Cantor Confusion
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From: mueckenh on 22 Oct 2006 03:56 cbrown(a)cbrownsystems.com schrieb: > * we "can get to" (show the existence of) a set whose members satisfy > the requirements described by omega in AoI, > > then we wouldn't "need" (have to separately assume) the *axiom* of > infinity in order to talk about omega actually being a set. It would > simply /logically follow/ as a *theorem* from the other axioms. Of course we cannot get to mega by counting. According to set theory we get to omega by the limit: lim [n-->oo] {1,2,3,...,n} = N. lim [n-->oo] {1 + 1/2^2 + 1/3^2 + ... + 1/n^2} = (pi^2)/6. lim [t-->oo] X(t) = X(omega). This limit is some existing entity which can be used, not something that only can be approached as closely as you want. > > ZFC excluding AoI does /not/ say "you can get to omega" in the sense > you are using "can get to" here. Therefore ZFC excluding INF is not sufficient to prove the existence of numerical representations of irrational numbers, which in fact do not exist. Don't misunderstand me: The ratio of circumference to diameter of a circle is pi. But pi is not representable as a number. > Instead, ZFC /with/ AoI says, "since, > if we are honest, we have to admit that you /can't/ 'get to' omega > using the other axioms, we must therefore /assume/ omega's existence, > in order to talk logically about arguments that assume omega is a set > in the first place". In order to assume that an irrational number does exist with its omega digits in decimal representation. > > But so what? In a mathematical discussion amongst set theorists, you > /don't have to/ accept that it's true or obvious or reasonable or > sensible or accords with current dogma or is religious law or is > required under punishment of banishment or /whatever/, that "omega is a > set". > > Of course that assumes that you have /some/ axioms in mind: otherwise > it is simply not a mathematical question whether your statements > actually follow, one from the other, in your argument. It is instead an > argument of philosophy. True mathematical questions can be answered by physical experiments. For matheology you need belief or axioms. > > To clarify: suppose we agree that "since there is a set having the > properties of omega, therefore ..." is not a valid for a correct > argument. Do you have complaints about the remaining assumptions and > axioms of ZFC? Or do you find them (and the logical conclusions that > follow from them) to be acceptable as being true, obvious, logical, > correct, etc. in mathematical discourse? Finite set theory is fine and true and useful . The axiom of choice then is true too although it is no longer needed. The only fault, and one of the most damaging in the scientific history of mankind, is the introduction of actual infinity, i.e., finished infinity. Regards, WM
From: mueckenh on 22 Oct 2006 07:24 Virgil schrieb: > > The identity of 1 and 0.999... in analysis is proven. The difference > > 10^(-n) disappears for n --> oo (and not earlier!). The identity is not > > proven in case of the diagonal number because it cannot be proven, > > because of the lacking factor 10^(-n). Even for the digit with index n > > --> oo the difference is as important as for the fist digit. > > Except for such cases as 1.000... = 0.999..., and others involving an > endless string of 0's or an endless string of 9's, two reals differ if > they digger in ANY digit, regardless of place value. Why not in such cases as 1.000... = 0.999...? You, as always, assert something just as you like it, without proof. This 10^(-n) is the key to the inconsistency. Represent a number by 0.a_1,a_2,a_3,...,a_n,... with digit a_k eps {0,1,2,3,...,9}. In the limit {n-->oo} we have 0.a_1,a_2,a_3,...,a_n,... = 0.a_1,a_2,a_3,...,1+a_n,... Therefore Cantor's diagonal argumen fails. > So a number whose > decimal representation does not contain any occurrence of the digits 0 > or 9 will differ in value from any other number if it differs at digit > from the other number. If such a number really was defined, then the digits of pi could form a natural number. > > Since the Cantor diagonal rules all exclude use of 0 or 9, such a number > can equal another only if they have the same unique decimal expansion, > matching at every decimal place, with every digit equal. > > > > Here is the schisma, the point where mathematics is inconsistent: > > Either the digit becomes negligible in both cases or in none of the > > cases, i.e., either 1 =/= 0.999... in analysis or the later digits of > > the diaonal number become more and more unimportant such that for an > > infinite number there are negligible digits and Cantor's argument > > breaks down. > > Not in any real mathematics. In real mathematics, there is no non-zero > difference so small as to allow numbers differing by that amount to be > equal. lim {n-->oo} 10^(-n) > 0? Yes, in realistic mathematics this is the case because omega does not exist. But in that what you call mathematics, lim {n-->oo} 10^(-n) = 0 as a prerequisite for the unique definition of irrational numbers. > The set of constructed numbers is finite. Yes, it is always finite but unbounded, i.e., potentially infinite. > The set of constructible numbers is countable. That is the same if interpreted correctly. What do you think how an actually infinite set of Turing machines could be realized? Even by touring through the whole universe you would not find enough matter and space. > WM has three quite > distinct limit situations no one of which justifies or is justified by > any of the others. The existence of all required natural numbers n is guaranteed by the axiom of infinity for all three cases or for none of them. > >lim {t --> oo} X(t) = X(omega) >Then X(omega) = 0. LOL. You will not shilly-shally like a reed before the wind. Even if all the balls in the vase at noon were bricks and would fall down on you, I am sure you would refuse to believe that. Stay with your belief. Regards, WM
