The Definition of Points
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From: Bob Kolker on 13 Apr 2007 10:08 Bob Kolker wrote: > Mike Kelly wrote: > >> >> a) A consecutive set of naturals starting with 1 with size X can not >> have any maximum other than X. >> b) A consecutive set of naturals starting with 1 with size X has >> maximum X. > > > Whoa! If a -consectutive-(!!!) set of naturals with 1 as its least > element has cardinality X (an integer), then its last element must be X. > > A simple induction argument will show this to be the case. > > Can you show a counter example? I should have said, for I assumed it, that X is finte. Sorry about that. Bob Kolker
From: Mike Kelly on 13 Apr 2007 10:16 On 13 Apr, 15:06, Bob Kolker <nowh...(a)nowhere.com> wrote: > Mike Kelly wrote: > > > a) A consecutive set of naturals starting with 1 with size X can not > > have any maximum other than X. > > b) A consecutive set of naturals starting with 1 with size X has > > maximum X. > > Whoa! If a -consectutive-(!!!) set of naturals with 1 as its least > element has cardinality X (an integer), then its last element must be X. > > A simple induction argument will show this to be the case. > > Can you show a counter example? > > Bob Kolker N -- mike.
From: Mike Kelly on 13 Apr 2007 10:16 On 13 Apr, 15:08, Bob Kolker <nowh...(a)nowhere.com> wrote: > Bob Kolker wrote: > > Mike Kelly wrote: > > >> a) A consecutive set of naturals starting with 1 with size X can not > >> have any maximum other than X. > >> b) A consecutive set of naturals starting with 1 with size X has > >> maximum X. > > > Whoa! If a -consectutive-(!!!) set of naturals with 1 as its least > > element has cardinality X (an integer), then its last element must be X. > > > A simple induction argument will show this to be the case. > > > Can you show a counter example? > > I should have said, for I assumed it, that X is finte. Sorry about that. > > Bob Kolker Ah. Fair enough. -- mike.
From: Alan Smaill on 13 Apr 2007 11:10 Lester Zick <dontbother(a)nowhere.net> writes: > On Thu, 12 Apr 2007 14:23:04 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: >> >>That's okay. 0 for 0 is 100%!!! :) > > Not exactly, Tony. 0/0 would have to be evaluated under L'Hospital's > rule. Dear me ... L'Hospital's rule is invalid. > ~v~~ -- Alan Smaill
From: stephen on 13 Apr 2007 12:52
In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > Lester Zick wrote: >> On Mon, 2 Apr 2007 16:12:46 +0000 (UTC), stephen(a)nomail.com wrote: >> >>>> It is not true that the set of consecutive naturals starting at 1 with >>>> cardinality x has largest element x. A set of consecutive naturals >>>> starting at 1 need not have a largest element at all. >>> To be fair to Tony, he said "size", not "cardinality". If Tony wishes to define >>> "size" such that set of consecutive naturals starting at 1 with size x has a >>> largest element x, he can, but an immediate consequence of that definition >>> is that N does not have a size. >> >> Is that true? >> >> ~v~~ > Yes, Lester, Stephen is exactly right. I am very happy to see this > response. It follows from the assumptions. Axioms have merit, but > deserve periodic review. > 01oo Everything follows from the assumptions and definitions. People have been telling you this for well over a year now. If you change the axioms, or change the definitions, you will get different results. However the old axioms, definitions and results remain just the same as before. N has a cardinality. If "size" is defined as cardinality, N has a "size". If "size" is defined differently, N still has a cardinality. Stephen |