The Definition of Points
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From: Tony Orlow on 31 Mar 2007 11:46 Mike Kelly wrote: > 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. There is no successor in a pure set. That only occurs in a discrete linear order. > >> Good enough? > > You tell me. Did you mean to say "a sequence is a set"? If so, good > enough. > > -- > mike. > Not exactly, and no, what I said is not good enough. A set with an order where each element has a unique successor is a forward-infinite sequence. Each can have a unique predecessor, and then it's backward-infinite. And if every element has both a unique successor and predecessor, then it's bi-infinite, like the integers, or within the H-riffics, the reals. One can further impose that x<y ->~y<x, to eliminate circularity. Good enough? Probably not yet. tony.
From: Tony Orlow on 31 Mar 2007 11:47 Mike Kelly 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? > > -- > mike. > As I said to Brian, it's provably the size of the set of finite natural numbers greater than or equal to 1. No, there is no last finite natural, and no, there is no "size" for N. Aleph_0 is a phantom. tony.
From: Tony Orlow on 31 Mar 2007 11:54 stephen(a)nomail.com wrote: > In sci.math Tony Orlow <tony(a)lightlink.com> 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. > > Who said anything about uncountable? You said: >>>>> An actually infinite sequence is one where there exist two elements, one >>>>> of which is an infinite number of elements beyond the other. > No mention of uncountable. > I have said in the past, and you will remember if you rub those neurons together, that in my parlance, "countably infinite" is equivalent to the old "potentially infinite", and that "actually infinite" really means "uncountable". Sound familiar? I was asked what my definition of "actual infinity" was. That's a really old term, older than "uncountable". It implies fact, rather than process. >> 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 you think there are only a finite number of elements between 1 and > w? What is that finite number? 100? 100000? 100000000000000000? > 98042934810235712394872394712349123749123471923479? Which one? > Aleph_0, which is provably a member of the set, if it's the size of the set. Of course, then, adding w to the set's a little redundant, eh? > It should be obvious that the number of elements between 1 and w is > larger than any finite natural number. Let X be a finite > natural number > 1. Then {2, 3, .. X, X+1, .. 2X } is a subset > of the elements between 1 and w that has more more than X elements. > > As I said, even you do not accept your own definition of "actually > infinite". > > Stephen > If you paid attention, the apparent contradiction would evaporate. The number of elements up to and including any finite element of N is finite, and equal to that element in magnitude. If the number is n, then there's an nth, and its value is n. As Ross like to say, NeN. We are not alone. :D Tony
From: Bob Kolker on 31 Mar 2007 12:23 Tony Orlow wrote: > > As I said to Brian, it's provably the size of the set of finite natural > numbers greater than or equal to 1. No, there is no last finite natural, > and no, there is no "size" for N. Aleph_0 is a phantom. No. It is the cardinality of the set of integers. Deep in your heart you want everything to be finite. That will limit mathematics to totally up grocery bills and such like. Mathematics based on infinities has made physics possible. Bob Kolker
From: Brian Chandler on 31 Mar 2007 12:41
stephen(a)nomail.com wrote: > In sci.math Tony Orlow <tony(a)lightlink.com> 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. > > Who said anything about uncountable? You said: > >>>> An actually infinite sequence is one where there exist two elements, one > >>>> of which is an infinite number of elements beyond the other. > No mention of uncountable. No, but learning mathematical terminology has proved to be beyond Tony. In Tonyspeak, "countable" means you can count it, and obviously the count (result) is where you get to when you finish counting. If counting the natural numbers (ours, not the Orlovian ones), then the count is the last natnum (which in itself proves set theory is inconsistent), which is an Elusive number. Plainly adding w to the end (!) of the seqence 1, 2, 3, ... only gets to said Elusive number plus one, which is another Elusive number. > > > 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 you think there are only a finite number of elements between 1 and > w? What is that finite number? 100? 100000? 100000000000000000? > 98042934810235712394872394712349123749123471923479? Which one? None of the ones you've mentioned. Although it is, of course, a perfectly ordinary natural number, in that one can add 1 to it, or divide it by 2, its value is Elusive. Only Tony could actually write it down. > It should be obvious that the number of elements between 1 and w is > larger than any finite natural number. Let X be a finite > natural number > 1. Then {2, 3, .. X, X+1, .. 2X } is a subset > of the elements between 1 and w that has more more than X elements. > > As I said, even you do not accept your own definition of "actually > infinite". Reminds me of what Wittgenstein said about lions, somehow. Brian Chandler http://imaginatorium.org |