The Definition of Points
in [Math]
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From: Tony Orlow on 31 Mar 2007 08:41 Mike Kelly wrote: > On 30 Mar, 18:25, Tony Orlow <t...(a)lightlink.com> wrote: >> Lester Zick wrote: >>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <t...(a)lightlink.com> >>> wrote: >>>>>> If n is >>>>>> infinite, so is 2^n. If you actually perform an infinite number of >>>>>> subdivisions, then you get actually infinitesimal subintervals. >>>>> And if the process is infinitesimal subdivision every interval you get >>>>> is infinitesimal per se because it's the result of a process of >>>>> infinitesimal subdivision and not because its magnitude is >>>>> infinitesimal as distinct from the process itself. >>>> It's because it's the result of an actually infinite sequence of finite >>>> subdivisions. >>> And what pray tell is an "actually infinite sequence"? >>>> One can also perform some infinite subdivision in some >>>> finite step or so, but that's a little too hocus-pocus to prove. In the >>>> meantime, we have at least potentially infinite sequences of >>>> subdivisions, increments, hyperdimensionalities, or whatever... >>> Sounds like you're guessing again, Tony. >>> ~v~~ >> An actually infinite sequence is one where there exist two elements, one >> of which is an infinite number of elements beyond the other. >> >> 01oo > > Under what definition of sequence? > > -- > mike. > A set where each element has a well defined unique successor within the set. Good enough? tony.
From: Tony Orlow on 31 Mar 2007 08:48 stephen(a)nomail.com wrote: > In sci.math Virgil <virgil(a)comcast.net> wrote: >> In article <460d4813(a)news2.lightlink.com>, >> Tony Orlow <tony(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. Try (...000, ..001, ...010, ......, ...101, ...110, ...111) Tony
From: Brian Chandler on 31 Mar 2007 10:07 Tony Orlow wrote: > stephen(a)nomail.com wrote: > > In sci.math Virgil <virgil(a)comcast.net> wrote: > >> In article <460d4813(a)news2.lightlink.com>, > >> Tony Orlow <tony(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. Perhaps you might care, Tony, to list some properties of this "some step" you have referred to above? I tell you what, I'll give you a start - let's call this 'step' (actually this is the wrong word, since step is normally the gap between two adjacent elements**, so let's call this element) Q. ** I'm sure you understand that being described as more logically coherent (orwhateveritwas) than Lester Zick is rather like being called more caring than Jack the Ripper, but I take the sentiment to mean that you will probably agree with this nitpick about 'step' terminology. So: Q has the property of being the last element in an endless sequence Q has the property of nonexistence, actually Now it's your turn. > Try (...000, ..001, ...010, ......, ...101, ...110, ...111) Why? What is it, anyway? Brian Chandler http://imaginatorium.org
From: Mike Kelly on 31 Mar 2007 10:27 On 31 Mar, 13:41, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > > On 30 Mar, 18:25, Tony Orlow <t...(a)lightlink.com> wrote: > >> Lester Zick wrote: > >>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <t...(a)lightlink.com> > >>> wrote: > >>>>>> If n is > >>>>>> infinite, so is 2^n. If you actually perform an infinite number of > >>>>>> subdivisions, then you get actually infinitesimal subintervals. > >>>>> And if the process is infinitesimal subdivision every interval you get > >>>>> is infinitesimal per se because it's the result of a process of > >>>>> infinitesimal subdivision and not because its magnitude is > >>>>> infinitesimal as distinct from the process itself. > >>>> It's because it's the result of an actually infinite sequence of finite > >>>> subdivisions. > >>> And what pray tell is an "actually infinite sequence"? > >>>> One can also perform some infinite subdivision in some > >>>> finite step or so, but that's a little too hocus-pocus to prove. In the > >>>> meantime, we have at least potentially infinite sequences of > >>>> subdivisions, increments, hyperdimensionalities, or whatever... > >>> Sounds like you're guessing again, Tony. > >>> ~v~~ > >> An actually infinite sequence is one where there exist two elements, one > >> of which is an infinite number of elements beyond the other. > > >> 01oo > > > Under what definition of sequence? > > > -- > > mike. > > A set where each element has a well defined unique successor within the > set. So any set is a sequence? For any set, take the successor of each element as itself. > Good enough? You tell me. Did you mean to say "a sequence is a set"? If so, good enough. -- mike.
From: Mike Kelly on 31 Mar 2007 10:29
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? -- mike. |