The Definition of Points
in [Math]
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From: Lester Zick on 31 Mar 2007 15:28 On 31 Mar 2007 07:29:06 -0700, "Mike Kelly" <mikekellyuk(a)googlemail.com> wrote: >On 31 Mar, 13:48, Tony Orlow <t...(a)lightlink.com> wrote: >> step...(a)nomail.com wrote: >> > In sci.math Virgil <vir...(a)comcast.net> wrote: >> >> In article <460d4...(a)news2.lightlink.com>, >> >> Tony Orlow <t...(a)lightlink.com> wrote: >> >> >>> An actually infinite sequence is one where there exist two elements, one >> >>> of which is an infinite number of elements beyond the other. >> >> >> Not in any standard mathematics. >> >> > It is not even true in Tony's mathematics, at least it was not true >> > the last time he brought it up. According to this >> > definition {1, 2, 3, ... } is not actually infinite, but >> > {1, 2, 3, ..., w} is actually infinite. However, the last time this >> > was pointed out, Tony claimed that {1, 2, 3, ..., w} was not >> > actually infinite. >> >> > Stephen >> >> No, adding one extra element to a countable set doesn't make it >> uncountable. If all other elements in the sequence are a finite number >> of steps from the start, and w occurs directly after those, then it is >> one step beyond some step which is finite, and so is at a finite step. > >So (countable) sequences have a last element? What's the last finite >natural number? 46 ~v~~
From: Lester Zick on 31 Mar 2007 15:33 On Sat, 31 Mar 2007 12:23:48 -0400, Bob Kolker <nowhere(a)nowhere.com> wrote: >Deep in your heart you want everything to be finite. That will limit >mathematics to totally up grocery bills and such like. No that will limit arithmetic and not mathematics to totaling up grocery bills and the like. >Mathematics based on infinities has made physics possible. Mathematikers still can't say what an infinity is, Bob, and when they try to they're just guessing anyway. So I suppose if we were to take your claim literally we would just have to conclude that what made physics possible was guessing and not mathematics at all. ~v~~
From: Bob Kolker on 31 Mar 2007 16:13 Lester Zick wrote: > > > You talk about lines as if they were made up of points. In one model of Euclidean geometry that satisfies all of Hilbert's Axioms the lines are made up of points. Furthermore it can be shown Hilbert's Axioms are categorical, so all models are isometric. So a line is made up of points in -any- model for Hilbert's Axioms. Bob Kolker
From: Bob Kolker on 31 Mar 2007 16:14 Virgil wrote: > > I have never found Zick to be correct on any point. But then I have long > since stopped looking at Zick's posts. I suppose that it is marginally > possible that Zick may have been right about something since then. A stopped clock? Except Zick is not riight twice a day. Possibly twice a year though. Bob Kolker
From: Bob Kolker on 31 Mar 2007 16:18
Lester Zick wrote: > > Mathematikers still can't say what an infinity is, Bob, and when they > try to they're just guessing anyway. So I suppose if we were to take > your claim literally we would just have to conclude that what made > physics possible was guessing and not mathematics at all. Not true. Transfite cardinality is well defined. In projective geometry points at infinity are well defined (use homogeneous coordinates). You are batting 0 for n, as usual. Bob Kolker |