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From: Eckard Blumschein on 7 Apr 2005 04:07 Dear Dave Rusin, thank you for revealing your personal point of view. Being in the weakest possible position, I will nonetheless not swallow that set theory has a sound basis. The reason for me to deal with Cantor's original papers was not interest in history, religion, or philosophy. I am an experienced engineer, and I got furious because some experts objected against some simple and perhaps absolutely compelling conclusions of mine just in order to obey set theory. I should also tell you that I grew up in a frequently changing political environment. So I used to carefully check all sort of tenets. In the end, I am almost amused how many opponents of Cantor failed to show that he was wrong. You wrote > I confess that I don't know as much about math history as I should, I would not see any reason to deal with it if the fundamentals were sound. > but I offer no apologies for a lack of familiarity about religious > history or philosophy. This should be read as a serious reproach against Georg Cantor who was unable or unwilling or both to grasp the most basic rule oo+a=oo, in other word he did not grasp and/or deliberately denied the essence of what has been accepted the basic idea of infinity in all science since Aristotele and is still valid everywhere outside Cantorian mathematics. > In particular, I don't think it's really a _mathematical_ > question to ask for what Cantor's notions were. I see it just a temporary mathematical question. In principle you are right, Henri Poincarý wrote: Future Generations will consider set theory a disease to recover from. > > Certainly there is no justification for going from uncertainty about > Cantor's original ideas I felt forced to pretend uncertainty in order to hopefully reach mislead experts. Actually, Cantor's blunder is quite obvious. > to the question, > > Does set theory really lack a reliable basis? > since the "reliable basis" Thank you for putting "reliable basis" into quotation marks. > has been offered by (take your pick) > Zermelo-Frankl, Bernays, etc. I dealt not just with such mediocre proponents of a wrong concept like Zermelo and the remedies they introduced but also with Hilbert. You will find a few comments of mine at http://iesk.et.uni-magdeburg.de/~blumsche/M280.html Zermelo was just the editor of Cantor's work. In 1904, the Hungarian mathematician Julius Koenig objected against Cantor's claim of a well-ordered arrangement of the reals. Cantor was already mental ill, perhaps because he felt that his life-work was based on his fallacious notion of infinity, and he got again very stirred. Spontaneously, Zermelo came up with the blunt idea to just define without any proof his axiom of choice. What about Adolf Abraham Fraenkel, you should read his book "Einleitung in die Mengenlehre", Springer, Berlin (1923) in order to judge his anything but impartial attitude. > Are you really objecting to statements like > k + 1 = k for every infinite cardinal k > which is a _theorem_ of ZFC? Even if I am not a mathematician, I conclude from Cantor's violation of oo+a=oo that his whole concept of infinite cardinality is wrong. In ZFC, all sets are well-orderable, per definition. As far as I know, nobody was able so far to arrange the continuum of the reals like a well-ordered set. > (You can, of course, state that the axioms of ZFC do not model > anything which you find interesting, which is your choice. And you can > object that the axioms of ZFC may be inconsistent, which as I'm sure > you're aware is a possibility we cannot even disprove within ZFC. > And you can propose alternative axioms which you think better capture > your own intuitive sense of what sets are. I do not have any own ambition in that direction. I am just looking for a non-elusive basis of set theory. As I showed in M280, Even Cantor himself learned to some extent that creation ý la Dedekind has its limits due to the claim of mathematics to be self-consistent across its branches including geometry, analysis, etc. and to provide appropriate tools for application e.g. in physics. > But none of that takes away from the fact that "set theory" as > ordinarily studied is on as firm a foundation as any other branch of > mathematics.) I accept that "set theory" is on a firm but fundamentally wrong foundation, having no tenable basis at all. > dave (sci.math.research moderator) While I see a little chance to persuade you personally, I am fully aware that just very few personalities have enough courage as to purify mathematics from Cantor who (according to Kronecker) corrupted the students. > Having just read Lavine again and also original literature quoted in > http://iesk.et.uni-magdeburg.de/~blumsche/M280.html > I would like to ask for the insight behind what Lavine again and again > reiterated as Cantor's fundamental but for my feeling very strange > contribution to mathematics: oo, oo+1, oo+2, etc. > > I am aware of Cantor's distinction between what he called 'Infinitum > aeternum increativum sive Absolutum' referring to god and 'Infinitum > creativum sive Transfinitum' relating to nature. While god's infinitum > corresponds to the convincing definition by Spinoza, I am desperately > looking for any tennable definition or at least reasonable > justification of Cantor's own notion of infinity. Cantor's letter to > Cardinal Franzelin is not mathematically persuading to me. Does set > theory really lack a reliable basis? > > Dr.-Ing Eckard Blumschein With sincere sympathy, Eckard
From: Ross A. Finlayson on 7 Apr 2005 04:50 Hi, If your real numbers have basically a least positive real or iota-value, then they are naturally well-ordered by their normal ordering for positive reals or the widening spiral for positive and negative real numbers. If they don't, then nobody has an example for you, of a well-ordering of the reals. Many agree that there is a well-ordering of the real numbers or for that matter any set. They do. That's pretty simple. Where they do, the proof of nested intervals doesn't apply, and sets of numbers are measurable, with at worst non-standard measure. The antidiagonal result, that's kind of a different thing, and that basically resolves to ultrafinitism, or a dually minimal and maximal ur-element, that corresponds well to the Russell set, Burali-Fortian Ord, the Liar, the empty set, and the Ding-an-Sich and Being and Nothing, towards a theory that can be consistent, and Goedelianly complete, Quineanly. Using these proper names is concision, conciseness. If you get to looking at Turing, a variety of statements about Turing machines, that would seem to collide with, for example, what I say, in the infinite may be true because they're about the finite. I have a lot to learn about complexity and Kolmogorov. A lot of people base their proofs upon statements that ZF is consistent. Many of those theorems are still correct when ZF is determined to be inconsistent. Where the physical universe is all physical objects, and a physical object, that's similar, in a way, to a set being a set of all sets, about concreteness. That means it's empirical evidence of that kind of thing. Infinite sets are equivalent. Ross -- "This style is hard."