From: mueckenh on 22 Oct 2006 07:27 David Marcus schrieb: > I think you have this backwards. It is the Copenhagen interpretation > that suffers from this problem. In orthodox Quantum Mechanics, in > addition to the Schroedinger Equation, you need rules that tell you how > the wave function collapses when a measurement is made. (Without these > rules, it isn't clear that the theory actually predicts anything.) And, > this collapse must happen instantaneously. This is why the EPR authors > weren't satisfied with the Copenhagen interpretion. Unfortunately, > Einstein's catchy "God does not play dice" has given people the mistaken > impression that Einstein objected to a non-deterministic theory; this > isn't true. It is true! But as he failed to state it in a formal language, it cannot be understood by people understanding nothing else. Regards, WM
From: mueckenh on 22 Oct 2006 07:33 David Marcus schrieb: > Han de Bruijn wrote: > > Bob Kolker wrote: > > > > > The are mathematical systems we know for sure are consistent since they > > > have finite models. For example Just Plain Old Group Theory. No > > > contradictions can be inferred from the group postulates simpliciter. > > > > True. And that is because the axioms of group theory actually serve as > > a _definition_ of what a group _is_. Right? > > Just as the axioms of set theory define what a set is They don't define this at all. >, the axioms of the > natural numbers define what a natural number is, This knowledge has been present over centuries until someone set out to construct axioms to describe what was completerly superfluous because everybody knew. Regars, WM
From: mueckenh on 22 Oct 2006 07:47
Dik T. Winter schrieb: > > lim{n --> oo} 1/n = 0 proclaims "omega reached". > > It proclaims nothing of the sort. You want to read more in notation than > is present. You want to read less than present and necessary. lim{k --> oo} a_k = pi means omega reached. Or would you state that pi belongs to Q like every a_k for natural k? > > > But you are dishonest in transforming the vase problem (where the answer > > > was asked at t = 0) to another problem (where the answer was asked at > > > t = oo). > > > > It is nothing but just a simplification in notation! > > It is not. When giving a function from R to R, f(0) might exist, but > f(oo) *never* exists. So asking for f(0) in the first case is a > legitimate question. Not more legitimate than asking for f(oo) because at noon also infinitely many transactions must have happened. Therefore the distinction you make is false. > After the transformation you ask for f(oo), which > is *not* legitimate. So the transformation does not simplify notation, > it simply transforms the problem to an illegitimate problem. I hope you see your error. Without the actual existence of omega both problems with omega transactions are undefined. The proclaimed existence of omega is just the reason to raise this problem (and to show that omega does not exist). > > So is it not. Good heavens, that is unimportant for the present > > argument. Every constructed number is an element of a countable set. > > The set of all constructed numbers is countable. Every diagonal number > > belongs to this set. > > Pray define what you understand under constructable number. In mathematics > there is a precise definition. Do you think that e is constructible? Of course, by Sum 1/n! > It > can easily be constructed using continued fractions. Yes. You see, there are even more constructions than constructed numbers. > > > > On the other hand, I think you are meaning computable > > > > No, I was not meaning that. I mean: constructed (by list or by > > formula). > > Pray give a precise definition. A constructible number according to my view is a number which can be constructed, i.e., a number of which every digit can be known either by a formula or by a catalogue or a list or by whatever. > Not handwaving. What operations are > allowed in the construction of a constructable number in your view? As > I read this, (pi^2)/6 is a constructable number (in your opinion), > because it is constructed with a formula. Yes. > That deviates from the common > mathematical meaning of constructable. Please leave me alone with your common definition! The common mathematical definition of "realism" has nothing at all to do with reality and realism but is pure idealism. So, it seems, that common mathematical use of words is often the opposite of the usual meaning, but, alas, not in al cases. Then it were easy to talk this language. I mean that a constructible number can be constructed. > Also your formulation makes > 0.554445444444444444444445... > (which is Liouville's constant with 40/9 added to it) constructible. Of couse 0.110001000... and all its sums with rational numbers are constructible numbers in my sense. > Which "real" number is not constructable in your view? The great majority of real numbers does not belong to the set of constructible numbers because this set is countable. But the non-constructible numbers cannot be described completely, none of the, because this would mean constructing them. Regards, WM |