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From: Will Twentyman on 16 Apr 2005 17:17 Eckard Blumschein wrote: > On 4/13/2005 6:41 PM, Will Twentyman wrote: > > >>>Engineers contempt elusive precision. >> >>And if the precision is not elusive, but right there in their grasp? > > This is basic to engineering, too. Good. In that case I offer you a method of measuring the relative size of infinite sets, thus enabling you to compare them with more precision than you have in the past. Enjoy your infinite cardinalities. -- Will Twentyman email: wtwentyman at copper dot net
From: Will Twentyman on 16 Apr 2005 17:27 Eckard Blumschein wrote: > On 4/13/2005 6:52 PM, Will Twentyman wrote: >>I simply disagree with it at so many points >>that I consider it unlikely that we can come to any agreement on it. > > After a hundred years Cantor has been idolized. No. This is not an issue of Cantor, but your work. If you would like, I would be happy to copy M280 into a new thread with a critique of it. >>More specifically, reading it causes me to believe you do not understand >>what Cantor was doing or how mathematicians reason. > > I can retrace his reasoning except for the cheeky idea to quantify the > quality infinity. As long as he has a precise, consistent definition, I don't care how cheeky it was. It works. >>I agree with you >>that Cantor's original notation may not have been tidily presented, > > > It was clever presented in a demagogic manner. > > >>but >>his fundamental concepts were sound > > Quantifying a quality is not a sound concept. We do it all the time. Both ground beef and steak have the quality "tasty", yet I can quantify which is *more* tasty. One way is to establish a price/kg(or pound). Quantifying the qualities "useful", "desireable", etc are the basis of economics. >>and have been formalized. I note >>that you ignore ZF, ZFC, and other formalizations that may have >>eliminated any "rough edges" on Cantor's terminology or exposition in >>favor of the papers that serve as the basis for those works. Any >>particular reason why? > > I see most of the paradoxes just superficially remedied. Please name one that is not. I'll admit that some are not remedied in a way that is entirely satisfying, but if the paradox is eliminated, where is the superficiality? >>I'll give one example where you are simply wrong: "Cýs infinite alephs >>only distinguish between countable and uncountable sets." > > I am aware that they claim to manage much more. I referred to what they > really are able to perform. They distinguish between more than two classes of sets. >>The infinite >>alephs establish equivalence classes of sets which have a natural >>partial ordering based on surjections, and which also subdivide the the >>class of uncountable sets. > > Given it would be make sense to subdivide the class of uncountable sets. > Do you expect these subdivition nearly equally fundamental as the > division into countable and uncountable numbers? Yes. What is not fundamental about taking the notion of dividing infinite sets into the classes "countable" and "uncountable", and then repeating the process on "uncountable" to give aleph_1 and aleph_2, then aleph_3, then aleph_4...? The fact that the basic process can be repeated is nearly as fundamental as the initial split between countable and uncountable. -- Will Twentyman email: wtwentyman at copper dot net
From: mitch on 17 Apr 2005 15:43 "Eckard Blumschein" <blumschein(a)et.uni-magdeburg.de> wrote in message news:4254EA59.2040002(a)et.uni-magdeburg.de... > I am almost > amused how many opponents of Cantor failed to show that he was wrong. Right and wrong do not enter into a mathematics based on "self-consistency." Perhaps you should look at my most recent post "a model of succession using knight's tours." In any case, at least take the time to look up "paraconsistency." Kant's insights into "self-consistency" led him to take a position on mathematics different from his contemporaries. He is very explicit in "Prolegomena to Any Future Metaphysics" concerning the act of "visualization" in the practice of mathematics. The Stanford Encyclopedia of Philosophy has an entry on paraconsistency that is compatible with Kant's views. Now, not being a native speaker of German, my statements are grounded on particular translations. For "Critique of Pure Reason" I have typically used the Norman Kemp Smith edition without seeing any major discrepancies with other versions on these topics. For "Prolegomena to Any Future Metaphysics" I use an anthology by Wolff. I shall, however, reiterate: Right and wrong do not enter into a mathematics based on "self-consistency."
