Cantor Confusion
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From: David Marcus on 31 Oct 2006 19:03 mueckenh(a)rz.fh-augsburg.de wrote: > David Marcus schrieb: > > mueckenh(a)rz.fh-augsburg.de wrote: > > > > > > But which is obviously possible and correct under special > > > circumstances, while Zermelo's AC is obviously false under any > > > circumstances. > > > > What do you mean an axiom is "false"? > > An axiom leading to a contradiction when added to a set of axiom which > do not so, is false. The axiom that every straight line crosses itself > is false in Euclidian geometry. It is amazing that you seem unable to make a straight statement. Are you now saying that ZFC contains a contradiction? I.e., that you can prove a contradiction from within ZFC? There seems no other way to interpret what you just wrote, but you have denied in the past that you can prove a contradiction from within ZFC. > > Are you discussing philosophy or > > mathematics? > > I don't know what predominates in Cantor's theory - philosophy or > theology, but I am sure that there is no mathematics there. Considering most of us are discussing modern mathematics, not Cantor's theory, your statement is not an answer to my question. -- David Marcus
From: David Marcus on 31 Oct 2006 19:04 MoeBlee wrote: > mueckenh(a)rz.fh-augsburg.de wrote: > > David Marcus schrieb: > > > What do you mean an axiom is "false"? > > > > An axiom leading to a contradiction when added to a set of axiom which > > do not so, is false. > > Under that definition of 'false', EVERY non-logical sentence is false. > That is, under your definition, EVERY sentence that is not a theorem of > the predicate calculus alone is false. Good point. -- David Marcus
From: Virgil on 31 Oct 2006 19:06 In article <1162325467.755574.102840(a)h48g2000cwc.googlegroups.com>, Han.deBruijn(a)DTO.TUDelft.NL wrote: > stephen(a)nomail.com wrote: why I am > > > talking about physical problems anyway? The balls and vase problem > > is not a physical problem. Mathematics is perfectly capable > > of handling non physical problems, despite your cryptic insistence > > that "mathematics is not independent of physics". > > Sure. Case closed. As usual. And that's the end of our debate, huh? > > Han de Bruijn It could be if HdB paid any attention to anything but his own beliefs.
From: Lester Zick on 31 Oct 2006 19:10 On Tue, 31 Oct 2006 02:27:15 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> wrote: >In article <md5ak294kscg4uk48a276jktc64lf430rq(a)4ax.com> Lester Zick <dontbother(a)nowhere.net> writes: > > On Sun, 29 Oct 2006 02:01:16 GMT, "Dik T. Winter" <Dik.Winter(a)cwi.nl> > > wrote: >... > > >In all discussions you need a "domain of discourse". This, however, does > > >not mean that you need to flag every term with the "domain of discourse". >... > > Of course you need a domain of discourse. In mathematics that is > > established either by true definitions or axioms. But what I was > > specifically referring to is a process of particular definition rather > > than general definition. Particular definition is simple enumeration > > of some quality to be defined as properties of particular objects. > >(That may exist or may not exist.) Correct except there are two kinds of things which may not exist: happenstantially non existents and things which cannot exist. > A > > general definition just defines the subject without reference to > > objects defined in such terms. If I say "infinity is . . ." it's a non > > specific general definition applicable to whatever domain applies. > >But that is not a mathematical definition, but a philosophical definition. What makes you think it's not a mathematical definition? And more importantly what makes you think it's not a true definition whether mathematical or not? >Without domain of discourse, such a definition makes no sense, mathematically. Then perhaps mathematics makes no sense mathematically. > > Whereas if I try to define the quaity of infinity in specific objects > > as in "infinity(x) is . . ." I'm stuck with whatever x may turnout to > > be and without specifying what x is there is no way to determine > > whether the definition can be correct or not. > >This is wrong. You do not define the quality of infinity in specific >objects, but as a general property within a set of axioms (part of your >domain of discourse). Then why would anyone try to define infinite(x)? That's nonsense. > As far as I know, "infinity" in mathematics has >only been defined in the context of topology, where it is the single point >used in the one-point compactification (but it is long ago that I did all >this). And from there it has a derived definitions in projective geometry >(the point at infinity, and also the line at infinity). The term is used, >losely, in analysis, using limits, but is not really defined there as term. >You will see oo used in analysis, but when it is used it does *not* mean >"infinity". When that symbol is used the usage has a specific definition. Obviously because no one can define "infinity" in mathematically exhaustive