Obections to Cantor's Theory (Wikipedia article)
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From: Tony Orlow on 27 Jul 2005 13:49 Virgil said: > In article <MPG.1d500de16aaa9f18989f85(a)newsstand.cit.cornell.edu>, > Tony Orlow (aeo6) <aeo6(a)cornell.edu> wrote: > > > that is not at all my logic. Pay attention. > > > Since we have never seen anything from TO that qualifies as logic, there > has been nothing to pay attention to, at least of logic. > > TO claims that an iterative process, n -> n+1, that has no end must > achieve actual infinity, but cannot say at which iteration such a > transformation occurs. At the same time TO claims that the iterative > process, 1/n -> 1/(n+1), never actually reaches zero. > Again you snip everything and then make false statements. You really need to grow up. Inductive proof doesn't "achieve actual infinity" and I never said that, but it implicitly defines an infinite loop that covers all members of the infinite set using the transitive nature of logical implication. Why don't you try talking about what YOU think, instead of trying to malign others by making up lies about what they said, for a change? You DO think, don't you? Hmmmm....maybe not. -- Smiles, Tony
From: MoeBlee on 27 Jul 2005 13:49 Han de Bruijn wrote: > MoeBlee wrote: > > > Meanwhile, I'm still fascinated by your inconsistent theory, posted at > > your web site, which is: > > > > Z, without axiom of infinity, bu with your axiom: Ax x = {x}. > > > > Would you say what we are to gain from this inconsistent theory? > > No. But first we repeat the mantra: > > A little bit of Physics would be NO Idleness in Mathematics > > It is physically correct that every member of a set is at the same time > a _part_ of the set, meaning that a e A ==> a c A , where e stands for > membership and c for being a subset. The above axiom that a member of a > set cannot physically distinguished from a set which contains only the > member - the envelope {} means nothing, physically - is only a weaker > form of this idea. > > There has been a thread on this topic in 'sci.math' as well, called > "Set inclusion and membership". It is noted that a "set theory without > the membership" actually exists. Google it up as "mereology" and you > will find quite some clues. > > Now I simply add this axiom to ZFC and see what happens. Nothing else. Actually, you add your axiom to Z without infinity, and claim that this is consistent. I don't understand why you think it is consistent. I don't understand why you expect people to take you seriously when you propose a blatant contradiction. You're ready with mockery about a field of study of which you don't understand the most basic rudiments. That's egregious. You might as well enter a forum to deride concepts in harmony and counterpoint while not knowing even the notes in the C major scale. If you'd like to have a theory in which Ax x = {x}, then that's fine. But your claim - pretty much bragging - on your web site, that this is consistent with Z without infinity is false. > If the consequence may be that ZFC scrumbles into nothingness, then, > unfortunately, we are not going to gain anything. But I consider that > not to be my problem, because I'm just doing .. the mantra. > > No kidding: it IS a problem, but I find that first things come first. > > Han de Bruijn Meanwhile, I had posted to you a few days ago asking for substantiation of your claim that Hilbert said that mathematics is a senseless game. Unless I missed something, you did not respond. That is, of course, your prerogative, but another poster picked up on your post, so that the claim was iterated, thus contributing to the propogation of your unsubstantiated assertion. There is spread over the Internet a supposed quote of Hilbert saying that mathematics is a meaningless game of symbols. But Martin Davis's question whether this quote is in Hilbert's writings has so far not had an affirmative response. (Let alone that whatever Hilbert's remarks were, they must be taken in a context that includes Hilbert's explicitly stated concerns with intuition and content.) You've made an unsubstantitated and misleading claim about an important matter in the history of mathematics, and you haven't the intellectual responsibility to address this. But, hey, you've got mantras do... MoeBlee
From: Alan Morgan on 27 Jul 2005 13:42 In article <3kpuavFvls7jU3(a)individual.net>, Robert Low <mtx014(a)coventry.ac.uk> wrote: >Tony Orlow (aeo6) wrote: > > Proof that f(n), the number of strings in the set of all strings up >to and including length n in N, on a finite alphabet of size S, is finite: > > >But nobody has disagreed with that. The point of contention is >whether the union over all n of S^n contains an infinite string; >it doesn't, but you claim it does. > >When you answer the question: > >"How many elements does the set of all finite integers contain?" He's already claimed that the set of all finite integers is, in some wierd way, ill-defined. I think it's supposed to be finite, but of a completely undetermined and undeterminable size. Why this is supposed to be an improvement is beyond me. Alan -- Defendit numerus
From: Tony Orlow on 27 Jul 2005 13:53 Daryl McCullough said: > Tony Orlow (aeo6) says: > > >Except for the fact that somehow you got an infinite set in > >a finite number of steps, producing 1 element at a time. > >How does that work? > > No, if you produce one element at a time, then there is never > a time in which you will have produced an infinite set. Once > again, you're having quantifier problems. Stop saying that. It's bullshit. > > Let enum(e,s,t) mean "enumeration e produces element s at > or before step t". Then > > 1. S is finite > <-> exists enumeration e, exists step t, forall s in S, > enum(e,s,t) > > 2. S is countable > <-> exists enumeration e, forall s in S, exists step t, > enum(e,s,t) > > Note the difference between 1. and 2. The difference is in > the order of quantifiers. That difference is important. > The definition of finite says that there is some *maximum* > number of steps t, independent of the element s, at which > you know that you will have enumerated s. The definition > of countable says that for each s, there is a corresponding > t (depending on s) at which you will have enumerated s. > > -- > Daryl McCullough > Ithaca, NY > > That has nothing to do with it. Do you produce the naturals through this inductive process ala Peano, adding 1 each time to produce the next natural number? Is it an infinite set? So, are you not adding 1 an ifninite number of time sto generate an infinite number of naturals? I can't argue this anymore. It's too obvious. -- Smiles, Tony
From: Tony Orlow on 27 Jul 2005 13:57
Daryl McCullough said: > Tony Orlow (aeo6) wrote: > > > >imaginatorium(a)despammed.com said: > > >> Remind us how you determine the bigulosity of the set > >> { 1, 1/2, 1/4, 1/8,...} > > >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. > > So, you agree that for *finite* sets, two sets have the same > bigulosity if and only if there is a bijection between the two? > But that no longer holds for infinite sets? > > Then how is bigulosity an improvement over cardinality? Because Bigulosity takes into account the nature of the bijection in order to determine a precise relative size of infinity. The question of size for finite sets is trivial: count the elements. But mapping functions can also be used to exactly measure the size of a finite numeric set and an infinite numeric set equally well. It's a much better measure than bijection which, as we all know, even works on proper subsets, when they are infinite. A system that even makes proper subsets the same size as their supersets seems to be broken, to me. > > -- > Daryl McCullough > Ithaca, NY > > -- Smiles, Tony |