Cantor Confusion
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From: cbrown on 23 Oct 2006 18:33 David Marcus wrote: > cbrown(a)cbrownsystems.com wrote: > > David Marcus wrote: > > > cbrown(a)cbrownsystems.com wrote: > > > > I should state up front that I don't actually claim that "A and not A" > > is a good ontological description of entanglement; I was mostly just > > tweaking HdB. > > A worthy endeavor. > Well, at least it can be entertaining. I doubt it has any measurable effect on HdB's worldview. > > > But, if one theory is vague or ill defined, then it can be hard to say > > > whether an experiment really supports it. > > > > I donlt quite get what you are implying here. I assume you have some > > particular physical experiment in mind where this is the case as > > regards QM. Could you elaborate? > > Not a particular experiment, but the Copenhagen interpretation itself. > See "What is the meaning of the wave function" by Jean Bricmont. Google > turns up several copies on the Web. > I appreciate your comments, and will look up the indicated references. Thanks for your time. Cheers - Chas <large snippage>
From: cbrown on 23 Oct 2006 19:24 mueckenh(a)rz.fh-augsburg.de wrote: > cbrown(a)cbrownsystems.com schrieb: > > > > Of course we cannot get to mega by counting. According to set theory we > > > get to omega by the limit: > > > > No, according to most set theories, we "get to" omega by simply > > assuming it exists. > > We get omega by assuming it exists. We get to omega by limits. No, I mean we get to omega by assuming omega exists. We do not get to omega by limits; unless in some bizzare way, you think the phrase "We get to omega by limits" is logically equivalent to the statement "We get to omega by assuming that omega exists". > > > > Set theory doesn't define limits at all. > > Set theory defines the limits of sequences in analysis. > No, /Analysis/ is the branch of mathematics defines the limits of sequences in Analysis. It may choose to use the language of Set Theory in order to do this; and it is certainly the intention of set theorists that their results should be useful to Analysis; but Analysis is not equal to Set Theory. > > Topology and analysis define > > limits. > > They are based upon set theory. Some such theories are instead based on Category Theory. Category Theory is not Set Theory. > > > > > > > > lim [n-->oo] {1,2,3,...,n} = N. > > > > Here, one assumes you mean pointwise convergerence; and of course you > > are explicitly /assuming/ that N /is/ a set - otherwise, N has no > > meaning. > > Of course. That is why and how we get to omega. "Why" we get to omega is not a mathematical question. > > > > > lim [n-->oo] {1 + 1/2^2 + 1/3^2 + ... + 1/n^2} = (pi^2)/6. > > > > Here, one assumes you mean using the usual metric topology; > > No. This formula stems from Leonhard Euler. I works without any > topology. At that time topology did not exist. So, you claim that the above statement follows from no definitions at all? Instead it follows simply because Euler said it? > > > > > lim [t-->oo] X(t) = X(omega). > > > > > In what topology? Without specification, your statement is useless. > > Like that of Euler? I assume Euler had something specific in mind by "lim n->oo" when he stated the relation that you mentioned. On the other hand, I have no idea what you mean above; or whether you in fact intend your statement to mean anything specific at all. > > > > lim n->oo f(n) is /not/ in general defined as f(omega). If it were, > > then your first limit would not be N; it would be N union {N}. > > That observation shows but an inconsistency of set theory. No, it shows that if we are not specific about what we mean by "lim n->oo", it is possible for two people to have two entirely different definitions of "lim->oo", which result in two entirely incomatible results. That's why it is generally wise to state what you mean explicitly by a statement such as "lim n->oo", in order to prevent confusion. > Cantor, in > his first proof, used the notation a_oo. Later he chnaged oo to omega. And from this, we can draw what conclusions? > > > > For example, if we restrict ourself to the rationals only, then the > > sequence you mention above regarding pi^2/6 "converges" in some sense, > > It is called Cauchy-convergence. > Is that what Euler called it? > > but not to a rational number - the rational numbers are not complete > > with respect to the usual metric. > > Therefore the irrational numbers are not a subset of the rational > numbers. I agree. It follows of course more immediately from the definition of an irrational number: it is a number which is not rational. > > > > > > > > > > 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. > > > > > > > So, pi is a number; but it is not representable as a number? > > Pi is not a number, but an idea like beauty or justice which also are > not representable as numbers. When Euler stated that (something) = pi^2/6, do you think he considered pi to be a number? Do you think he considered that pi could be divided by the number 6? Do you think that he felt that he was justified in the conclusion that pi^2/6 = pi*(pi/6)? Conversely, do you think Euler would have considered that beauty could be divided by the number 6? Do you think he would have felt that it followed that beauty^2/6 = beauty*(beauty/6)? > > > > > > 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. > > > > First of all, there are other ways to describe irrationals such as > > sqrt(2) which don't rely on describing it as the limit of a sequence of > > approximations. For example, we can describe it as "that number which, > > when squared, equals 2" or "the number which is the ratio of the > > diagonal of a unit square to the number 1". > > Yes. That, however, does not yield a decimal representation as needed > in Cantor's argument. I am not claiming that it does or does not have any relevance to Cantor's diagonal argument. > > > > Secondly, we don't require that the set of all natural numbers exist in > > order to define the limit of a sequence of rational numbers. We merely > > require a definition of what it means to be a natural number; so that > > we can say "n is a natural". Once we have done that, we can say things > > like "there is a natural m such that, for all n, if n is a natural with > > n > m, then f(n) < f(m)"; > > That yields only rational numbers. It does not yield limits like pi. How so? Can you see how we can modify the usual delta-epsilon definition to change for "all n in N, it follows that..." to instead "if n is a natural number, then ..."? > Al > representations which are not actually infinite, i.e. have not omega > digits, are representations of rational numbers. > > Somewhat
