Cantor Confusion
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From: jpalecek on 17 Oct 2006 07:50 mueckenh(a)rz.fh-augsburg.de wrote: > jpalecek(a)web.de schrieb: > > > mueckenh(a)rz.fh-augsburg.de napsal: > > > Dave L. Renfro schrieb: > > > > > > > Peter Webb wrote (in part): > > > > > > > > >> This is a complete red herring. There is no question that > > > > >> the Real generated by Cantor's proof is computable (r. e,) > > > > >> if the original list is, [...] > > > > > > > > mueckenh(a)rz.fh-augsburg.de wrote (in part): > > > > > > > > > Of course. That's why the diagonal proof only proves the > > > > > existence of numbers which belong to a countable set i.e. the > > > > > set of constructible reals. This proof proves in essence that > > > > > the countable set of constructible real numbers is uncountable. > > > > > A fine result of set theory. > > > > > > > > You're overlooking Peter Webb's hypothesis "if the original > > > > list is". You need to have a list (x_1, x_2, x_3, ...) such > > > > that the function given by n --> x_n is computable. Thus, > > > > before you can conclude what you're saying (which sounds like > > > > a metalogic "proof by contradiction" to me, but no matter), > > > > you need to come up with a computable listing of the computable > > > > numbers (or at least, show that such a listing exists). > > > > > > One cannot compute a list of all computable numbers. By this > > > definition, > > > (1) the computable numbers are uncountable. > > > (2) There is no question, that the computable numbers form a countable > > > set. > > > This is a contradiction. It is not necessary to come up with a list of > > > all computable numbers. > > > > There is no contradiction. > > > > The fact that you cannot compute a list of all computable reals does > > not mean that there is no list of all computable numbers. There is one, > > > > and it is not computable. > > > The fact that you cannot compute a list of all reals does not mean that > there is no list of all reals. There is one, but it is not possible to > publish this list. Again, stop mixing computability theory with set theory if you can't figure out which is which. The theorems we're talking about are For ANY list of reals, there is a number which isn't in that list. For ANY COMPUTABLE list of COMPUTABLE reals, there is a COMPUTABLE real which isn't in that list. Can you spot the difference? > Every list of reals can be shown incomplete in exactly the same way as > every list of computable reals can be shown incomplete. No. The nonexistence theorems which follow are There isn't ANY complete list of reals. There isn't ANY COMPUTABLE complete list of COMPUTABLE reals. See how they correspond? Asking whether a given program defines a computable real is a Pi_2 question, therefore you can generate the list of all computable reals with a Pi_2-recursive function. This is far far less than uncountable. Regards, JP
From: jpalecek on 17 Oct 2006 08:01 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > > > > > Consider the binary tree which has (no finite paths but only) infinite > > > > > paths representing the real numbers between 0 and 1. The edges (like a, > > > > > b, and c below) connect the nodes, i.e., the binary digits. The set of > > > > > edges is countable, because we can enumerate them > > > > > > > > > > 0. > > > > > /a \ > > > > > 0 1 > > > > > /b \c / \ > > > > > 0 1 0 1 > > > > > ..................... > > > > > > > > > > Now we set up a relation between paths and edges. Relate edge a to all > > > > > paths which begin with 0.0. Relate edge b to all paths which begin with > > > > > 0.00 and relate edge c to all paths which begin with 0.01. Half of edge > > > > > a is inherited by all paths which begin with 0.00, the other half of > > > > > edge a is inherited by all paths which begin with 0.01. Continuing in > > > > > this manner in infinity, we see that every single infinite path is > > > > > related to 1 + 1/2 + 1/ 4 + ... = 2 edges, which are not related to any > > > > > other path. > > > > > > > > Are you using "relation" in its mathematical sense? > > > > > > Of course. But instead of whole elements, I consider fractions. That is > > > new but neither undefined nor wrong. > > > > > > > > Please define your terms "half an edge" and "inherited". > > > > > > I can't believe that you are unable to