Cantor Confusion
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From: David Marcus on 29 Oct 2006 11:04 mueckenh(a)rz.fh-augsburg.de wrote: > Dik T. Winter schrieb: > > In article <1161861368.657796.161130(a)k70g2000cwa.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > > Dik T. Winter schrieb: > > > If you mean that there are more finite definitions than numbers, you > > > are right. Therefore we have a countable set of constructable numbers. > > > > Pray, refrain from using non-standard terminology. WM-construcatble =3D > > computable. > > Definiert man die reellen Zahlen in einem streng formalen System, in > dem nur endliche Herleitungen und festgelegte Grundzeichen zugelassen > werden, so lassen sich diese reellen Zahlen gewi=DF abz=E4hlen, weil ja > die Formeln und die Herleitungen auf Grund ihrer konstruktiven > Erkl=E4rungen abz=E4hlbar sind. (Kurt Sch=FCtte, Hilbert's last pupil) > > Finite definitions! Call this set the set of computable numbers. > > > > > > When you substitute "computable" for "constructable", both are indeed > > > > right. But Cantor was wrong. It is only after Turing that this was > > > > solved. The finite definitions are countable, but not every finite > > > > definition gives a number. So there exists a list of finite definitions > > > > (this has been formalised using Turing machines), but this is not a > > > > list of computable numbers, because there will be non-halting Turing > > > > machines in the list, and you do not know how to separate them from > > > > the rest. Look up the halting problem. There is no computable list > > > > of computable numbers. > > > > > > Why do you insist on this obvious fact? It does not support your > > > position. There is a list of all finite constructions or definitions > > > (encoded by numbers, G=F6del). This is the definition of countablity. > > > > I thought that G=F6del encoded theorems? > > Encode theorems as well as formulas for constructing numbers. Encode > everything which can be expressed by a finite sequence of letters. Then > build the diagonal number. This is also a finite sequence of letters, > because it cannot be longer than the finite list numbers. > > > But it is trivial (you do not > > need G=F6del for that) to find that the number of finite definitions is > > countable. > > All finite sequences are countable. They yield another finite sequence. > Hence they are uncountable. Contradiction. 1. Are you saying that the contradiction is provable within some standard mathematical system, e.g., ZFC? 2. Are you saying that the contradiction is true within your own system? -- David Marcus
From: mueckenh on 29 Oct 2006 11:13 William Hughes schrieb: > > > There is no computable list > > > of computable numbers. > > > > Why do you insist on this obvious fact? It does not support your > > position. There is a list of all finite constructions or definitions > > (encoded by numbers, Gödel). This is the definition of countablity. > > The diagonal number of this list shows the listed numbers are > > uncountable. Contradiction. > > > There is indeed a list of all possible finite constructions or > definitions. Call this list A. However, this list must contain things > that look > like definitions but are not because the method given to produce > a number does not halt. We cannot get a diagonal number from A, > because some of the members of A do not give numbers. > So there is no contradiction. A contains all finite words (construction fromulas, theorems). The diagonal (cannot be longer than the lines and hence) is a finite word too. That is enough to obtain a contradiction. > > So let's take list B. We get list B from A by taking only the true > definitions > from A (that is we throw away anything that will does not halt). > So now we have the list B, so by definition B is countable. > Now we can get a diagonal number, d. This number cannot be a member > of B. > > But note. d cannot have a finite definition. A finite definition of d > would > include not only the diagonal procedure (finite description) but also > the list B (infinite description) or a finite description of a way to > get the list B from the list A (i.e. a solution to the halting > problem). > > So d is not a member of B. If we do not take > a contrucivist viewpoint this is not a contradiction, there are lots > of things that are not a member of B. > > But let's say we take a constructivist viewpoint. d is not computable, > so d does not exist. (Nothing that is not a member of B > exists.) We still do not have a contradiction because > if we take a constructivist viewpoint the list B does not exist so we > cannot use it to form the diagonal d. Funny, but not sufficient to confuse a clear thinking mind sufficiently. Regards, WM
From: mueckenh on 29 Oct 2006 11:17 Sebastian Holzmann schrieb: > Virgil <virgil(a)comcast.net> wrote: > > In article <1161883732.413718.244570(a)k70g2000cwa.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > >> What you propose, namely the infinity of ZF without the axiom INF would > >> not be an advance. But meanwhile you may have recognized that your > >> assertion (ZF even without INF is not finite) is false. > > > > It is, however, quite true that ZF without INF need not be finite. > > It is, more than that, quite true that ZF without INF _is_ infinite Are you really sure? > (the axiom schema of separation alone provides infinitely many axioms). Are you really sure? > The point is: ZF without INF does not prohibit the existence of infinite > sets, nor does it force them to exist. It prohibits to speak of infinite sets and to recognize such sets. So one cannot be sure, but you are? Regards, WM
From: mueckenh on 29 Oct 2006 11:20 stephen(a)nomail.com schrieb: > Can you describe a continuous version of the problem where each > "unit" of water has a well defined exit time? A key part of > the original problem is that the time at which each ball is > removed is defined and reached. This is crucial to the problem. It is > not just a matter of rates. If you added balls 1-10, then 2-20, > 3-30, ... but you removed balls 2,4,6,8, ... then the vase is > not empty at noon, even though the rates of insertions and removals > are the same as in the original problem. So you cannot just > say the rate is 10 in and 1 out and base an answer on that. > The answer for any time t *before noon* is independent of the chosen enumeration of the balls. Doesn't that fact make you think a bit deeper? Regards, WM
From: mueckenh on 29 Oct 2006 11:23
MoeBlee schrieb: > David Marcus wrote: > > Not sure if he ever said precisely "within Z set theory", but he > > certainly said things very similar. Below are a few messages that I > > found. There are probably others. > > > > In the first, he says that "standard mathematics contains a > > contradiction". as ar as standard mathematics is derived from set theory an comes to the conclusion of an empty vase at noon or an uncountable set of reals. Classical mathematics is free of such contradictions. > >In the next two, he states there are "internal > > contradictions of set theory". In the next, I say that he says that > > "standard mathematics contains a contradiction", and he does not > > dispute this. > > Thanks. And at least a couple of times I said that I was reading his > argument to see whether it does sustain his claim about set theory, as > I mentioned specifically Z set theory. My proof of the binary tree covers all possible theories. And it should not cost you too much time to see that it is true. Regards, WM |