Obections to Cantor's Theory (Wikipedia article)
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From: David Kastrup on 26 Jul 2005 16:51 Tony Orlow (aeo6) <aeo6(a)cornell.edu> writes: > The big problem in transfinite cardinality is the assumption that a > bijection necessarily indicates equal sizes for infinite sets, as it > does for finite sets. Wrong. There is no such assumption. Set sizes are _defined_ to be >= if a surjection exists. That is all. For convenience, a number of sizes are given names, for example, all finite sets are given the number of their elements as size. It turns out that a few infinite sets can also be given a consistent label for their size. > When the only way to form a bijection is with a mapping function, > then that function needs to be taken into account. This nonsense > about an infinite set of finite whole numbers is pretty bad too, Well, it seemingly is not a concept accessible to everybody, surprising though this may appear. Core at this problem is the inability to differentiate between infinite (a property of a single element) and arbitrarily large (a property of available elements from an infinite set), in short, the inability to distinguish between There exists an n such that for all k n>k (false, n can't be infinite) For all k there exists an n such that n>k (true, n can be arbitrarily large) is remarkably common among the set cranks. -- David Kastrup, Kriemhildstr. 15, 44793 Bochum
From: Virgil on 26 Jul 2005 16:54 In article <MPG.1d5046e4924ae60c989f95(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > imaginatorium(a)despammed.com said: > > > > > > Tony Orlow (aeo6) wrote: > > > Martin Shobe said: > > <big snip> > > > > > > BTW, there is a caveat on convergence. You have to assume the > > > > standard topology. In other topologies, that sequence can converge. > > > > > You mean with a ring? That's really not what we're talking about, unless > > > you > > > agree that the number line is a circle, and even then it's not relevant. > > > In > > > pure quantitative terms, a sum of infinite 1's is infinite. > > > > Tony, could you please clarify: when you use the word "ring", what do > > you mean? > > > > (a) The algebraic structure known by mathematicians as a ring > > (b) Something else (in which case please call it a T-ring) > > (c) You're sure it is (a), but cannot actually sketch the axioms for a > > ring (a) > > > > If you select (c), please confirm you really meant (a) by sketching the > > axioms. > > If you select (a), please suggest why you think the name "ring" is > > used. > > > > (Since I really have no idea, I'd be interested in informed comments on > > the last question.) > > > > Brian Chandler > > http://imaginatorium.org > > > > > > > > > > > > So, please make up your mind. Do we increment to get a successor an > > > infinite > > > number of times, or only a finite number of times, to get N? > > > > > > > > Martin > > > > > > > > > > > > > > -- > > > Smiles, > > > > > > Tony > > > > > I am not an expert in rings, nor am I going to sketch the axioms that you > know > better than I, nor does it help the conversation when you snip the original > statement was repsonding to, which had soemthing to do with numbers being > their > own multiples of more than 1, or something. It didn't make sense in the > context > of normal quantitative addition. It was a vague guess as to what the idea > was. > > The sum of an infinite number of 1's is infinite. That's all. The sum of an infinite number of 1's does not exist. Only finite sums exist. Limits may sometimes exist for sequences of finite sums, but it is an abuse of language to speak of that giving a sum of infinitely many terms.
From: Virgil on 26 Jul 2005 16:55 In article <MPG.1d504a2bd2e03311989f96(a)newsstand.cit.cornell.edu>, Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > Okay, I don't know what I was saying. It relies on bijections, but doesn't > consider any bijection to mean equality. The mapping functions are used to > determine relative size when it comes to numeric sets. Sorry. The beginning of wisdom?
From: Tony Orlow on 26 Jul 2005 17:00 Chris Menzel said: > On Wed, 20 Jul 2005 19:40:06 GMT, Stephen Montgomery-Smith > <stephen(a)math.missouri.edu> said: > > You do come across as sincere in your differing opinions. I can only > > suppose that in some strange manner that your brain is wired > > differently than ours are. What seems completely logical and sensible > > to us, seems to be nonsense to you, and conversely, what seems to be a > > proper argument to you, is so weird and strange to us that we seem > > unable to even know where to start it trying to disuade you from your > > point of view. > > Oh please. There isn't just a difference in points of view here. TO is > making demonstrable, elementary mistakes of both logic and mathematics. Untrue. > It would be one thing if he were to point out clearly some particular > principle (Axiom of infinity? Power set? Excluded middle?) that he > disagreed with -- we might then be able just to acknowledge a difference > in intuitions, in which case the idea of "different wiring" might have > some purchase. Bijection between infinite sets as proving equal size. I disagree with that assumption. Also, the use of induction, an implicitly infinite process, to prove finiteness. > But TO has at least implicitly acknowledged such things > as the existence infinite sets and cardinals, and the power set axiom, > and he reasons using excluded middle, so his rejection of the theorems > these principles entail shows that his views are at least implicitly > contradictory. Contradictory to what? Neither bijection as a direct proof of equality nor the implicitly infinite nature of induction depend on those concepts. > In addition, however, he has explicitly made many > elementary mathematical errors and committed numerous logical howlers, > all of which have been pointed out to him very clearly. I have been accused of quantifier errors, but that is due to you not paying attention to my arguments, not to any logical eror on my part. Cantorians claim "quantifier dyslexia" on a regular basis around here, without cause. I may not know all the areas you mention, like abelian groups or the topological context of infinite series, but any mathematical errors I made were typos at best. Care to point out any specific mistakes? You folks love to make these statements about your own successes and the failures of others, conveninetly leaving out any particular detail. > That may indeed > be a matter of wiring, but that would be a shame; I would rather hope it > is a curable combination of ignorance and rather appallingly shameless > hubris. I am quite happy with my wiring, thank you. > > Chris Menzel > > -- Smiles, Tony
From: Tony Orlow on 26 Jul 2005 17:02
MoeBlee said: > Tony Orlow: > > What is your logistic system, your primitive terms, and your axioms? > > MoeBlee > > I am trying to get together some pages concerning this, but don't have a lot of time. By "primitive terms" I assume you mean definitions, and I know what axioms are, but what exactly do you mean by "logistic system"? -- Smiles, Tony |