Cantor Confusion
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From: David Marcus on 12 Nov 2006 17:13 Lester Zick wrote: > On Sat, 11 Nov 2006 15:41:39 -0500, David Marcus > <DavidMarcus(a)alumdotmit.edu> wrote: > >Lester Zick wrote: > >> On Fri, 10 Nov 2006 17:33:16 -0500, David Marcus > >> <DavidMarcus(a)alumdotmit.edu> wrote: > >> >A silly question. You haven't even bothered to learn the current meaning > >> >of the word "definition". If you would actually read a textbook or take > >> >a math course at a university, you might learn something. > >> > >> Curious coming from one whose definitions aren't demonstrably true. > > > >Care to define "demonstrably true"? > > Already have in the root post to the thread "Epistemology 201: The > Science of Science". Your main problem seems to be a willingness to > assume the truth of whatever you're talking without demonstration. You > repeatedly use the term "true" in a collateral reply to Gene Ward > Smith while unable to demonstrate the truth of what you claim to > prove. 'Tis a puzzlement indeed for one trained in precise and exact > meanings of modern mathematics. The word "true" has different meanings in different contexts. Its meaning in mathematics is somewhat unusual. Usually in mathematics, it just means "provable". With this meaning, it makes no sense to say something is true, but not provable. It also makes no sense to ask whether a definition is true, since we don't prove definitions. They are simply abbreviations. In logic, "true" has a technical meaning. Outside mathematics, "true" has an entirely different meaning. I don't know what you mean by "demonstrably true", but I am reasonably sure you don't mean what the word "true" means in mathematics. I'm not particularly interested in reading old threads. If you care to give a concise explanation again, I'll read it. Mathematics is a language. If you take Mathematics and assume it is English, you will misunderstand what the writer means. -- David Marcus
From: David Marcus on 12 Nov 2006 17:23 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > David Marcus schrieb: > > > > > > > > We need no experts in trees. This tree does nothing else but represent > > > > > the real numbers of an interval. Here it is in a form which can be > > > > > understod by ever mathamatician. > > > > > > > > > > Consider a binary tree which has (no finite paths but only) infinite > > > > > paths representing the real numbers between 0 and 1 as binary strings. > > > > > The edges (like a, b, and c below) connect the nodes, i.e., the binary > > > > > digits 0 or 1. > > > > > > > > > > > > > > > > > > > > 0. > > > > > > > > > > /a \ > > > > > > > > > > 0 1 > > > > > > > > > > /b \c / \ > > > > > > > > > > 0 1 0 1 > > > > > > > > > > .......................... > > > > > > > > > > > > > > > > > > > > The set of edges is countable, because we can enumerate them. 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. > > > > > > > > Half of the paths include edge a. One quarter of the paths include edge > > > > b. > > > > > > > > Suppose we have a finite tree of height n. Then there are 2^n paths. > > > > Consider a path (e1,e2,...,en). Edge e^j is contained in 2^-j of the > > > > paths. We can define a real-value function on paths by g(e1,e2,...,en) = > > > > sum_{j=1}^n 2^-j = 1 - 2^-n. Not sure what to do next... > > > > > > > > > Continuing in this > > > > > manner in infinity, we see by the infinite recursion > > > > > > > > > > f(n+1) = 1 + f(n)/2 > > > > > > > > What is f(n)? > > > > > > f(n) is the number of edges related to the initial segment of one path > > > which has passed through the first n edges. > > > > Are you saying that f(1) is the number of edges "related" to the paths > > that contain edge a? What is the value of f(1)? > > f(1) = 1 is the number of edges related to initial segment (of the > path) that contains edge a. Still don't get it. Let's try an example. Consider the path where we always go left at each node. This path has initial segments. Each such initial segment contains a. But, what is f(1) supposed to be? > We start with the first edge a of a path. Then we see the path splits > into two paths. So half of edge a is related to each one and so on. Of > course we never get ready, but there is no edge which remains > unconsidered. So, when the height of the tree is 1, there is one path containing edge a. When the height is 2, there are 2 paths containing edge a. When it is 3, there are 4 paths. In general, when the height is n, there are 2^{n- 1} paths containing edge a. If we let the tree go on forever, then there are an infinite number of paths that contain edge a. Is that what you mean? If so, what is the next step in your argument? -- David Marcus
