An uncountable countable set
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From: Virgil on 17 Sep 2006 15:09 In article <450d4da5(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Mike Kelly wrote: > Han, if I prove inductively, say, that 2^x>2*x for all x>2, do you find > it objectionable to say that this also applies to any infinite value It is objectionable, and false, to say that any infinite case follows from finite cases by reason of finite induction. What holds for finite cases may or may not hold for "infinite" cases, but which will be the case in any particular instance cannot be determined by finite induction. , > > But we are discussing whether there exists a uniform distribution over > > the naturals. If you don't think this claim means anything at all then > > why do you dispute it? If you reject the existence of the set of > > natural numbers then you reject the set theory probability is based on. > > So why bother to argue against individual theorems? You don't accept > > *any* of probability theory. > > Just because someone disagrees with the transfinite portions of set > theory doesn't mean they reject all of set theory. Clearly those of us > who object do so on the basis of the conclusions drawn in infinite case, > which derive from the axiom of infinity and/or the axiom of choice. As > far as probability goes, it certainly depends on the concept of sets, > since probability more or less measures a subset of events with respect > to the entire set of possible events. However, the same question remains > as with the rest of transfinitology - is the cardinality generalization, > based solely on raw bijection, really the most appropriate > generalization from the finite to the infinite for sets? Since that question is irrelevant in the issue of whether there can be a uniform probability distribution on a countably infinite set, it is a diversionary tactic, the fallacy of the straw man, to bring it up. > > If you refuse the idea of infinite sets, what does it mean to you to > > say a function has domain and range R? > > > > What does it mean to you, if not that one can use that range as a > constant for infinite sets? Why can't we say that, over the entire range > of R, the naturals have twice the density of the evens, and so are twice > as large a set? One can say almost anything one likes, else TO would long since have been stifled, but with infinite sets "more" should mean that no matter how one paired thing off, the larger set would have extra members, and that is not the case for TO's theories. Now if TO wishes to present a theory only valid for standardly ordered subsets of his "extended set" of reals... But any theory that depends on having to use any particular order relation on any set to find its size is a theory about order relations, not a theory about set sizes. So TO only has, at most, a theory of order relations, not a theory of set sizes.
From: Virgil on 17 Sep 2006 15:35 In article <450d53f4(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > it doesn't seem to me that one CAN measure > an infinite set with any accuracy without involving some notion of > measure into the properties of the elements which define the set. According to both ZF and NBG, the only properties that a set can have are entirely dependent on which objects are members or not members, and any properties of those members other than their mere membership are irrelevant. >Trying > to derive measure from a structure where no measure has been introduced > is bound to fail. Cardinality gives a certain gross measure of > complexity, but to consider the alephs to be any exact numbers of any > sort is unjustified, in my opinion. But anything with finer distinctions than cardinality, at any of those suggested by TO, require attention be paid to properties other than membership, such as a required order relation being imposed on the set. So that what TO is measuring is not the size of the set but the way is is ordered. TO has a measure of order relations only, and not a measure of set sizes. > Yes, there are some concepts such as limiting density and Lebesgue > measure, which come to different conclusions regarding the same sets. And are also dependent on structure other than mere membership. TO has not come up with any measure dependent only on set membership and on no other property. Cardinality, is still the only one which does this. > That is, they detect differences between some sets where cardinality > does not. They detect differences, but between additionally imposed properties other than mere membership. > Doesn't this mean that bijection alone misses real > distinctions in element count, or size, which can only be detected using > more sophisticated methods? Cardinality does not miss anything that can be determined by mere membership, with the only allowable relation between members that of equality. > If general set theory comes to a conclusion > that contradicts the conclusions of a theory that takes into account > more details of the situation, then hasn't the generalization failed in > the specific case? The general theory, cardinality, is the one based on the relation of equality of members and no other relation. Adding anything else, like TO's dependence on order relations, produces a different measure of "size", not appropriate for plain sets but only for such specially structured sets as ordered sets. Thus TO's measures are not measures of ordinary sets at all, but are only measures of order relations.
