An uncountable countable set
in [Math]
|
Prev: integral problem
Next: Prime numbers
From: Tony Orlow on 17 Sep 2006 10:00 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. ;) Tony
From: Tony Orlow on 17 Sep 2006 10:03 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. I understand that axioms are necessary, but they should not be arbitrary. > >>>> 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. That depends what state you're in. ;)
From: Tony Orlow on 17 Sep 2006 10:10 Virgil wrote: > 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. No, that's the point I have been making. Without the additional structure which Aatu suggests, bijection may show some kind of equivalence, but it cannot be considered any kind of exact analog for the size of finite sets. You're trying to extract measure from something with no measure in it, like blood from a stone. Notice that my two major suggestions, dealing with real quantities and symbolic languages (IFR and N=S^L) employ just that kind of additional structure required to get a formulaic and exact comparison between the sets. I haven't found myself rejecting one thing that Aatu has said so far. Tony
From: Aatu Koskensilta on 17 Sep 2006 10:12 Tony Orlow wrote: > Do you think he would have disagreed with you, Aatu? I think there's much I disagree about with Virgil. In particular his conception of what mathematics is about seems extremely wrongheaded. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: Tony Orlow on 17 Sep 2006 10:15
Aatu Koskensilta wrote: > Virgil wrote: >> In article <Yz2Pg.13567$VX1.6175(a)reader1.news.jippii.net>, >> Aatu Koskensilta <aatu.koskensilta(a)xortec.fi> wrote: >> >>> 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. > > Quite possibly. Since he's an obvious crank there's really very little > point in caring about what he thinks or rejects, and even less point in > engaging him in endless "debates". Of course, this is USENET and there's > very little point to anything in any case; hence my few observations on > the rhetorical tactics in these debates. > Okay, well, I reject that. Tony |