terms. > > >The same holds for the Godbach conjecture. When you read current accounts, > > >they are clear, but when you read the original, it is nonsense. Until you > > >realise that the original was written when 1 was considered to be prime. > > >According to the tables of D. H. Lehmer, there are five primes smaller > > >than 10. > > > > But don't forget the process of comprehension itself improves vastly > > with time and experience of expression. When I look back over my own > > posts the level and sophistication of expression and application has > > improved enormously. > >No, there is something else. In time definitions in a particular field >of mathematics can change over time. So when you are reading old books >or articles you need to know whether the definitions actually have been >changed since then or not. Sure but that can be a result of improved comprehension of the subject. > Especially Wolfgang Mueckenheim does not >allow for such changes of definitions. But also in physics definitions >do change in time. The definition of a metre (as measure) has changes >at least two times, and I think three times, in the course of time. >That definition changed, due to more sophistication; newer definitions >could be more precise, and objects could be measured more correctly >using the new definitions. In mathematics it is a bit different. At >a certain time, 1 was considered a prime because it satisfied the >naive definition (a prime is divisible only by 1 and itself). This >became unsatisfactory because it complicated the formulation of major >results (Unique Prime Factorisation), especially so when it was >attempted to use it in other places than the natural numbers. So now, >1 is a unit and 2 is the smallest prime in the natural numbers. And I'm still left wondering why 0!=1. > > >You need some assumptions about the domain of discourse in which the > > >terminology sits. In general that is clear, but some people bemuddle > > >that and try to proof some inconsistency of (say) ZFC using something > > >that is completely false in ZFC, and can not be proven within that > > >theory. >... > > Yet on the other hand those with a great deal of experience in any > > system often claim the system itself is true when it is in fact only > > problematic at best. > >I think you are wrong. At best they tell that the system is not shown >to be inconsistent. Which tells us what exactly? Certainly not that it's true. > > The idea of a "standard" system really only > > refers to whatever happens to be conventional and not what is true or > > even best. > >That is false, at least in mathematics. There is no single standard >system at all. There are, of course, conventional standard systems >that are taught at universities because it is thought that they give >the more direct information for progress. Only until now did they >show better that they produced results than other systems. So what "standard" do you use for "standard mathematics"? > > It is often argued that one cannot argue against a system unless one > > understands the system and one cannot understand the system without > > having studied the system, the implication being that one cannot argue > > against a system as long as there is anyone who understands and has > > studied the system longer than critics of it. This is not necessarily > > true if one argues against isolated pieces or parts of the system and > > not the system as a whole.
From: David Marcus on 31 Oct 2006 19:18
imaginatorium(a)despammed.com wrote: > David Marcus wrote: > > <snip> > > > I wonder: do Lester, Ross, Han, Tony, and WM all agree that noon doesn't > > exist? It is such an odd thing to say. Imagine walking up to someone in > > the street and trying to convince them that noon doesn't exist. > > > > It is so hard to keep the nonsense straight. It all seems to run > > together--although there are stylistic differences. > > More than stylistic, I think. For example, don't offhand recall Lester > ever saying anything mathematical that was even wrong, whereas our Tony > is reponding in another thread, saying something that makes sense, > seems to be correct, and (gasp!) includes use of the f-word. > > http://groups.google.com/group/sci.math/browse_frm/thread/ea85cc0bd40c0ca8/43d661d74974c55d#43d661d74974c55d > > Question: > If a_n is such that sum(a_n) converges is it true that sum((a_n)^3) > converges? > > Tony's answer: > It would seem so, at least assuming all positive terms. The number of > a_n greater than 1 must be finite if sum(a_n) converges, so the number > of a_n^3 greater than 1 would be finite as well, and that portion > converge. For all a_n less than or equal to 1, a_n^3 is less than or > equal to a_n, and so that portion would converge as a series of lesser > terms. > > I think he may have missed the point slightly, since I suppose the > nontrivial question is about the case where all terms are not positive, > but let's not be negative. But notice his style: first he gives what he thinks the answer is, then he points out that he's adding an assumption! -- David Marcus |