Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 1 Dec 2006 05:17 On 11/30/2006 1:34 PM, Bob Kolker wrote: > Giants, such as Hilbert, welcomed Cantorian mathematics. It was David Hilbert who declared Cantor's CH the first out of 23 problems of mathematics, and he did so in 1900 at 2nd International Congress of Mathematicians when he was 38 years old, full professor for 7 years and chair of mathematics at U of Goettingen for 5 years. Cantor was founder and president of the Society. Both Hilbert and Cantor benefitted from their close friendship with Hurwitz. Because Hilbert successfully applied axioms to geometry in 1999, and set theory did not have a reliable basis, he may have felt in position and was perhaps invited to provide axioms for set theory, too. However, despite of beeing desined for this task, he did lower his standard and invent the masterpiece of delusion which has been ascribed to Zermelo: claiming the existence of infinite sets. Hilbert wrote in 1923: "waehrend es fueher ohne die aximatische Methode naiv geschah, dass man an gewisse Zusammenhaenge wie an Dogmen glaubte, so hebt die Axiomenlehre diese Naivitaet auf, laesst uns jedoch die Vorteile des Glaubens". Whether or not Hilbert himself did really belief in these tenets might be questionable. At least in his speech in honour of Weierstrass in 1925, he probably tried to cover up some of the most nonsensical elements of Cantor's theory by wording like "einfaches Hinueberzaehlen" and too emphatic rhetoric.
From: Bob Kolker on 1 Dec 2006 06:57 Eckard Blumschein wrote:> > Serious mathematicans have to know the pertaining confession. Dedekind > wrote: "bin ausserstande irgendeinen Beweis f�r seine Richtigkeit > beizubringen". In other words, he admitted being unable to furnish any > mathematical proof which could substantiate his basic assumption. > Consequently, any further conclusion does not have a sound basis. > Dedekind's cuts are based on guesswork. Dedikind cuts are well defined objects which have exactly the algebraic properties one wishes real numbers to have. So it is quite sensible to identify the cuts with real numbers. We can define addition, mutliplication, subtraction and division for cuts and the cuts satisfy the postulates for a an ordered field. Furthermore every set of cuts (identified with real numbers) with an upperbound has a least upper bound. Bingo! Just what we want. Bob Kolker
From: Eckard Blumschein on 1 Dec 2006 07:26 On 11/30/2006 5:13 PM, Bob Kolker wrote: > Eckard Blumschein wrote: >> >> The argument Cantors transfinite numbers are somthing positive something >> progressive is old and has proven wrong. Not even aleph_2 has found an >> application. > > So what? The criterion for goodness in pure mathematics is consistency, > not usability. After that aesthetic issues dominate. Are the systems > interesting. Do they have a kind of beauty? etc. etc. > > Bob Kolker I like mathematical beauty very much. Is Buridan's donkey really beautiful? Are more than indefinitely many numbers beautiful? You should read Foundations of Set Theory by Fraenkel et al. in order to get an impression how uggly set theory is. Maybe it is harmless. At least the authors do not intend killing people. Mathematics will survive set theory.
From: Eckard Blumschein on 1 Dec 2006 08:44 On 11/30/2006 5:15 PM, Bob Kolker wrote: > Eckard Blumschein wrote: >> >> >> Either discrete or continuous. Nothing in between. > > You obviously have no knowledge of fractal dimension or Hausdorf > dimension. For example the Peano space filling curves have a dimension > between 1 and 2. > > Bob Kolker WM wrote a nice booklet Die Geschichte des Unendlichen. Chapter V. "Unbegrenzt" is devoted to such stuff like Moebius's band and Peano's line. When I used Matlab, I got aware of the Mandelbrot fractal. So you are not telling me something new. I still tend to consider any continuous line including the Sierpinski triangle like something which has just one dimension, regardless of selfsimilarity. Fractal dimensions as suggested by Hausdorff (with double f like iff) do not invalidate this view.
From: Tony Orlow on 1 Dec 2006 10:50
Eckard Blumschein wrote: > On 11/29/2006 6:37 PM, Bob Kolker wrote: >> Tony Orlow wrote: >>> It has the same cardinality perhaps, but where one set contains all the >>> elements of another, plus more, it can rightfully be considered a larger >>> set. >> Not necessarily so, if it is an infinite set. >> >> Bob Kolker What, is it not necessarily so that it CAN rightfully be considered a larger set, with some justification? It doesn't have a right to be considered that way? Why? Because it contradicts theoretical transfinitology? > > This time I agree with Bob. > You do? Do you mean that the addition of elements not already in a set doesn't add to the size of the set in any sense? Hmmm... Can x+y=x, when y>0? That basic tenet of what "addition" means seems like a fundamental concept which should be preserved, but is not reflected in the results derived from the definition of cardinality. In other words, there are fundamental concepts of addition and subtraction which are lost in the infinite case, when only looking at bijections, without regard to whether they map one set of points in a sequence/continuum with another set in the same sequence/continuum. That shared measure, of the real line or the set of naturals, for instance, provides some means of mathematically justifying the notion of various infinities, whether over a standard countable range, or a non-standard uncountable range, with actually infinite distances between elements, such as we find with 0.3333... - 0.2222... = 0.1111..., an infinite distance, in terms of string successorship. It seems absurd to claim, for instance, that there is no justification for wanting to see results like, for instance, that the even naturals are a set half the size of the naturals, or any other of an infinite number of subset relations not reflected in the numbers. :) Have a nice day. Tony |