Cantor Confusion
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From: MoeBlee on 27 Nov 2006 15:25 mueckenh(a)rz.fh-augsburg.de wrote: > William Hughes schrieb: > > > > > > > If both, W and N, are countable, then renaming the elements of one of > > > > > them leads to an identity map. > > > > > > > > You are confusing bijections within the model to bijections > > > > outside of the model. > > > > > > And who told you that you were outside? > > And who told you that you were outside? > > > You are noting that it is not possible to construct a bijection > > in a nonstandard model and that it is > > possible to construct a bijection in the standard model. This > > is true, however, it is not a contradiction. > > In particular because there is neither a stadard model nor a > non-standard model of ZFC. You haven't proven that there is no model of ZFC. > And in particular because the expression "contradiction" is not pat of > ZFC. In Z set theory, we can formulate definitions of 'S is a contradiction' and 'T is inconsistent'. MoeBlee
From: Virgil on 27 Nov 2006 16:10 In article <456AD842.5000906(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/25/2006 7:59 PM, Virgil wrote: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > The form of induction I use is based on having, as one has within ZF and > > NBG, an infinite set of finite naturals. If one has the first natural > > and one has the successor to every natural, then one has all of the > > members of that infinite set of finite naturals. > > Isn't this Cantor's childish naive alias selfcontradictory assumption? It is Peano's assumption. > > What makes any number finite? Depends on one's definition of "finite". What is your definition of "finite" as applied to "numbers". For that matter, what is your definition of "number". > Do not confuse any with all. > The expression "all natural numbers" does not have a conceivable > meaning. Millions of people quite regularly conceive of a meaning for it. > Since the natural numbers count themselves That is easily refuted by in NBG, in which no natural number counts itself. > > You may have any natural you like. You may never get them all just by > counting. In order to have them all, you have to move onto the opposite > quasi devine point of view on the expense of loosing the ability to see > them like single elements. It is called assuming the axiom of infinity, which both ZF and NBG do.
From: Virgil on 27 Nov 2006 16:12 In article <456AD89D.1060001(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 11/25/2006 8:03 PM, Virgil wrote: > > In article <1164446180.867559.37730(a)l39g2000cwd.googlegroups.com>, > > mueckenh(a)rz.fh-augsburg.de wrote: > > > >> Philosophy of mathematics belongs to mathematics. > > > > > > If that is so then those like WM and EB who try to impose the philosophy > > of physics on math are out of bounds. > > Geometry is not outside mathematics. It is doable within set theory, and does not require physics.
From: Virgil on 27 Nov 2006 16:17 In article <1164634255.699055.138450(a)n67g2000cwd.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > I am only interested in sets of he form {1,2,3,..., n}. Only such sets > can be generated by induction. Induction does not "generate" any sets. The sets exist a priori as deducible from the axioms, and in ZF or NBG the axioms require an infinite set >Their finiteness is sufficient for my > purposes. WM has no purposes.
From: Lester Zick on 27 Nov 2006 17:13
On Mon, 27 Nov 2006 01:50:38 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >In article <t7pjm2pnrbh3ut0s2ll5ehvo1mi1nh26je(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > > On Sun, 26 Nov 2006 03:32:30 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > > wrote: >... > > > > If you begin by assuming this you wind up by assuming that > > > > and pretty soon you find yourself assuming what you were supposed to > > > > demonstrate in the first place, that square circles are conceivable. > > > > > >Well the latter was not an assumption. There was an explicit question. > > > > But no demonstrated answer. You just claimed square circles are > > conceivable according to a metric you can't demonstrate in the same > > Euclidean terms you claim to rely on. > >I did. But I think you did not understand it. Of course I didn't understand in the Euclidean terms you claim to rely on. The more important question is whether anyone can understand it. > > > > On the > > > > other hand if all you're describing are variable measures of distance > > > > then you'd have to explain how you obtain those measures without an > > > > underlying Euclidean metric and corresponding assumptions. You can't > > > > just assume them as modern mathematikers are wont to do. > > > > > >You read more in the word "metric" than is in there. > > > > Not at all. Even in the restricted sense of "distance measure" a term > > like "metric" has to be self consistent with underlying assumptions of > > measure and what is measured and how. > >The measure I give is consistent. Any taxi-cab driver knows the metric. Well I rather doubt that. Maybe Manhattan taxicab drivers know the metric. And maybe that's why they're taxicab drivers. > > >Now you require that without assumptions rather than metrics... That > > >is something completely different. Let us recap the Euclidean postulates: > > >(1) Two points determine a line > > >(2) Any line segment can be extended in a straight line as far as desired, > > > in either direction > > >(3) Given any length and any point, a circle can be drawn having the length > > > as radius and that point as center > > >(4) All right angles are congruent > > >(5) The parallel postulate > > >Now, that is precisely what I did use (after translation to analytical > > >geometry). > > > > The problem is you have a host of undefined terms for things like > > points, lines, length, circles, right angles, etc.If Euclid understood > > these things one way and you understand and employ them another > > they're not Euclidean assumptions at all just revisionist private > > definitions. To justify your claim that square circles are conceivable > > you also claim a metric you don't demonstrate on the basis of > > Euclidean assumptions you claim to rely on. > >Balderdash. Personally I prefer the Bronx cheer metric. Occasionally the Brooklyn metric comes to mind. But then I'm often swept away by the majesty of the Yonkers metric.I think I'll leave you to the Staten Island metric. > > > > >In that case you should use better formulations in your