From: Virgil on 17 Sep 2006 15:37 In article <450d54dc(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Aatu Koskensilta wrote: > > Virgil wrote: > >> In article <450c71a1(a)news2.lightlink.com>, > >> Tony Orlow <tony(a)lightlink.com> wrote: > >> > >>> Aatu Koskensilta wrote: > >> > >>> Given the axioms and rules of inference, the conclusions are provably > >>> true or false. > >>> > >>> Soundness is another issue, regarding the fundamental justification > >>> for the logical axioms themselves, and whether they are "correct", > >>> meaning "objectively verifiable". > > > > I didn't write the above, nor did Tony in his post partially quoted by > > Virgil claim I did. Do be careful with the attributions and quotations. > > > > Actually, if you look back, it says at the top "Tony Orlow said". The > little bit with your name referred to something else snipped, and should > have been removed, but in Virgil's post I think it was clear those were > my words. Do you think he would have disagreed with you, Aatu? It's me > he's after. ;) It is TO's silly claim that his "measurements" of order relations are measures applicable to arbitrary sets that I am after.
From: Ross A. Finlayson on 17 Sep 2006 15:43 Han.deBruijn(a)DTO.TUDelft.NL wrote: > Mike Kelly wrote: > > > [ ... snip ... ] It's not clear to me that providing finite examples then > > saying "obviously this holds for infinite cases too" without any > > justification whatsoever should be at all convincing to anyone. > > It may be not clear to any mathematician, but it is clear to any > scientist. The reason is that infinities do not really exist. > They only exist as an attempt to make the "very large" rigorous > in some sense. The moment you forget this, you get into trouble. > > Han de Bruijn Hi Han, Hey how you doin buddy. I told you I got the management summary it's called "The Handbook of Differential Equations." Then, there are also some symmetry textbooks which I find readily accessible. Then, with your chaos theory and fractals and so forth, basically I'm trying to understand what that means in terms of metastability. Ullrich, Dave, are you reading this? Go back to posting! Have a nice day! Ross
From: Virgil on 17 Sep 2006 15:47
In article <450d5597(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <450c87cc(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <450c71a1(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Aatu Koskensilta wrote: > >>>> Given the axioms and rules of inference, the conclusions are provably > >>>> true or false. > >>>> > >>>> Soundness is another issue, regarding the fundamental justification for > >>>> the logical axioms themselves, and whether they are "correct", meaning > >>>> "objectively verifiable". > >>> If axioms were ever objectively verifiable they would not need to be > >>> assumed in the first place, but would be objectively verified. > >>> > >> In the mathematical world, the greater framework can be considered > >> relatively objective. > > > > Greater than what? If one wnats something in one's system, either it is > > provable in terms of other things in the system or it must be assumed > > without being provable in terms of other things in the sysem, and just > > like with having to have undefined terms, at some point you have to have > > unproven assumptions. > > > > In mathematics, when you get to that point, you call those unproven > > assumptions axioms. > > > > TO seems to want to do without any axioms by some sort of daisy chain > > circle of proofs lifting the whole mess up by its bootstraps. > > How on Earth do you read all that from what I said. The "greater > framework" is mathematics in general. If a particular axiom or theory > contradicts enough other math, then it's trouble. There's no reason that > all of mathematics can't be consistent. That's the greater framework. The axioms system of Euclidean geometry is inconsistent with that of various non-Euclidean geometries. and there are a lot of other places where one system contradicts another. What mathematics allows is any system of axioms which does not appear to contain any self-contradictions, at least for as long as it maintains that appearance. > > I understand that axioms are necessary, but they should not be arbitrary. Which axioms in which mathematical systems does TO think have not been judicially chosen? > >>> Which brings up the issue of TO's sanity. > >>> > >> Or conviction. > > > > Of what crimes? So far, TO's form of insanity is not illegal. Illogical, > > and possibly even immoral, but not illegal. > > That depends what state you're in. ;) If TO wishes to plead guilty of some illegality, it is entirely his own volition. |