Galileo's Paradox
in [Math]
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From: Eckard Blumschein on 11 Dec 2006 11:16 On 12/7/2006 1:20 AM, David Marcus wrote: > 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. Cantor himself has shown with DA2 that they are not such field. >> >> 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. Where is your argument concerning the issue? >> 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. I clearly said that Dedekind cuts did not define any new number. He just declared the irrational sqrt(2) an irrational number. Where can I find "the set of Dedekind cuts"? Fine. Try saying what you mean in > normal mathematical English, so we have a hope of understanding it. Who does not intend to understand may hide behind formalisms. > >> >> 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". Are you not intelligent enough as to understand the meaning of actual infinity. Even Cantor understood it. > > Language was invented to facilitate communication. For it to do this, it > helps to use words with their usual meanings. Usual is not always appropriate. >> I do my best, >> and so far my puzzle fits together. > > Stringing gibberish together is not how most people solve puzzles. I referred to some so far open questions in mathematics.
From: Eckard Blumschein on 11 Dec 2006 11:26 On 12/6/2006 6:30 PM, Lester Zick wrote: > On Wed, 06 Dec 2006 08:57:24 -0500, Bob Kolker <nowhere(a)nowhere.com> > wrote: > >>Han de Bruijn wrote: >> >> >>> >>> Where w = 2 . See: >> >>Typo. The 'w' key is just below the '2' key. Fumble fingers. > > Fumble fingers or fumble mind? > > ~v~~ What do you expect from someone who would purify the Orient including Israel from predominantly muslim population by means of an atomic bomb, and ascribe Cantor's mental illness his perhaps low quality hereditary genetic code?
From: Eckard Blumschein on 11 Dec 2006 11:30 On 12/6/2006 8:48 PM, Virgil wrote: > In article <4576EB55.4040803(a)et.uni-magdeburg.de>, > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: > >> On 12/5/2006 11:41 PM, Virgil wrote: >> > In article <457593EB.9030809(a)et.uni-magdeburg.de>, >> > Eckard Blumschein <blumschein(a)et.uni-magdeburg.de> wrote: >> > >> >> >> >> Rationals are p/q. This system cannot be improved by adding genuine >> >> (i.e. rational) numbers. One can merely move to the fictitious >> >> continuous alternative. >> > >> > All numbers are equally genuine in any meaning of "genuine" other than >> > EB's improper meaning of "irrational". >> >> Kronecker was correct in that irrationals are no genunine numbers. > > That may have been thought to be the case in the ninteenth century, but > this is the twenty first century. > > > Kronecker died in 1891. Mathematics has progressed since then. Concerning its putative fundamentals obviously with no avail. On the contrary.
From: Eckard Blumschein on 11 Dec 2006 11:43 On 12/7/2006 1:27 AM, David Marcus wrote: > Eckard Blumschein wrote: >> 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? Reals as defined e.g. by nested intervals are still rationals: Bijection. Reals corresponding Meray: fictitious limits and therefore also corresponding to DA2 do not have an approachable numerical address. They can only be injected (addressed) via their defining tasks. > >> > 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. Not never, or not at all? > >> > 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". So far I was told aleph_1 means the continuum of the reals.
From: Tony Orlow on 11 Dec 2006 11:54
stephen(a)nomail.com wrote: > Tony Orlow <tony(a)lightlink.com> wrote: >> David Marcus wrote: >>> Tony Orlow wrote: >>>> David Marcus wrote: >>>>> Tony Orlow wrote: >>>>>> David Marcus wrote: >>>>>>> Tony Orlow wrote: >>>>>>>> Why do you have a problem with the mere suggestion of an infinite value? >>>>>>> I don't have a problem with infinite values. However, you have to do >>>>>>> more than merely say "positive infinite n". Assuming your positive >>>>>>> infinite things aren't the same as something that we already know about >>>>>>> (in which case, you should just say so), you either need to give a >>>>>>> construction of these positive infinite things or you need to specify >>>>>>> their properties. You haven't done either. For example, how many of >>>>>>> these things are there? How do they relate to each other? How do they >>>>>>> interact with the natural numbers? Are the operations of addition and >>>>>>> multiplication defined for them? >>>>>> Well, I have been through much of that regarding such specific language >>>>>> approaches as the T-riffic digital numbers, but that's not necessary for >>>>>> this purpose. It suffices to say that, if a statement is proved true for >>>>>> all n greater than some finite k, that that also includes any postulated >>>>>> infinite values of n, since they are greater than any finite k. I don't >>>>>> need to construct these numbers. Consider them axiomatically declared. >>>>> Then list the axioms for them. >>>> (sigh) >>>> >>>> infinite(x) <-> A yeR x>y >>> Is that your only axiom? If so, then state your first theorem about them >>> and give the proof. >>> > >> That's the only one necessary for what defining a positive infinite n. A >> whole array of theorems pop forth from infinite-case induction and IFR, >> such as that the size of the even naturals is half that of the naturals. >> That's a no-brainer. Go back to where I first answered your question at >> length, and read again, at length. It's not transfinitology, but it's >> also not nonsense. > >> Allow me to add another: > >> |{ x| yeR and 0<(x-y)<=1}|=Big'un. > >> That's the unit infinity. > > What is the definition of | | ? > > Stephen "Size" of the set. Or, is "|...|" reserved only for "cardinality"? I think I can adopt the previous symbol for absolute value, as set theorists have, without stepping on too many toes. Tony |