Galileo's Paradox
in [Math]
|
From: Michael Press on 6 Dec 2006 16:46 In article <MPG.1fe01bc338e764a9899c8(a)news.rcn.com>, David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > Tony Orlow wrote: > > Virgil wrote: > > > In article <456f334d$1(a)news2.lightlink.com>, > > > Tony Orlow <tony(a)lightlink.com> wrote: > > >> Virgil wrote: > > >>> In article <456e4621(a)news2.lightlink.com>, > > >>> Tony Orlow <tony(a)lightlink.com> wrote: > > > > > >>>> Where standard measure is the same, there still may be an infinitesimal > > >>>> difference, such as between (0,1) and [0,1], if that's what you mean. > > >>> The outer measure of those two sets is exactly the same. > > >> Right, and yet, the second is missing two elements, and is therefore > > >> infinitesimally smaller in measure. > > > > > > Except that in outer measure there are no infinitesimals, and the outer > > > measure of the difference set, {0,1} is precisely and exactly zero. > > > > So, you're saying infinitesimals cannot be considered? You're saying one > > is not ALLOWED to consider the removal of a finite set from an ifninite > > set to make any difference in measure? I say you're wrong. > > I guess you still haven't figured out that in mathematics we make > precise statements and then make deductions. We don't just decide the > theorems based on what we wish. If a certain concept (e.g., cardinality, > measure, outer measure, Hausdorff measure, continuity, absolute > continuity, differentiability) doesn't do what we want, then we come up > with a new concept, state it precisely, and see if we can prove that it > does what we want. > > The term "outer measure" has a precise meaning that is given in courses > and books on Real Analysis. I mean to play devil's advocate. Mathematicians _do_ figure out what theorems they want, then construct minimal (hopefully) axiom sets to get those theorems. The intermediate value property was taken as the definition of continuous until folks started to examine functions like f(x) = sin 1/x, x <> 0; f(0) = 0. f has the ivp at 0, but does not satisfy the intuitive notion of continuous. Now ivp it a theorem, which is what we wanted, be it axiom or theorem. -- Michael Press
From: Michael Press on 6 Dec 2006 16:58 In article <jack-3EEBB8.13464706122006(a)newsclstr02.news.prodigy.co m>, Michael Press <jack(a)fake.net> wrote: > In article <MPG.1fe01bc338e764a9899c8(a)news.rcn.com>, > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > > Tony Orlow wrote: > > > Virgil wrote: > > > > In article <456f334d$1(a)news2.lightlink.com>, > > > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > > > >>> In article <456e4621(a)news2.lightlink.com>, > > > >>> Tony Orlow <tony(a)lightlink.com> wrote: > > > > > > > >>>> Where standard measure is the same, there still may be an infinitesimal > > > >>>> difference, such as between (0,1) and [0,1], if that's what you mean. > > > >>> The outer measure of those two sets is exactly the same. > > > >> Right, and yet, the second is missing two elements, and is therefore > > > >> infinitesimally smaller in measure. > > > > > > > > Except that in outer measure there are no infinitesimals, and the outer > > > > measure of the difference set, {0,1} is precisely and exactly zero. > > > > > > So, you're saying infinitesimals cannot be considered? You're saying one > > > is not ALLOWED to consider the removal of a finite set from an ifninite > > > set to make any difference in measure? I say you're wrong. > > > > I guess you still haven't figured out that in mathematics we make > > precise statements and then make deductions. We don't just decide the > > theorems based on what we wish. If a certain concept (e.g., cardinality, > > measure, outer measure, Hausdorff measure, continuity, absolute > > continuity, differentiability) doesn't do what we want, then we come up > > with a new concept, state it precisely, and see if we can prove that it > > does what we want. > > > > The term "outer measure" has a precise meaning that is given in courses > > and books on Real Analysis. > > I mean to play devil's advocate. Mathematicians _do_ > figure out what theorems they want, then construct > minimal (hopefully) axiom sets to get those theorems. > The intermediate value property was taken as the > definition of continuous until folks started to examine > functions like f(x) = sin 1/x, x <> 0; f(0) = 0. f has > the ivp at 0, but does not satisfy the intuitive notion > of continuous. Now ivp it a theorem, which is what we > wanted, be it axiom or theorem. s/axiom/definition/g Please read `definition' where I wrote axiom. Thanks. -- Michael Press
From: David Marcus on 6 Dec 2006 19:20 Eckard Blumschein wrote: > On 12/6/2006 5:19 AM, David Marcus wrote: > > Eckard Blumschein wrote: > > >> >> Why do you think that the diagonal argument defines the reals? > >> > >> You all know that DA2 shows by contradiction that real numbers are > >> uncountable. I carefully read how Cantor made sure that the numbers > >> under test are real numbers. He did not use Dedekind cuts, nested > >> intervals or anything else. > > > > Well, of course he did't use Dedeking cuts, etc. > > Cantor explained why he preferred his own definition. > Read how he made sure that the numbers under test actually were real > numbers. I just got through telling you that it is irrelevant how the real numbers are defined. All that matters is that they are a complete ordered field. You will never learn anything if you insist that you already know everything. > >> He assumed numbers with actually > >> indefinitely much rather than many e.g. decimals behind the decimal > >> point. Strictly speaking, he did not immediately show that the reals are > >> uncountable but that these representation like never ending decimals is > >> uncountable. > > > > That's because anyone who took an analysis course in college (or maybe > > even freshman calculus) can prove (starting from the properties of a > > complete ordered field) the existence of the decimal representation of > > real numbers. > > Cantor took no analysis