Galileo's Paradox
in [Math]
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From: Virgil on 12 Dec 2006 15:52 In article <457EB7E4.505(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/11/2006 10:19 PM, Virgil wrote: > > In article <457D75A0.1060201(a)et.uni-magdeburg.de>, > > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > > > >> On 12/6/2006 9:10 PM, Virgil wrote: > >> > In article <4576F816.5060809(a)et.uni-magdeburg.de>, > >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> > > >> >> On 12/6/2006 12:03 AM, Virgil wrote: > >> > > >> >> > DA2 does not define anything. But if they were to b e defined by a > >> >> > theorem, they would already be defined by what I will call DA1. > >> >> > >> >> DA1 dealt with rationals. > >> > > >> > By DA1 I was referring to Cantor's first proof of the uncountability of > >> > the reals, and it deals with far more than the rationals. > >> > >> ??? What paper? Do you mean Cauchy's zig-zag diagonalization? > > > > As anyone but a fool would have known I meant: > > > > http://en.wikipedia.org/wiki/Cantor%27s_first_uncountability_proof > > DA means diagonal argument. AS Cantor only produced one diagonal argument, what is DA2? > This early attempt did not get famous. But it did get the job done. Which was the point.
From: MoeBlee on 12 Dec 2006 15:54 Tony Orlow wrote: > Granted, I'm not very far through it, but so far I see no need for it. Not very far through it? You read the cover and skipped to page 51 and then stopped reading, apparently. You didn't even read the VERY FIRST SENTENCE OF THE BOOK: "In the fall of 1960 it occcured to me that the concepts and methods of contemporary Mathematical Logic are capable of providing a suitable framework for the development of the Differential and Integral Calculus by mean of infintely small and infinitely large numbers." - Abraham Robinson, first sentence of the preface of his book 'Non-Standard Analysis'. And in the fall of 2006 it occurred to Tonly Orlow that he could, without understanding that mathematical logic, take one sentence out of the book and use it to spew whatever nonsense he likes about Robinson's work. >From the first sentence of chapter II: "The text is written in such a way that it can be understood by anyone with a rudimentary knowledge of Mathematical Logic and Abstract Set Theory." But Orlow has no such rudimentary knowledge. Abstract Set Theory. Robinson requires that the reader understand its rudiments. And Robinson uses the set of natural numbers and countable infinity all over the place in the developments. And Robinson even uses omega in a formula to describe the order structure of the non-standard *N. But Tony Orlow spews, for post after post after post, that Robinson's work precludes omega. Tony, you really are a jerk. MoeBlee
From: Virgil on 12 Dec 2006 15:56 In article <457EBB88.4080902(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/6/2006 8:18 PM, David Marcus wrote: > > Eckard Blumschein wrote: > > >> He assumes that his list of all reals is complete and shows that this is > >> not the case. From this contradiction he was forced to conclude that the > >> reals are uncountable > >> but he intentionally misinterpreted the outcome by > >> claiming there are more real than rational numbers. > > > > Please define the word "more" in this context. What exactly are you > > saying is not correct (or misinterpreted)? > > The word more is nearly synonymous to the > relation. > There is no number x of all real "numbers". > There is also no number y of all rational numbers. > There is not even a number z of all natural numbers. > > So it may be intuitively obvious but it is not justified to guess x>y>z. Cantor had previously defined "more" to mean that there was an injection from the lesser set to the greater but none in the reverse direction. In the Cantor sense of "more", Cantor proved twice that there are more reals than rationals.
From: Virgil on 12 Dec 2006 16:10 In article <457EC360.4000405(a)et.uni-magdeburg.de>, Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > On 12/6/2006 8:29 PM, Virgil wrote: > > In article <4576E233.7030802(a)et.uni-magdeburg.de>, > > >> >> You all know that DA2 shows by contradiction that real numbers are > >> >> uncountable. > >> > > >> > WRONG! It is not, at least in Cantor's version, a proof by contradiction. > >> > >> He assumes that his list of all reals is complete and shows that this is > >> not the case. > > > > > > Not so. He assumes one has an arbitrary list, with no other > > qualifications than that it is a list of reals, and then shows it is not > > complete. > > You wrote not so. Why? Did you read his original text? If I understand things correctly, Cantor himself never applied the diagonal argument directly to reals, he only applied it to binary sequences, The modification to apply it to reals was done by others. > > > There is no necessity to assume it complete and convert a > > straightforward direct proof into a proof by contradiction. > > Do you refer to literature or is this just your own comment? It refers to Cantor's binary sequence argument. > > > > In is only those who misread the proof that themselves supply that > > additional assumption that Cantor himself does not require. > > I do not intend to question historical sources. > > > > >> From this contradiction he was forced to conclude that the > >> reals are uncountable but he intentionally misinterpreted the outcome by > >> claiming there are more real than rational numbers. > > > > In the sense in which Cantor defined "more", there are more. > > I am not aware of Cantor's definition of "more". Then you argue from ignorance. Cantor said that there were more members in set B then in set A if there were injections from A to B but none from B to A. > > If EB > > wishes to use his own meanings for words in place of the clearly defined > > mathematical ones, he will, as here, repeatedly come a cropper. > > You are not in position to grade me. Then cease submitting you ideas to be graded. > >> I do not understand what you mean in this contex regarding Dedekind > >> cuts. I do not refer to any flaw in DA2 itself, just the interpretation > >> is wrong. > > > > It is the interpreter making that interpretation who is in error because > > he insists on using his own non-mathematical meanings for in place of > > mathematically standard definitions. > > Standard definitions may be inappropriate. Then produce alternate words and their definitions rather than misusing words with firm meanings. > In contrast to the bizarre > theory of more than infinitely many numbers, my interpretation is > logical and makes not just naive set theory, CH, AC and many other > worries unnecessary. It has only advantages. Perhaps, being your interprtation's mother, you are as blind to its faults as mothers are prone to be. When you can give any reason, other then your prejudices and unwarranted assumptions, against ZFC or NBG, we will listen. But of your unreasoning prejudices we have heard more than enough.
