An uncountable countable set
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From: Virgil on 16 Sep 2006 20:37 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. > >> That means that when we try adopting a set of > >> axioms, we follow them to their conclusions, deductively. Then, we > >> evaluate the axioms given the discrepancies between what they conclude > >> and what we expected intuitively. If we find that our expectations are > >> not met, then we must revise our theory and adjust our axioms or, even > >> possibly, our rules of inference. > > > > Sane people, as they grow up, recognize that they must occasionally > > revise their expectations. TO seems impervious to this reality of life. > > > > 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.
From: Virgil on 16 Sep 2006 20:54 In article <450c8974(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <450c6449$1(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <450c3d37(a)news2.lightlink.com>, > >>> TO keeps harping on the "length" of the real line as a part of his > >>> mythology. > >> Does it exist? It's the number of naturals on the line. > > > > Between what two points does TO find that measure of length? > > Between which two naturals does Virgil find a difference of aleph_0-1? The difference (distance?) between any two naturals is a natural, which Aleph_0 - 1 is not. The distance between any two "points" on the real line is a real number, the (absolute) difference between the real numbers for these points. > > >> Is it fixed, or does it stretch and shrink at whim? > > > > The "length" of the real line does not exist at all. > > Then neither does the count of naturals, What TO means by "the count of naturals" only he knows, but the cardinality of the set of naturals exists. > > >>> Length measurements on a line are distances on that line between two > >>> points on that line. > >> Measurements of line SEGMENTS, yes. > > > > What other sorts of "length" measurements does TO claim he can make on a > > line? > > I claim that, even if no length can be determined due to endlessness of > a given line, that if that line exists, it is always as long as itself. > Can you disagree? > > >>> Which points on the real line does TO use to measure the length of that > >>> line? > >> It is not measurable, but like all objects and properties, is equal to > >> itself. > > > > If the line is not measurable then it does not have any length at all, > > just as TO has no sense at all. > > If you think the line is not measurable, then quit trying to pretend you > are providing any kind of measure for sets which traverse the line. There is a standard measure for the distance between any two numbers on the real line which is given by the absolute difference between those real numbers. But that in no way gives any measure for the line in its entirety, because it has no end point numbers from which to take an absolute difference. If TO's mind is too perplexed to see that, he needs a shrink. > My logic is clear and simple. And wrong!
From: Aatu Koskensilta on 16 Sep 2006 22:39 Tony Orlow wrote: > Hi Aatu - > > I appreciate your desire to accommodate the simply naive and confused by > addressing issues in terms they can understand regarding mathematical > questions. I think that's very human and generous of you, and good > advice to anyone trying to teach. I do think that once we get into > foundational arguments of the sort going on here, we really can't avoid > such technicalities, since the validity, if not the soundness, of the > arguments hinges on such fundamental issues. First order logic, rules of inference, and all that form a mathematical tool that correctly captures, to an extent, the informal notion of something logically following from a set of premises. This is shown by the completeness theorem for first order logic with a few conceptual considerations - famously due to Kreisel in his informal rigour paper. Now, one might consider the basic idea of logical consequence in first order context suspect or inadequate in some way, in which case an alternative notion of logical validity etc. must be provided - most likely not in the form of any formal system of logic with rules of inferences and formal axioms, but in informal terms similarly as one presents the classical or the intuitionistic picture. It is then up to the individual mathematicians to evaluate the fruitfulness, plausibility, coherence, applicability and so forth of the presented idea of logic - in this process formalization might or might not be of some help, e.g. by enabling a proof that the alternative picture is in some sense incompatible but still intertranslatable with the classical picture, or enabling one to establish conclusively that arguments of this or that kind are not valid under the alternative conception of logic. Now, in addition to the question of logic one might simply choose to reject this or that mathematical principle either as outright false or simply as unjustified. Of course, without tweaking one's logic it is not possible to accept a mathematical principle and reject some of its consequences. Here too formal theories might be of some limited use, e.g. allowing one to establish that some principle is indeed independent of some other principle (over some suitable background assumptions). > Yes, this is my point. When we speak of the "size" of a set, for finite > sets it's the count. For infinite sets, some generalization becomes > necessary. The issue for me is which tenets of finite sets do we > consider most important to preserve as we generalize. The concept of "size" when applied to non-finite sets bifurcates into many different concepts that just happen to agree on finite sets. The most general such notion of size is provided by cardinality in the Cantorian sense, since it doesn't presuppose any numerical ordering or a notion of density or some such be provided along with the sets compared. It also leads to a highly fruitful and beautiful mathematical theory with applications in almost every area of modern mathematics. This is not to say that other notions of "size" as applied to sets are ignored; the idea that there are twice as many naturals as there are odd naturals can be captured mathematically, although this notion is less general and applies only in case the sets in question are equipped with additional structure. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Aatu Koskensilta on 16 Sep 2006 22:42 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. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Virgil on 17 Sep 2006 00:54
In article <Yz2Pg.13567$VX1.6175(a)reader1.news.jippii.net>, Aatu Koskensilta <aatu.koskensilta(a)xortec.fi> wrote: > Tony Orlow wrote: > > Hi Aatu - > > > > I appreciate your desire to accommodate the simply naive and confused by > > addressing issues in terms they can understand regarding mathematical > > questions. I think that's very human and generous of you, and good > > advice to anyone trying to teach. I do think that once we get into > > foundational arguments of the sort going on here, we really can't avoid > > such technicalities, since the validity, if not the soundness, of the > > arguments hinges on such fundamental issues. > > First order logic, rules of inference, and all that form a mathematical > tool that correctly captures, to an extent, the informal notion of > something logically following from a set of premises. This is shown by > the completeness theorem for first order logic with a few conceptual > considerations - famously due to Kreisel in his informal rigour paper. > Now, one might consider the basic idea of logical consequence in first > order context suspect or inadequate in some way, in which case an > alternative notion of logical validity etc. must be provided - most > likely not in the form of any formal system of logic with rules of > inferences and formal axioms, but in informal terms similarly as one > presents the classical or the intuitionistic picture. It is then up to > the individual mathematicians to evaluate the fruitfulness, > plausibility, coherence, applicability and so forth of the presented > idea of logic - in this process formalization might or might not be of > some help, e.g. by enabling a proof that the alternative picture is in > some sense incompatible but still intertranslatable with the classical > picture, or enabling one to establish conclusively that arguments of > this or that kind are not valid under the alternative conception of logic. > > Now, in addition to the question of logic one might simply choose to > reject this or that mathematical principle either as outright false or > simply as unjustified. Of course, without tweaking one's logic it is not > possible to accept a mathematical principle and reject some of its > consequences. Here too formal theories might be of some limited use, > e.g. allowing one to establish that some principle is indeed independent > of some other principle (over some suitable background assumptions). > > > Yes, this is my point. When we speak of the "size" of a set, for finite > > sets it's the count. For infinite sets, some generalization becomes > > necessary. The issue for me is which tenets of finite sets do we > > consider most important to preserve as we generalize. > > The concept of "size" when applied to non-finite sets bifurcates into > many different concepts that just happen to agree on finite sets. The > most general such notion of size is provided by cardinality in the > Cantorian sense, since it doesn't presuppose any numerical ordering or a > notion of density or some such be provided along with the sets compared. > It also leads to a highly fruitful and beautiful mathematical theory > with applications in almost every area of modern mathematics. This is > not to say that other notions of "size" as applied to sets are ignored; > the idea that there are twice as many naturals as there are odd naturals > can be captured mathematically, although this notion is less general and > applies only in case the sets in question are equipped with additional > structure. Exactly the point that Tony Orlow rejects. |