questions. > > > > > > > > And you should use better formulations in your answers. > > > > > >How could I give a better formulation than stating that in the Manhattan > > >measure it is possible? > > > > Your stating it has nothing to do with whether it's true. > >Later on I did show that it is true. You didn't show anything is true. > > > I did state that explicitly, while you did not > > >state that you did use Eucledian measure. > > > > No reason I should if the words themselves rely on the same Euclidean > > assumptions you claim to rely on. > >They do not. They don't what? Rely on the same Euclidean assumptions you claim to rely on? Or rely on some other assumptions you don't claim to rely on? > > > > Except that your answer relies on non Euclidean assumptions that > > > > circles are squares. > > > > > >No. I show that with a particular measure all circles are squares. There > > >is no assumption. > > > > So you say but so you don't demonstrate. What you say is nothing more > > than a claim. > >And I did show why it works. But you don't show the metric relies on the same Euclidean assumptions you claim to rely on. > > > > If I ask why one sided quadrangles are > > > > unimaginable and you reply that they aren't if you start counting from > > > > four would you consider your answer responsive to the question asked? > > > > > >You are using a few words that you did not define, and again using some > > >assumptions. Quadrangle I can understand. What is one-sided? Does it > > >mean that the top side is different from the bottom side? In that case > > >I can show you quadrangles on a Moebius strip that have only one side. > > >On the other hand, I think you are referring to the edges. In that case, > > >in topology a connected set (as a quadrangle is) has a single boundary. > > > > So you wind up with a four sided figure with one side? I don't see > > that you're taking your words or mine seriously. How is it you get > > from one side to another regardless of topology and how do you get > > past the points of intersection? > >In topology those points are not interesting. They're kind of interesting when you try to get around them without specifying how the trick is done. But then maybe we should consult a taxicab driver. Except I have no desire to insult taxicab drivers. > Try to learn a bit about >mathematics. I will if you will. > > > > All you're doing is answering a question that wasn't asked in terms > > > > employed by the original question. > > > > > >The terms employed by the original question did not refer to Euclidean > > >metric. So my answer was valid. > > > > Terms like "circle" and "square" don't refer to Euclidean metric? Okay > > then. > >I did define those terms in the article where I did show how it works. >And those definitions are not too uncommon: > A square is a rectangle where the distances between successive points > are equal. >and: > A circle is a figure where the distances from each point on the figure > to a central point are equal. >What more do you want? How about a little less casuistry for a change? I have no idea where you get these nonsensical definitions for squares and circles. "A square is a rectangle"? So what is a rectangle? Are the sides of rectangles straight lines? Are circles composed of straight lines? > > > > I can make up private definitions > > > > just like everyone else does but that doesn't make definitions true. > > > > > >A definition can not be false. > > > > Another interesting and rather self serving claim from someone who > > obviously prefers false definitions to true definitions. > >Indeed. Isn't it? From someone who prefers square circles it certainly is. > > > It can be the case that there is nothing > > >(within the current context) that satisfies the definition, but it is > > >still not false. > > > > So is it the case that definitions cannot be self contradictory or > > that self contradiction is not false? > >If there are contradictory requirements in the definition that does not >make the definition false, it only tells us that there is nothing that >satisfies the definition. It tells us considerably more about people who prefer the definition. > Consider the following definition: > a is the smallest Fermat prime larger than 65537 >Is that a false definition? If so, why? And when it is shown that >there are no Fermat primes larger than 65537 it suddenly becomes a >false definition? Sure. Why not? > > > > In fact my personal favorite private definition for distance metrics > > > > is one I made up for the real number line which runs 1, 2, e, 3, pi, > > > > 4, 5, . . but I don't try to pretend that when I'm trying to analyze > > > > real numbers that that is a true definition. > > > > > >That is not enough for a definition. > > > > Sure it is. I just said so and you just said that definitions cannot > > be false. > >I do not say the definition is false, I say the definition is insufficient. Then you say a lot of things which are insufficient. >What is the distance between 1 and sqrt(2)? If you can not tell from your >statement above, the definition is insufficient to supply a metric. Only in your dreams. I can most definitely show the exact distance between 1 and the square root of two using rac construction. We're still waiting for you to show us this magic Manhattan metric using anything but your daydreams. > > > > Besides as far as I can tell you still haven't answered my question as > > > > to whether there are any curves at all with the Manhattan measure. In > > > > fact I can't even find it reprinted above. > > > > > >Pray define "curve". > > > > Don't need to. For purposes of the present discussion it is enough to > > understand that circles are curves and squares aren't. > >But in that case my answers were sufficient. But in that case your answers are most definitely sufficient just not demonstrably true. > There are indeed curves: >the circles. They are at the same time also non-curves, but that is >not a problem. (Like in topology there are sets that are both open and >closed.) Yes well I daresay so: "There are indeed curves but they are also non curves but that is not a problem" because modern mathematikers are too lazy or stupid to consider the truth of what they're talking about. > > If you maintain > > otherwise I'd like to know how and whether there are other curved > > figures besides circles in the Manhattan metric which are squares. > >I still do not know what you mean by "curved figures". Then learn some mathematics. Maybe your true calling is indeed driving taxicabs. ~v~~ |