An uncountable countable set
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From: mueckenh on 26 Aug 2006 12:35 Dik T. Winter schrieb: > In article <1156364184.155913.12090(a)74g2000cwt.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: > > Dik T. Winter schrieb: > > > > What is the sum of infinitely many finite numbers? What is the sum of > > > > all finite numbers? > > > > > > I state that it holds for infinitely many 'n'. Not that it holds for > > > infinite 'n' (whatever that may be). > > > > Infinitely many have infinitely many differences. If you sum them up > > then you get an infinite sum. > > Well, if you give me a proper definition of the sum of infinitely many > numbers, I may have some idea about your meaning. In mathematics such > a thing is not defined. Your statement above is not a definition. If there are infinitely many numbers and, hence, infinitely many differences, then there is a definition of an infinite sum. Addition, as a mathematical procedure, is a precursor of subtraction. > > > > > > But I am doing so in order to show that his arguing concerns > > > > impredicative definitions and is inconclusive.f is the mapping, n is a > > > > natural number, M_f(n) is a set which contains all nongenerators, > > > > including n if not including n which is mapped on M. > > > > > > Yes. Such triples do not exist. And that precisely shows why Hessenberg's > > > proof was right. > > > > That it is false. An argument which makes use of the last digit of pi > > is void, because the last digit of pi does not exist. > > Again avoiding the issue by going on with completely different things. No. Only showing by an example what you desire or believe. > > > > If there is a surjective mapping f from N to P(N) it is > > > a requirement (of surjectivity) that such a triple *does* exist. > > > > I gave an example that this set cannot exist, independent of the > > surjectivity, independent of the cardinalities of the sets involved in > > the mapping. > > But it is trivial that such a triple can not exist. Of course. It is as trivial as the fact that the last digit of pi does not exist. Therefore I used this example. > But if there is > a surjective mapping from N to P(N), such a triple *must* exist. No. If a number is indexed, then it is covered (by my list numbers). You deny this. So you can believe also other absurd ideas like this: The set M(f) may exist, but not defined by M(f) !?!? I did not say that a surjective mapping would exist. I only deny that this proof was valid. Regards, WM
From: mueckenh on 26 Aug 2006 12:37 Tony Orlow schrieb: > MoeBlee wrote: > > Tony Orlow wrote: > >> MoeBlee wrote: > >>> Tony Orlow wrote: > >>>> Set theory contradicts with: > >>>> > >>>> (1) E y e N, A x>y, x< 2*x < x^2 < 2^x (y=2) > >>>> > >>>> because: > >>>> > >>>> (2) A y e N, aleph_0>y > >>> I don't know what you intend '<' to stand for. For the domination > >>> relation? The less than relation on ordinals? > >> The standard "less than" operator, commonly used for finite reals. > > > > This is in set theory, right? w is not in the field of the less than > > relation on the reals. Therefore, your (2) is incoherent. Also, if were > > in set theory, 'finite' adds nothing in description of real numbers. > > So, you cannot say that an infinite count like aleph_0 is greater than > any finite count? That's fairly lame. Sorry, I did not follow the complete discussion. But that is also wrong according to standard ZFC set theory. The transfinite ordinals and cardinals stand in trichotomy with the finite numbers. If this is denied, then my first goal is achieved. Regards, WM
From: mueckenh on 26 Aug 2006 12:42 David R Tribble schrieb: > > Don't call me a finitist. There is no largest number but only a largest > > set of less than 10^100 elements. > > I see now. You allow that for any natural k there is also a k+1, so > there is no upper bound on the naturals. Hence there is no > largest natural. > > On the other hand, you place an arbitrary upper bound of 10^100 > elements for any set. (Apparently, in order for a set to exist there > must exist a corresponding particle in the known physical universe > to correspond to each of its members. If we run out of particles, > we run out of place-holders in our set, I suppose.) There are no > cardinal numbers greater than 10^100. Of course there are such numbers. 10^10^10^10 is one of them. (In finity cardinal number = ordinal number = number.) On the other hand, the natural number consisting of the first 10^100 digits of pi (in case it is a normal irrational) does not exist. > > Several conclusions follow from these premises. One is that > there is no set that contains all the naturals. Correct. More than 10^100 elements of any set cannot be distinguished. But elements which cannot be identfied and distinguished are not different elements and cannot be counted as such. > Likewise, intervals > such as [0,1] cannot be represented by sets of reals (or sets of > points) because they are too big to be sets. Another is that > the series f(x) = sum{n=0 to oo} (-x)^n/n! cannot exist because > we can't make a set from the partial sums of the sequence. > Which means, in turn, that irrational numbers cannot exist, > because we can only approximate them with a Cauchy sequence > of finite length. Correct. An irrational number cannot be approximated to better than 1/10^100 of its value, unless it has some underlying building-formula like 0,101010001... > > The list goes on and on. Correct. In particular, there is no aleph and no omega. Regards, WM