From: Ross A. Finlayson on 17 Apr 2005 20:58 I think it may be possible to illustrate to you why there are infinitely many values of at least one thing in the universe. Consider any two masses, there is the force of gravity between them, as a function of their distance. Because it is seen as a force, the velocity, acceleration, and third and fourth and so on derivatives of position with respect to t, time, exist, infinitely many of them, and because the force varies continuously as a function of the distance, none of the higher derivatives are ever zero. The units of those higher dimensions, or rather, derivatives, are m/s, m/s^2, m/s^3, ..., ad infinitum. The universe is infinite, infinite sets are equivalent. The schools of logicism, intuitionism, formalism, those were at the time modernisations (modernizations). The goals behind their expression and formation are reasonable, basically to satisfy unification of concept. The specifics are trivia: meaningful and in the high levels, utilitarian, for our grubby human needs. >From a philosophical standpoint, I find that Kant's Ding-an-Sich reflects some structural observations found in what would be the theoretical ur-element, as does the Hegelian Being and Nothing. You might notice I say theoretical instead of set-theoretical, because I exploit the duality of the singular proper class and ur-element to form theories of at once sets or collections, numbers, and concrete things. That makes me a Platonist, or platonist; in a very real sense those concrete things are just theoretical constructs, and counting on your fingers moves mountains. The singular proper class with no non-logical axioms enables extension of metatheoretical statement, within the first order, for a theory to be complete. When we discuss quantification, in set theory, it's over all sets, the set of all sets, because there are only sets in a set theory. Anyways, the infinite exists, or, we don't. With some very obvious evidence that we do, it does. Ross
From: Albert Wagner on 17 Apr 2005 22:13
Ross A. Finlayson wrote: > I think it may be possible to illustrate to you why there are > infinitely many values of at least one thing in the universe. Consider > any two masses, there is the force of gravity between them, as a > function of their distance. Because it is seen as a force, the > velocity, acceleration, and third and fourth and so on derivatives of > position with respect to t, time, exist, infinitely many of them, and > because the force varies continuously as a function of the distance, > none of the higher derivatives are ever zero. > > The units of those higher dimensions, or rather, derivatives, are m/s, > m/s^2, m/s^3, ..., ad infinitum. > > The universe is infinite, infinite sets are equivalent. Yes. I fully understand what you are saying. And I understand the utility of such a point of view. But, it is just a point of view. What is happening is continuous, not a sequence of jumps. The infinity you describe is arbitrary and invented, a human construct imposed on a continuity, merely as a convenience in calculation. > The schools of logicism, intuitionism, formalism, those were at the > time modernisations (modernizations). The goals behind their > expression and formation are reasonable, basically to satisfy > unification of concept. The specifics are trivia: meaningful and in > the high levels, utilitarian, for our grubby human needs. > >>From a philosophical standpoint, I find that Kant's Ding-an-Sich > reflects some structural observations found in what would be the > theoretical ur-element, as does the Hegelian Being and Nothing. You > might notice I say theoretical instead of set-theoretical, because I > exploit the duality of the singular proper class and ur-element to form > theories of at once sets or collections, numbers, and concrete things. > > That makes me a Platonist, or platonist; in a very real sense those > concrete things are just theoretical constructs, and counting on your > fingers moves mountains. > > The singular proper class with no non-logical axioms enables extension > of metatheoretical statement, within the first order, for a theory to > be complete. > > When we discuss quantification, in set theory, it's over all sets, the > set of all sets, because there are only sets in a set theory. > > Anyways, the infinite exists, or, we don't. With some very obvious > evidence that we do, it does. It isn't the infinite that exists in Nature; it is continuities. The infinite exists only as a human abstraction. The abstraction is required because we have discovered no better way to talk about continuities. -- "I know that most men, including those at ease with problems of the greatest complexity, can seldom accept even the simplest and most obvious truth if it be such as would oblige them to admit the falsity of conclusions which they have delighted in explaining to colleagues, which they have proudly taught to others, and which they have woven, thread by thread, into the fabric of their lives." - -- Tolstoy |