From: MoeBlee on 12 Dec 2006 16:10
Tony Orlow wrote: > Can you cite where he uses omega in the development of NSA please? The existence of the set of natural numbers is used all over the place in the development, as a well as countable infinity, and on page 52, omega is used in a formula of ordinal arithmetic to describe the order type of the non-standard *N. You are just FLAT OUT INCORRECT that non-standard analysis is incompatible with omega. > >>> What are you TALKING ABOUT? Read Robinson (which means reading the > >>> actual development, not just isolated passages), why don't you, instead > >>> of ignorantly spouting about what YOU THINK he does and does not need. > > > >> There is no need for omega in nonstandard analysis. > > Robinson works in classical mathematical logic and set theory, in which > > omega exists. IST includes Z set theory, in which omega exists. Or, if > > you want to point to so other treatment of non-standard analysis in > > which treatment does not also entail the existence of omega, then > > you're welcome to do it, but it ain't Robinson and it ain't IST. > > > > He ignores it. It would contradict his internally consistent theory, and > that would bother set theorists. No, he doesn't, you jerk. Read the damn book. And read his other works, and read his summary of his mathematical philosphy in his essay "Formalism 64". > >> There is no smallest > >> infinite allowed at all. > > > > It's not a question of "allowed". You really understand NOTHING about > > this. In particular sets and systems that are proven to exist, ordinals > > are not members. So what? The ENTIRE theory in which this takes place > > DOES prove the existence of ordinals. Look, no ordinal is a complex > > number, but we construct the complex numbers in a theory in which > > ordinals do exist, even if ordinals are not complex numbers. No ordinal > > is a non-standard real. But the theory in which non-standard reals are > > proven to exist does also prove the existence of ordinals. > > > > Ahem. He "proves" it cannot exist, just like the monkeys prove there's > no largest banana. What a complete loser you are. He's ALREADY taken set theory, and the existence of omega, which is proven in set theory, as RUDIMENTARY. > There's no tiniest giant. He's talking about > something on the same continuum as the reals and naturals and rationals. > He's not talking about any extra dimensions of specification like with > complex numbers. You competely missed the point of my analogy. No surprise there. > Ordinals and cardinals are naturals in their finite > state. Do the infinite forms of them jut off in other directions? > Apparently so. Robinson's do not. > > Isn't it time Nonstandard Analysis became the Standard? > > >> He makes reference to "countablility" but I > >> haven't seen any alephs about yet. > > > > The ordinals themselves are not members of the non-standard number > > system, but DERIVING the existence of a non-standard system takes place > > in a theory in which ordinals do exist. > > Yeah, over there, and they can't play this game. They don't belong to > this continuum. You're hopeless. Apparently, you demand just to pontificate about an author's work of which you even ADMIT you don't know the rudiments that the author himself stipulates. > > You can't just rip one part of > > a theory, like a shard, out of a whole theory. Perhaps there is a > > non-standard analysis that can be devised without classical > > mathematical logic and ZFC, but Robinson's work does NOT do that. He > > uses classcial mathematical logic and set theory all over the place in > > connection with results in non-standard analysis. And IST includes > > EVERY SINGLE theorem of Z set theory. > > Ae you saying there is no contradiction between standard and nonstandard > analysis, no cnclusions that are different? They are conclusions about DIFFERENT objects and different kinds of objects that still exist in a single theory. Why do you REFUSE to understand this? > > Read the VERY FIRST SENTENCE in Robinson's book, why don't you. > > Did you read "contemporary mathematical logic" to mean "transfinite set > theory"? You can do better than that. Contemporary mathematical logic is inextricably woven with set theory. The languages mathematical logic usually talk about have an infinite set of symbols. The theorems of mathematical logic are flying all over the place with resuts about countability and uncountability and infinite sets of symbols, formulas, theorems, and axioms. Robinson uses the set of natural numbers and countable infinity throughout his developments. And he mentions understanding of "Abstract Set Theory" as rudimentary for the book, and it is used, if you even bothered to READ the damn book. MoeBlee |