From: Lester Zick on 23 Oct 2006 19:24 On 23 Oct 2006 08:48:07 -0700, "Randy Poe" <poespam-trap(a)yahoo.com> wrote: > >Lester Zick wrote: >> On Mon, 23 Oct 2006 02:00:06 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> >> wrote: >> >> >In article <1161518242.756958.103660(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: >> >> [. . .] >> >> > Virgils definition, although not wrong, definitions >> >can not be wrong, does not lead to desirable results. >> >> Since when can definitions not be wrong? > >A definition is a statement by a person that a certain >symbol will be used by them to stand for some concept. > >What is there in such a statement that can be wrong? The concept? >Example: I will use the term "gleeb" to refer to an integer >which is divisible by 2. > >How can that statement be wrong? How would you >define "wrong" for such a statement? Self contradictory predicates defining the concept in the definition. Ex: "squircles are square circles" "x is an even, odd" "gleeb is a finite integer divisible by 0". ~v~~
From: Lester Zick on 23 Oct 2006 19:25 On Mon, 23 Oct 2006 16:08:37 +0000 (UTC), stephen(a)nomail.com wrote: >Randy Poe <poespam-trap(a)yahoo.com> wrote: > >> Lester Zick wrote: >>> On Mon, 23 Oct 2006 02:00:06 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> >>> wrote: >>> >>> >In article <1161518242.756958.103660(a)h48g2000cwc.googlegroups.com> mueckenh(a)rz.fh-augsburg.de writes: >>> >>> [. . .] >>> >>> > Virgils definition, although not wrong, definitions >>> >can not be wrong, does not lead to desirable results. >>> >>> Since when can definitions not be wrong? > >> A definition is a statement by a person that a certain >> symbol will be used by them to stand for some concept. > >> What is there in such a statement that can be wrong? > >> Example: I will use the term "gleeb" to refer to an integer >> which is divisible by 2. > >> How can that statement be wrong? How would you >> define "wrong" for such a statement? > >> - Randy > >But it is horribly confusing when somebody insists >on using private definitions for common words, especially >when they do not explicitly state their definition. >If someone insists on using "even" to mean "numbers divisible >by 3" they are not likely to be well received. Exactly. The definition can be self contradictory and hence wrong. ~v~~
From: David Marcus on 23 Oct 2006 21:43
Sebastian Holzmann wrote: > David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > > Sebastian Holzmann wrote: > >> David Marcus <DavidMarcus(a)alumdotmit.edu> wrote: > >> > Sebastian Holzmann wrote: > >> >> Let's assume that ZFC - AoI is consistent, otherwise the statement is > >> >> void. Let \phi_n be a sentence in the language of set theory that > >> >> encodes the statement "There exists a set that has at least n elements" > >> >> (you can formulate this one as a homework problem). > >> >> > >> >> Since for every finite subset of the set of \phi_n, there exists a model > >> >> of ZFC - AoI where this subset is true (any model of ZFC - AoI should do > >> >> the trick), by compactness, there exists a model of ZFC - AoI where > >> >> _all_ \phi_n are simultaneously true. ("True" might not be the correct > >> >> English word for it, please do someone correct me in this.) A set that > >> >> has, for any n, at least n elements cannot be finite. Therefore it must > >> >> be infinite. > >> > > >> > Are you sure this argument works in ZFC - AoI? I really don't know, but > >> > I suspect you might have trouble proving the compactness theorem. Also, > >> > the fact that there is a model where the set is infinite doesn't mean > >> > ZFC - AoI knows the set is infinite. > >> > >> ZFC is a FO theory. Compactness for FO has been established a long time > >> ago. I might have enormous problems trying to prove that there is a > >> model of ZFC (with or without AoI) and hopefully I'll never succeed. But > >> apart from that I don't see a problem. > > > > I certainly don't claim to be an expert on what you can prove without > > AoI, but here is what I was thinking: I don't see a problem arguing that > > a proof of a contradiction is finite, so if each finite subset of axioms > > is consistent, then the entire set of axioms is consistent. But, can you > > prove completeness without AoI? I.e., can you prove that if a set of > > formulas is consistent, then it has a model? I don't know. > > Oh, I think I begin to see your problem here. But before we can speak of > ZFC as a theory, we must first have some sort of "background set theory" > available. And if we do not allow that background theory to "have" > infinite sets (in some naive way), we cannot even formulate Z, because > it consist of infinitely many sentences... Yes, that is what I meant. Mueckenheim apparently wants to redo all of Mathematics without using infinity. Of course, since he has no understanding of logic or mathematics, he doesn't realize how difficult this would really be to do. -- David Marcus |