course. You are thinking backward. Now you have moved from nonsense to absurdity. Cantor of course knew analysis. His motivation for developing set theory was his study of Fourier analysis. However, the more relevant question is whether you ever took an analysis or calculus course and what grade you got. > >> Being uncountable is the common property of these numbers under test. > >> To my knowledge, sofar nobody was able to show that the numbers > >> allegedly defined by Dedekind's cut or nested intervals are uncountable. > > > > Saying that to your knowledge no one has proved that the set of Dedekind > > cuts is uncountable > > I did not say this. Please quote me carefully. It is nearly impossible to quote you "carefully" since you speak your own made-up language. Try using mathematical English. > The set of existing Dedekind cuts is finite. The set of feasible cuts is > countable. There you go again, making up words ("existing", "feasible"). In mathematics, we are allowed to make up words, but only if we define them. You said that "sofar nobody was able to show that the numbers allegedly defined by Dedekind cuts are uncountable". The natural translation of this into normal language is "no one has showed that the set of Dedekind cuts is uncountable". Clearly, this statement is false. So, you now say that this is not what you meant. Fine. Try saying what you mean in normal mathematical English, so we have a hope of understanding it. > >> If we need the notion real numbers at all, then in connetion with the > >> common property to be uncountable. > >> > >> You might wondwer that there is no chance to define the reals at will. > >> Cantor made a false promise when he said the essence of mathematics just > >> resides within its fredom. > >> > >> Do you still not yet understand why DA2 lets no room as to define the > >> reals accordingly? > > > > Of course I don't understand it! What does "no romm as to define" mean? > > DA2 only works for actual infinity. So, "DA2 lets no room as to define the reals" means "DA2 only works for actual infinity"? Well, it is nice to collect sentences that mean the same thing, but none of them are intelligible. On the one hand, we have the unintelligible "no room to define". On the other, the unintelligible "actual infinity". Language was invented to facilitate communication. For it to do this, it helps to use words with their usual meanings. > >> but I don't see where you got this particular > >> >> nonsense from. Did you read it in a book? > >> > >> I read several original papers by Cantor. The rest is reasoning. > > > > Well, I guess that explains it. If you want to understand/learn > > mathematics, you pretty much have to take courses, read books, and do > > the exercises. Kind of arrogant to think you can rediscover centuries of > > mathematics on your own. Even Ramanujan read whatever books were > > available to him. > > Be not so lecturing to me. Why? Can't you learn from lectures? > Perhaps you are pretty young. Huh? > I do my best, > and so far my puzzle fits together. Stringing gibberish together is not how most people solve puzzles. > I feel myself by far not so arrogant > how I consider Cantor who actually ignored many many centuries of > science. I just try to revitalize the golden ideas by Archimedes, > Aristotele, Galilei, Newton, Spinoza, Leibniz, ..., Gauss, Kronecker, > Poincar� and many many others. Fine. Prove you actually understand some mathematics. You already showed that you don't know what it means for a function to be continuous or what the words "countable" or "uncountable" mean. Do you know any mathematics that Newton, Leibniz, Gauss, Kronecker, or Poincare did? If so, please tell us. Give us a lecture on a standard topic of your choosing. -- David Marcus
From: Bob Kolker on 6 Dec 2006 19:24 David Marcus wrote: > > There you go again, making up words ("existing", "feasible"). In > mathematics, we are allowed to make up words, but only if we define > them. or constrain them with postulates. Some terms must be undefined, but their meaning is implicit in the postulates they obey. Bob Kolker
From: David Marcus on 6 Dec 2006 19:27
Eckard Blumschein wrote: > On 12/6/2006 5:27 AM, David Marcus wrote: > > Eckard Blumschein wrote: > >> So it would not make any difference when the order aleph_0 and aleph_1 > >> were reversed or more reasonably the correct denotation was preferred: > >> countable instead of aleph_0 and uncountable instead of aleph_1. > > > > Yes, yes, I know you like to take standard words and give them new > > (vague) meanings. > > This is a praxis introduced by mathematicians to an extent that even > rich languages get short of terms without distorted meaning. Nonsense. We aren't about to run out of words any time soon. You've shown you are quite good at making up new terms. The problem is you use standard terms with your own personal meaning and you use nonstandard terms without telling anyone what they mean. So, the problem is all you! > > But, I was asking you whether the cardinality of the > > reals is greater than the cardinality of the integers if we use the > > standard meanings for the words "cardinality", "reals", and "integers". > > I consider cardinality nonsense. Just say countably infinite instead of > a_0 and uncountable instead of a_1 and forget the rest. I wasn't asking you what you "consider". Let's try a simpler question. Is there an injection from the integers to the reals? Is there an injection from the reals to the integers? > > As for aleph_2 itself, because of the > > independence of the continuum hypothesis, it doesn't come up too often > > in mathematics. > > = never? Of course, not. > > However, there were some interesting articles recently > > in the Notices of the AMS that discussed axioms to add to ZFC. There > > seemed to be good reasons to add an axiom which would make the > > cardinality of the reals equal to aleph_2. > > And which role has been envisioned for aleph_1? Kind of a silly question. aleph_1 is the first cardinal after aleph_0. That's its "role". -- David Marcus |