understand what "half" or > > > "inherited" means. > > > I rather believe you don't want to understand it. Therefore an > > > explanation will not help much. > > > > In standard terminology, a "relation between paths and edges" means a > > set of ordered pairs where the first element of a pair is a path and the > > second is an edge. Is this what you meant? > > Yes, but this notion is developed to include fractions of edges. > > > > I am at a loss as to how "half an edge" can be "inherited" by a path. > > An edge is related to a set of path. If the paths, belonging to this > set, split in two different subsets, then the edge related to the > complete set is divided and half of that edge is related to each of the > two subsets. If it were important, which parts of the edges were > related, then we could denote this by "edge a splits into a_1 and a_2". > But because it is completely irrelevant which part of an edge is > related to which subset, we need not denote the fractions of the edges. Sorry, but your "proof" doesn't work. Imagine an infinite path in the tree. Which is the edge it inherits as a whole? Whenever you give me that edge, I can tell you're lying because if a path inherits an edge as a whole, it means that the path terminates by that edge. This is impossible for infinite paths. The same argument applies to other terms in the sum. (That edge is inherited by an infinite path by 1/1024! Ok, but that means that the path terminates 10 levels lower). This means that infinite path inherit zero edges in your proof. Regards, JP
From: Han de Bruijn on 17 Oct 2006 08:23 stephen(a)nomail.com wrote: > You have not answered the question about how one determines > if a thing in the real world is true. I can guess you > will say something about measurements, but how does one > know that your measurements are "true", or that they truly > correspond to "a thing in the real world", and so on. > It is a big ugly kettle of philosophical fish. Not only philosophical fish. Also religious fish. And political fish. And scientific fish. Actually everyday's life fish. You are right! > I agree that it is sensible to assume that the Universe > is consistent, but given how strange and unintuitive > the Universe can be, who knows. I think we agree on the above. But it doesn't mean that we cannot answer _part_ of the question: do INFINITIES exist or not. Are they true or are they false? And IMO _that_ can be decided _now_, without rocket science. Han de Bruijn
From: Han de Bruijn on 17 Oct 2006 08:25 stephen(a)nomail.com wrote: > Han.deBruijn(a)dto.tudelft.nl wrote: > >>Virgil schreef: > >>>In article <290c1$45333e14$82a1e228$8972(a)news2.tudelft.nl>, >>> Han de Bruijn <Han.deBruijn(a)DTO.TUDelft.NL> wrote: >>> >>>>Dik T. Winter wrote: >>>> >>>>>In article <1160857746.680029.319340(a)m7g2000cwm.googlegroups.com> >>>>>Han.deBruijn(a)DTO.TUDelft.NL writes: >>>>> > Virgil schreef: >>>>>... >>>>> > > I do not object to the constraints of the mathematics of physics when >>>>> > > doing physics, but why should I be so constrained when not doing >>>>> > > physics? >>>>> > >>>>> > Because (empirical) physics is an absolute guarantee for consistency? >>>>> >>>>>Can you prove that? >>>> >>>>Is it possible to live in a (physical) world that is inconsistent? >>> >>>The consistency of the physical world did not guarantee the consistency >>>of the Phlogiston theory of combustion. Being a physicist is not a >>>guarantee of being right, or of being consistent. Every physical theory >>>must be, at least in theory, falsifiable, so that none of them can be >>>held to be infallibly consistent. > >>I'm not talking about a theory. I'm talking about the world as it IS. > > And exactly how IS the world? The secret is in the word "exactly". > We have nothing but theories about the world. We do not, > and cannot, know the world as it IS. On the contrary. We CAN know the world as IS, because it IS NOT EXACTLY. Han de Bruijn
From: Han de Bruijn on 17 Oct 2006 08:29
Virgil wrote: > In article <1161029391.305685.141910(a)b28g2000cwb.googlegroups.com>, > Han.deBruijn(a)DTO.TUDelft.NL wrote: > >>Sure, theories. Can't you talk about something else but "theories"? > > Isn't the point of physics to come up with theories? AND experiments. All physical theories are judged by experiments. > And now a physicist wants to outen them? Han de Bruijn |