Am I a crank?
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From: Lester Zick on 6 Sep 2006 12:38 On Tue, 05 Sep 2006 20:31:35 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <rnvrf2dhbfd5d87ht13c238o53vnvmfvqk(a)4ax.com>, > Lester Zick <dontbother(a)nowhere.net> wrote: > >> On Tue, 05 Sep 2006 15:54:22 -0600, Virgil <virgil(a)comcast.net> wrote: >> >> >In article <vckrf21eftu9dec4bndsritu590geljcak(a)4ax.com>, >> > Lester Zick <dontbother(a)nowhere.net> wrote: >> > >> > >> >> I have no idea what >> >> Virgil was trying to say but that's nothing unusual. >> > >> >As Zick usually does not even have any idea what Zick is trying to say, >> >one is hardly surprised at his inability to understand what others say. >> >> Oh don't go getting all bent out of shape, Virgil. > >What sort of shape is Zick bending himself into, then? > >> Isn't >> there anyone else you can talk to? > >Zick seems to be the one determined to talk to me. Yeah really. Can't imagine why I'm determined to talk to someone who has nothing to say. ~v~~
From: Virgil on 6 Sep 2006 15:00 In article <p8utf2t0ae4d19nktl1ro8mnq9frqfjhkt(a)4ax.com>, Lester Zick <dontbother(a)nowhere.net> wrote: > On Tue, 05 Sep 2006 18:22:30 -0600, Virgil <virgil(a)comcast.net> wrote: > > >In article <agvrf21km3cu96aq3sn94hb65mb1ttc8lj(a)4ax.com>, > > Lester Zick <dontbother(a)nowhere.net> wrote: > > > >> On Tue, 05 Sep 2006 15:51:32 -0600, Virgil <virgil(a)comcast.net> wrote: > >> > >> >Better than truth because Zick says so" > >> > >> Hardly better than truth, Virgil, just better than fantasy land. > > > >Zick would know better that I about what goes on in the latter. > > You would certainly know better what goes on in neomathematiker > fantasy land. As I am totally unfamiliar with any such thing, Zick is, as usual, wrong. Must be one of Zick's own many fantasies.
From: Virgil on 6 Sep 2006 15:03 In article <nbutf21mlpqn895jluk8vf2sd75m2pgdsm(a)4ax.com>, Lester Zick <dontbother(a)nowhere.net> wrote: > On Tue, 05 Sep 2006 20:31:35 -0600, Virgil <virgil(a)comcast.net> wrote: > > >In article <rnvrf2dhbfd5d87ht13c238o53vnvmfvqk(a)4ax.com>, > > Lester Zick <dontbother(a)nowhere.net> wrote: > > > >> Isn't > >> there anyone else you can talk to? > > > >Zick seems to be the one determined to talk to me. > > Yeah really. Can't imagine why I'm determined to talk to someone who > has nothing to say. Logorrhea?
From: Aatu Koskensilta on 6 Sep 2006 17:26 MoeBlee wrote: > Tony Orlow wrote: >> Hi Aatu - >> You say here that induction follows from our mathematical picture of the >> naturals, > > He's talking about a particular axiom of induction in PA. I agree that > it essential to our understanding of the naturals. But you should > understand that induction is even more general. Inductive sets are a > certain kind of set. Roughly, an inductive set is a set that is the > intersection of a class of sets each of which is closed under a given > relation. Yes. But it seems our understanding of such inductively generated sets is much more basic than our understanding of any set theoretic ideas. We readily understand natural numbers, various formal expressions in formal languages, and so forth, without any recourse to set theoretic thinking of these things as the smallest set closed under this or that rule. Such set theoretic reasoning and understanding only becomes important when we reach inductive definitions - such as arithmetical truth or being an ordinal notation in O - of more generalized nature. Taking these things to fall under the more generalized notion of inductive definition is mathematically fruitful, but does not reflect anything of conceptual or "foundational" significance. > I don't speak for Koskensilta, but this may work in many directions - > first order PA may be thought of as a formalization of our mathematical > understanding, so that it is not required to reverse this as you are > doing. In some sense we can take PA as a "formalization of our mathematical understanding" - in that it does indeed incorporate a part of our mathematical knowledge, or beliefs - and in others such a identification would be horribly wrong. What is central to our understanding of the naturals is that they are generated from a single element, 0, by repeated application of the successor operation. As said, abstracting on this we get the informal induction principle, applying to any property we regard as determinate. In particular, if we accept the language of arithmetic as meaningful, induction will apply to properties definable in that language, giving us PA. Of course, if we accept the language of arithmetic as meaningful, we should accept as meaningful also properties definable by quantification over properties thus definable, and so forth. There is no formal theory we could recognize, only on basis of our acceptance of the induction principle and the intelligibility of the language of arithmetic, as correct, that would cover all principles that are acceptable on such basis, for G?delian reasons. Pondering on such things, we recognize that such is the case even in case of set theory, and indeed any basis we can come up for mathematics. Formalized theories can only capture a portion - perhaps the only interesting portion! - of what our informal ideas lead us to accept, or what is acceptable on basis of our informal ideas. > Generally, I don't see existence of naturals as arrived upon inductively. > In first order PA, the natural numbers are not mentioned in the theory itself. True indeed, but first order PA is entirely irrelevant to our understanding of natural numbers. > We do have transfinite induction in Z set theories. It just does not > prove what you want it to prove here. So get some axioms that prove > what you want them to prove. A silly request. People like Orlow are not interested in mathematical proofs or derivations as usually understood, but in heroic resistance to the evil establishment. -- Aatu Koskensilta (aatu.koskensilta(a)xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
From: MoeBlee on 6 Sep 2006 17:49
Aatu Koskensilta wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Hi Aatu - > >> You say here that induction follows from our mathematical picture of the > >> naturals, > > > > He's talking about a particular axiom of induction in PA. I agree that > > it essential to our understanding of the naturals. But you should > > understand that induction is even more general. Inductive sets are a > > certain kind of set. Roughly, an inductive set is a set that is the > > intersection of a class of sets each of which is closed under a given > > relation. > > Yes. But it seems our understanding of such inductively generated sets > is much more basic than our understanding of any set theoretic ideas. We > readily understand natural numbers, various formal expressions in formal > languages, and so forth, without any recourse to set theoretic thinking > of these things as the smallest set closed under this or that rule. Such > set theoretic reasoning and understanding only becomes important when we > reach inductive definitions - such as arithmetical truth or being an > ordinal notation in O - of more generalized nature. Taking these things > to fall under the more generalized notion of inductive definition is > mathematically fruitful, but does not reflect anything of conceptual or > "foundational" significance. My remarks are to offer Orlow an appreciation that induction is more general (and 'general' not 'foundational' is the word I used) in mathematics than one particular (though, of course, most salient) axiom in one particular theory. And without claiming that this genearality is necessarily intuitively or philosphically foundational, an understanding of this generality does greatly enhance one's mathematical understanding of any particular induction schemata. MoeBlee |