arithemetic with those numbers, using the axioms. > (3) define negative numbers as pairs of two elements, the first is a single > bit (the sign), the second is a natural number. One can extent from the naturals to the integers in other ways as well. One way is as equivalence classes of pairs of naturals (a,b), a and b naturals, with (a, b) is equivalent to (c,d) if and only if a+d = b+c. > (4) provide arithemetic with the negative numbers. Show that the natural > numbers from (2) can be embedded in this, Call this the integers. > (5) assume pairs of integers, and provide equivalence classes amongst > these pairs (i.e. (a, b) ~ (p, q) iff a.q = b.p). > (6) provide arithmetic with such pair. Show that the integers can be > embedded in this. Call these the rationals. > (7) assume sequences of rationals. Create equivalence classes amongst those > sequences (a_n ~ b_n if |a_n - b_n| goes to 0; but this is losely > speaking and quite a few other methods are known, all equivalent). > (8) provide arithmetic for those sequences. Show that the rationals can > be embedded in this. Call these the reals. > > So, at what stage in this process is the limit of a function used to > define the irrationals? > > > > The irrational numbers are by definition the limits of particular > > > sequencens. And that is not set theory. > > > > Everything is a set. Will you object that irrational numbers are > > defined on the basis of set theory? > > No, not in principle, because also the natural numbers are defined on the > basis of set theory. Unless you can give a proper mathematical definition > that goes beyond handwaving like: III = apple+orange+pear = 3.
From: cbrown on 21 Oct 2006 00:32 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > <snip> > > I do not understand you. You cannot count all naturals and at some time > > saying that you are finished. > > (1) The set of natural numbers exist, by the axiom of infinity. > > (2) You will never terminate when counting through the natural numbers. > > If we could not get to omega, we would not need it. On the contrary. /Because/ we cannot "get to" omega, we absolutley "need" it (if we decide that we want it at all). And if we /could/ "get to" omega, we would /not/ "need" to say anything special about it, except to define it. More precisely: If, * starting with the empty set and then repeatedly deducing that for each succesive set x, we "can get to" (show the existence of) the set x union {x}, it then /logically followed/ that: * we "can get to" (show the existence of) a set whose members satisfy the requirements described by omega in AoI, then we wouldn't "need" (have to separately assume) the *axiom* of infinity in order to talk about omega actually being a set. It would simply /logically follow/ as a *theorem* from the other axioms. ZFC excluding AoI does /not/ say "you can get to omega" in the sense you are using "can get to" here. Instead, ZFC /with/ AoI says, "since, if we are honest, we have to admit that you /can't/ 'get to' omega using the other axioms, we must therefore /assume/ omega's existence, in order to talk logically about arguments that assume omega is a set in the first place". But so what? In a mathematical discussion amongst set theorists, you /don't have to/ accept that it's true or obvious or reasonable or sensible or accords with current dogma or is religious law or is required under punishment of banishment or /whatever/, that "omega is a set". Of course that assumes that you have /some/ axioms in mind: otherwise it is simply not a mathematical question whether your statements actually follow, one from the other, in your argument. It is instead an argument of philosophy. To clarify: suppose we agree that "since there is a set having the properties of omega, therefore ..." is not a valid for for a correct argument. Do you have complaints about the remaining assumptions and axioms of ZFC? Or do you find them (and the logical conclusions that follow from them) to be acceptable as being true, obvious, logical, correct, etc. in mathematical discourse? Cheers - Chas
From: stephen on 21 Oct 2006 01:50
Han de Bruijn <Han.deBruijn(a)dto.tudelft.nl> wrote: > Bob Kolker wrote: >> Han de Bruijn wrote: >>> >>> True. And you haven't seen any binary tree either. >> >> Bullshit. One can trivially construct finite binary trees. To "see" one >> is to think one. We can think binary trees as simply as we can think of >> a triangle with one of its sides removed. Three points, two sides. V for >> victory. > Did I say that _you_ haven't seen any binary tree? I thought this was > a response to David Marcus, who hasn't seen any, it seems. > Han de Bruijn You are just resorting to lame insults Han. You must be out of arguments. Stephen |