From: MoeBlee on 12 Nov 2006 19:28 Lester Zick wrote: > On 10 Nov 2006 15:18:07 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > > >Lester Zick wrote: > >> Your way or the highway huh, Moe(x). > >If he has an argument that he thinks can be put in set theory, then I'm > >interested in his argument; If he doesn't think his argument can be put > >in set theory, then I'm not interested. He can post about his argument > >all he wants, but I'm not obligated to study his argument. > > No one suggests you are. The problem I see is that one might cast an > argument in such terms as are acceptable to you and still not satisfy > exactly the same criteria on the part of others. I mean unless you are > the generally acknowledged expert in the field. Otherwise it would > look to me like you're just trying to take control of the discussion > in terms you find acceptable whether or not others do. Nothing of the kind. HE suggested to ME that I can see if his argument can be put into set theory. If I am take MY time and effort to do that, then I have every prerogative to set my own terms for doing it. > Let me see if I can simplify how the issue is or ought to be argued. > You posit certain properties of an infinite however you define it. So > the question then becomes whether your claim is or can be true. Now > one way to show it's actually true would be to produce some entity > with the properties you posit of an infinite. Otherwise you'd have to > find some other way to get at the truth of what you claim unless you > just intend to claim it's true because you or others say so. > > Now as I understand WM's argument he suggests you can never actually > produce any physical infinite because the physical universe is finite. WM and were talking about his tree argument. The finititude of the physical universe is a separate subject. > However he then apparently concludes from this that there can be no > infinites at all because there can be no physical infinites if the > universe is finite. > > Now personally I find most of the arguments disingenuous on both > sides.And I see no special merit to your definition for the properties > of infinites you recommend to the exclusion of others. But they are > specific properties you can't demonstrate through exemplification so > if you wish to show that the characteristics you assign to infinites > can be true you have to approach the proof some other way than > empirical exemplification. MoeBlee
From: MoeBlee on 12 Nov 2006 19:40 mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > > p. 118, considering that the set of all sets does not exist, they > > > write: a reasonable way to make this conform to a Platonistic point of > > > view is to look at the universe of all sets not as a fixed entity but > > > as an entity capable of "growing". > > > > > > Why should they write "reasonable" if they found it unreasonable? > > > > Because, as YOU SKIPPED RESPONDING to my explanation for you, and as > > would be clear had you bothered to read and UNDERSTAND the remarks in > > that section of the book, theyr'e NOT claiming that sets in general > > grow or are indeterminate. They are pointing out that if we adopt > > different axioms, then the entire universe of all sets is something > > different depending on which axioms we adopt. > > I will not discuss this topic further. Only a last remark for the > lurkers: In fact, there are axiom systems which tolerate the set of all > sets. Therefore, if different axiomatic systems were discussed by > Fraenkel et al., no capability of "growing" was required at all. That's a non sequitur. They discussed the options of different axioms and mentioned that the results are that we can prove the existence of certain sets in some theories and in other theories we can't prove the existence of those sets. > Further there is no talk about "something different" sets but about > growing sets. No, "growing" in the AUTHOR'S own scare quotes to convey a notion you ignore as you ignore the entire context of the section of that book. > Therefore exactly this growing of sets is meant as a > reasonable view. The authors expound no endorsement that sets themselves grow or indeterminate. The axiom of extensionality is used throughout the book and the author's comments in that particular section do not contradict that a set is compelely determined by its members; rather, the authors remark that we get a different UNIVERSE of sets depending on what axioms we adopt, so that the UNIVERSE (not just sets in general) "grows" (notice scare quotes) depending on what axioms we adopt. "Grow" is used in scare quotes by the authors to convey that we get different universes from different axioms; and there is NOTHING in the book to suggest that the authors endorse a notion that sets themselves grow over time or through any process or that sets do not have a determinate membership as per the axiom of extensionality. > Unfortunately MoeBlee is far from understanding > anything behind his formalism. "The formalists are like a watchmaker > who is so absorbed in making his watches look pretty that he has > forgotten their purpose of telling the time, and has therefore omitted > to insert any works." (Bertrand Russell) I've never posted a committment to an UNDEFINED formalism, and not to any such extreme formalism that holds that mathematics is only symbol manipulation. MoeBlee
From: MoeBlee on 12 Nov 2006 19:44
mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > > So let's be definite here. Do you have a strong belief that your > > argument can be expressed in the language of set theory and uses only > > first order logic applied to the axioms of set theory? > > I need no belief. If all consistent mathematics can be expressed in > ZFC, then my argument can be expressed in ZFC too. First you said that your argument is about set theory. Then said your argument has "nothing to do with" set theory. Now you're saying that if "all consistent mathematics can be expressed in ZFC" then your argument can be put in ZFC. If you would at least say clearly just what your argument about trees is supposed to be about - to what theory it is supposed to apply - then that would help. MoeBlee |