From: Eckard Blumschein on 7 Apr 2005 06:31 On 4/7/2005 10:50 AM, Ross A. Finlayson wrote: > Hi, > > If your real numbers have basically a least positive real or > iota-value, then they are naturally well-ordered by their normal > ordering for positive reals or the widening spiral for positive and > negative real numbers. I would appreciate you reading M280 first before commenting. Let me explain why I am objecting against the possibility of arranging the real numbers like a well-ordered set of numbers. I do not deny that the continuum corresponds to an ascending order. However, it is impossible to identify two immediately subsequent real numbers. After performing border crossing into the actual infinity of continuum, any number, even the embedded natural ones lost their property of being numerically approachable. I know that the least value was (wrongly) considered the problem, and Zermelo came up with the axiom of choice for that reason. > > If they don't, then nobody has an example for you, of a well-ordering > of the reals. Many agree that there is a well-ordering of the real > numbers or for that matter any set. > > They do. That's pretty simple. Where they do, the proof of nested > intervals doesn't apply, Nested intervals belong to the rational numbers. When I refer to real numbers I am referring to Cantors definition demanding lim n->oo. > and sets of numbers are measurable, with at > worst non-standard measure. I commented on non-standard analysis elsewhere in German language. On request I will translate my comment. > > The antidiagonal result, that's kind of a different thing, and that > basically resolves to ultrafinitism, I just share Kronecker's finitist position in so far that, while I do not deny the continuum of the reals, I see it outside the realm of possible representation by numerals. > or a dually minimal and maximal > ur-element, that corresponds well to the Russell set, Burali-Fortian > Ord, the Liar, the empty set, and the Ding-an-Sich and Being and > Nothing, towards a theory that can be consistent, and Goedelianly > complete, Quineanly. Using these proper names is concision, > conciseness. Being aware of these meanings, I consider most of them not relevant to my basic question. ZFC has no ur-elements (no atoms). I am not a fan of Russell because he declared causality a relic of bygone time like monarchy, etc. If I recall Lavine correctly it was Russell not Cantor himself who is to blame for introducing the reals into set theory. So he dealt with his own paradoxes. Well, Burali-Forte's paradox has to do with the notion of infinite sets. However, in general, I do not intend to remedy antinomies. I rather prefer clear statements like that by Brouwer: The exclusion tertium non datur is invalid with infinity. Except for an estimated 40 more or less famous opponents of Cantor's theory, I do not know anybody from Kant to W.V. Quine who dealt with Cantor's silly idea of counting oo+1, oo+2,... Even Wittgenstein's objections did perhaps not hit the nail on its head. > > If you get to looking at Turing, a variety of statements about Turing > machines, that would seem to collide with, for example, what I say, in > the infinite may be true because they're about the finite. I have a > lot to learn about complexity and Kolmogorov. Turing would also distract us. Kolgomorov is one out of several important constructivists. None of them tackled Cantor's fallacy at its roots. > > A lot of people base their proofs upon statements that ZF is > consistent. Many of those theorems are still correct when ZF is > determined to be inconsistent. Do not get me wrong. I do not attack details of set theory. I am questioning the very basis of Cantor's theory. > > Where the physical universe is all physical objects, and a physical > object, that's similar, in a way, to a set being a set of all sets, > about concreteness. That means it's empirical evidence of that kind of > thing. Cantor's ideas on physical matters have meanwhile proven far from being reasonable. Take this as food for thought for those physicists who still believe in Cantor's correctness. > Infinite sets are equivalent. To finite ones? I agree to some extent. Eckard
From: Dave Rusin on 7 Apr 2005 11:18 In article <4254EA59.2040002(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > Dear Dave Rusin, [Inexplicable mix of personal and public communication deleted.] It is tremendously bad form to quote personal email in public. (Not that I wouldn't have said any of this in a public forum, THIS TIME. But not every letter I write is intended for public consumption. Save it for my collected works when I'm dead.) dave "Just another uncourageous mathematician"
From: Eckard Blumschein on 7 Apr 2005 12:29
On 4/7/2005 5:18 PM, Dave Rusin wrote: > In article <4254EA59.2040002(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> Dear Dave Rusin, > > [Inexplicable mix of personal and public communication deleted.] > > It is tremendously bad form to quote personal email in public. > > (Not that I wouldn't have said any of this in a public forum, THIS TIME. > But not every letter I write is intended for public consumption. > Save it for my collected works when I'm dead.) > > dave > "Just another uncourageous mathematician" You are quite right. I have to apologize for not asking you for permission. On the other hand, as far as I can judge, you did not write anything that was not ready to be published. Being a moderator, you are fulfilling a very demanding job. I understood that you cannot continue email discussions with those whose posting you were forced to reject. My intention was to show that there are no factual arguments against the let's say suspition that Cantor's transfinite numbers are lacking any justification. When I am mocking about Cantor in the threads "who is the most brilliant mathematician who ever lived?" and "Cardinality question" I do so not because I am intending to hurt people but because Cantor's fallacy was treated like something to firmly believe and admire during a century of biased agitation. Those who are really interested in clarification may benefit from what I tried to excerpt from of the original papers into http://iesk.et.uni-magdeburg.de/~blumsche/M280.html Sincerely, Eckard Blumschein |