From: Tony Orlow on 26 Aug 2006 13:50 MoeBlee wrote: > Tony Orlow wrote: >> So, is it technically incorrect to say 1 = 3/3 = 1.000... = 0.999...? I >> never thought so. > > Those are notations that can be understood precisely only in context of > the particular treatment in which they occur. BLAM!! That's exactly what I'm saying. If a particular treatment cannot distinguish between two elements, then they are the same, GIVEN that particular treatment. Another treatment may well succeed in distinguishing the two, demonstrating that there is a property of the two objects not accounted for with the first treatment. > I said that IN A SYSTEM > that constructs the reals as either Dedekind cuts or as equivalence > classes of Cauchy sequences, then the naturals and rationals are > isomorphically embedded but are not actual subsets of the reals. (That > is BASIC undergraduate mathematics. I mean, even I know that, and I > don't even have an education in mathematics! It's appalling how > ignorant you are WHILE you so categorically opine.) How does the constructive treatment of the various number systems affect the fact that the set of points representing the naturals is a subset of the set of points representing the rationals or reals? If you claim to defend set theory, then it's pointless of you to claim that the naturals aren't a subset of those two sets. > If the system and > construction are of a different sort, then we'd have to evaluate upon > the specifics of that system and construction. But, again, as I said, > for contexts that do not need to be so pedantic, such notation is > understood well enough that the equations you mentioned do of course > hold. If that's the case, then the set of natural poitns on the real line is a subset of those other sets, and there is nothing i what I said to object to. > > By the way, a while ago I posted a very rigorous explication of decimal > notation in a post to you. Yes, I appreciated that. Hopefully I can extract it from my Outlook archive. > >>> Because it's more a philosophical issue or an issue of terminology >>> outside the system. The purpose of set theory is not to address the >>> question of what is and is not a number. Rather, among the purposes of >>> set theory is to axiomatize and construct the various number systems >>> that are of interest. >> So, how do you know when you're doing math, and when you've strayed into >> other territory? > > That's a question for philosophy. I have my own views on what > mathematics is, but I don't hold that my own concept of mathematics is > definitive. > Do you have an opinion you'd care to express? >>>>>> you don't see scientists accepting transfinitology either. >>>>> Of course, you have a survey of scientists to support your claim. >>>> Uh, yes, right here. Why don't you survey thise that object, regarding >>>> what they do? >>> That's quite an unscientific survey method of yours. >> Yo, man, it's empirical evidence. Ask around. Life's an experiment. > > It's anecdotal evidence only. Check it out. A scientific experiment is > not just any life experience. I depends how you live life. :) > >>> Numbers are among the primary concern of mathematics. Set theory >>> axiomatizes and constructs number systems. >> Which are the other primary concerns, if any? (how did we define >> "number" again?) > > I don't define 'number' in a formal theory. But in informal discussion > about mathematics I don't demur from using the word 'number' in its > ordinary dictionary senses. > > Other concerns of mathematics are sets, relations, spaces, geometries, > topologies, algebras, for example, and more. > >> I'll take as an "I dunno". > > Then you'll, AS USUAL, take incorrectly. > > MoeBlee > :)
From: Virgil on 26 Aug 2006 15:51
In article <1156584896.722145.65290(a)b28g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > MoeBlee schrieb: > > > mueckenh(a)rz.fh-augsburg.de wrote: > > > In real mathematics (as opposed to matheology) > > > we have the valid foundation: What cannot be addressed, that does not > > > exist. > > > > Do you have axioms for this "real mathematics"? > > > Abolish Axioms. Acquire And Ask An Abacus. > > What it says and what you can derive from that by pure logic: That is > mathematics. From only abacuses and sorobans, one must conclude that there are only natural numbers, and no such things as negatives or rationals or reals can exist, and that while for each abacus there is a largest natural, it is abacus/soroban dependent as to which number is largest. Also, if all our mathematics is to be so limited, everything but small natural number arithmetic is out the window. No geometry, no calculus, not much algebra. Even the ancient Greeks had more that abacus math. |