The Definition of Points
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From: Lester Zick on 21 Mar 2007 18:44 On Wed, 21 Mar 2007 01:25:42 -0600, Virgil <virgil(a)comcast.net> wrote: >> So, the quest is for the basic axioms that give rise to all the theorems >> we want, and more. For you, that's ZFC, but for some, that's not >> satisfying. Oh well. > >While ZFC is a nice axiomatic system, I am not at all sure that it is >what you claim for it. NBG is a nice system too, and there are a variety >of others of considerable virtue. Do any of them produce straight lines? ~v~~
From: Lester Zick on 21 Mar 2007 18:46 On Wed, 21 Mar 2007 09:37:42 -0500, Wolf <ElLoboViejo(a)ruddy.moss> wrote: >> Hilbert didn't just pick statements out of a hat. >> Rather, he didn't do so entirely, though they could have been >> generalized better. > >Oh my, another genius who understands math better than Hilbert, et al. Well perhaps not better than tables and beer bottles. ~v~~
From: Lester Zick on 21 Mar 2007 18:49 On Tue, 20 Mar 2007 20:53:56 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >Virgil wrote:> >> >> Read up on Pontryagin and his ilk before making such idiotic claims. > >For those who do not know it, Leon Pontrfyagin was a blind topologist. And you're a blind empiric. So what's your point? ~v~~
From: Lester Zick on 21 Mar 2007 18:58 On 21 Mar 2007 04:38:34 -0700, "hagman" <google(a)von-eitzen.de> wrote: >On 21 Mrz., 01:29, Lester Zick <dontbot...(a)nowhere.net> wrote: >> On 20 Mar 2007 16:15:06 -0700, "hagman" <goo...(a)von-eitzen.de> wrote: >> >> >> >> >On 20 Mrz., 20:24, Lester Zick <dontbot...(a)nowhere.net> wrote: >> >> On 20 Mar 2007 03:11:41 -0700, "hagman" <goo...(a)von-eitzen.de> wrote: >> >> >> >On 20 Mrz., 00:30, Lester Zick <dontbot...(a)nowhere.net> wrote: >> >> >> >> > I'm saying that a >> >> >> >model is just a model. The properties of the model do not cause >> >> >> >the thing it's modeling to have those properties. >> >> >> >> Oh great. So now the model of a thing has properties which don't model >> >> >> the properties of the thing it's modeling. So why model it? >> >> >> >> ~v~~ >> >> >> >You are trying to walk the path in the wrong direction. >> >> >E.g. 0:={}, 1:={{}}, 2:= {{},{{}}}, ... >> >> >is a model of the naturals: The Peano axioms hold. >> >> >> However we don't have a model of straight lines except by naive >> >> assumption. >> >> >Wrong. >> >The field of reals provides a very good model of a straight line, >> >also in the form of an affine subspace of a higher dimensional real >> >vector space. >> >> Please can the neomathspeak for a moment to explain how application of >> the Peano and suc( ) axioms and a resultant succession of integers and >> straight line segments, even if we allow their assumption, defines >> straight lines. I mean are we just supposed to assume the sequence of >> line segments is necessarily colinear and lies on a straight line? >> >> >> >However, in this model we have "0 is a set", which does not follow >> >> >from Peano axioms. >> >> >Thus the model has some additional properties. What's wrong with that? >> >> >> It isn't a model of what we wish to model. >> >> >You made a general statement against models (or rather grossly >> >misunderstood >> >someone elses general statement about models). >> >Thus I am allowed to pull out any model I like. >> >> Go right ahead. I'd just like to see you pull out straight lines >> first. > >I did that below: R is as straight as can get. How so? Are we supposed to accept your opinion these lines are straight just because you say so? I don't see any evidence to support your contention that these lines are straight if they're generated by association with integers and integers are generated by the Peano and suc( ) axioms because as far as I can tell these axioms only generate straight line segments and not straight lines. >To get back to the problem of excess properties: >R identified with Rx{0} as a subset of RxR "is" a straight line in a >plane. >Again, the model shares all properties of the abstract line in a >plane, >but it also has additional proerties, e.g. there is a special point >(0,0). > >Btw, {42}xR is another line of that plane model and it happens to be >the case >that >- lines are sets of points in this model by construction >- two non-parallel lines determine a unique point of intersection >(e.g. (42,0)) > >> >> >> >However, the model shows that the Peno axioms are consistent (provided >> >> >the set theory we used to construct the model is). >> >> >> Consistency is only a prerequisite not a final objective for a model. >> >> >On the contrary. Producing a model for a theory is the common way to >> >show that >> >the theory is consistent (provided the theory used to construct the >> >model >> >is consistent) >> >> Well you know, hagman, this is curious. I complain that consistency is >> not the final objective of a model. And your reply is to assure me >> that some models are consistent. Do you consider that responsive? > >Am I fooled by some subtle notion because I'm not a native speaker of >English? I wasn't aware that you weren't. However even if you're not I don't see your comments as responsive to my observation. >For me "final objective" means something like "purpose". >So your claim is: > purpose of model =/= consistency "Not only consistency" is different from "=/=" consistency. "Consistency" is a minimum requirement not a sufficient requirement. >My reply: > purpose of model == demonstration of consistency of modeled theory >Your paraphrasing of my reply: > Some models are consistent > > >> >> >A: Let's consider a theory where 'lines' consist of 'points' and >> >'lines' determine 'points'... >> >B: Hey, isn't that nonsense? Can such a theory exist? >> >A: Of course. Take pairs of real numbers for points and certain sets >> >of such pairs as lines... >> >B: I see. >> >> Except you aren't even bothering to demonstrate the existence of >> numbers, lines, straight lines, etc. You're just claiming they're >> there. How can you take something that isn't there? Then how can you >> demonstrate they are there? Let's get real for a change shall we. >> > >Of course we have to start *somewhere*. >It depends on the point where B really says "I see". >The usual accepted startiong point today is ZF(C) set theory, >which allows to construct a model of the natural numbers N. >Given N, one can construct a model for the field Q of rationals. >Given Q, one can construct a model for the complete archimedean field >R. >Given R, well, we had that already... > >So you have to redirect any attack to ZFC and, yes, that's about the >point where I stop bothering. >And maybe I don't even claim "they're there" but only "it makes sense >talking about them". ~v~~
From: Lester Zick on 21 Mar 2007 19:00
On Wed, 21 Mar 2007 11:55:08 +0000 (UTC), stephen(a)nomail.com wrote: >In sci.math hagman <google(a)von-eitzen.de> wrote: >> On 21 Mrz., 01:29, Lester Zick <dontbot...(a)nowhere.net> wrote: >>> >>> Well you know, hagman, this is curious. I complain that consistency is >>> not the final objective of a model. And your reply is to assure me >>> that some models are consistent. Do you consider that responsive? > >> Am I fooled by some subtle notion because I'm not a native speaker of >> English? > >No, your English is not the problem. The problem is Lester's. >Lester speaks his own language where words mean whatever he >wants them to mean, so are in fact meaningless. Often he is just >parroting back phrases he does not understand in a vain attempt to >appear knowledgable. Well I can tell as soon as boobirds like Stephen come out the cause is hopeless. At least I'm witty. ~v~~ |