The Definition of Points
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From: Tony Orlow on 31 Mar 2007 18:20 Brian Chandler wrote: > Tony Orlow wrote: >> Brian Chandler wrote: >>> Tony Orlow wrote: >> Hi Imaginatorium - > > That's not my name - for some reason Google has consented to writing > my name again. The Imaginatorium is my place of (self-)employment, so > I am the Chief Imaginator, but you may call me Brian. > >>>> 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'll give *you* a start, Brian, and I hope you don't have a heart attack >> over it. It's called 1, and it's the 1st element in your N. The 2nd is >> 2, and the 3rd is 3. Do you see a pattern? The nth is n. The nth marks >> the end of the first n elements. Huh! >> >> So, the property I would most readily attribute to this element Q is >> that it is the size of the set, up to and including element Q. > > Euuuughwh! Gesundheit! I seeee! Q is really Big'un, and this all jibes with my > previous calculation that the value of Big'un is 16. Easy to test: is > 16 the size of the set up to and including 16? Why, 1, 2, 3, 4, 5, 6, > 7, 8, 9, 10, 11, 12, 13, 14, 15, 16 - so it is!! > > Well, that's an interesting analysis, but something tells me there may be another natural greater than 16.... >> That is, >> it's what you would call aleph_0, except that would funk up your whole >> works, because aleph_0 isn't supposed to be an element of N. Take two >> aspirin and call me in the morning. > > I would call aleph_0 16, or I would call 16 aleph_0? > If you postulated that there were 16 naturals, that would be a natural conclusion. >> <snip> See above for a characterization of Q. > > Just to be serious for a moment, what do you understand > "characterization" to mean? In mathematics it usually implies that the > criterion given distinguishes the thing being talked about from other > things. But plainly your "characterization" applies perfectly to 16. > (Doesn't it? If not please explain.) What's more, even you agree on a > good day that there is no last pofnat - so your claim that Q is > somehow something "up to which" the pofnats go is not comprehensible. > > The set of all pofnats up to and including 16 constitutes 16 elements. The nth is equal to n. >>> 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. >>> >> n has the property of being the size of the sequence up to and including n. >> >>>> Try (...000, ..001, ...010, ......, ...101, ...110, ...111) >>> Why? What is it, anyway? >> Google 2-adics. >> > > Yes, I know what the 2-adics are. You have written an obvious left- > ended sequence ...000, ...001, ... then two extra dots, a comma and an > obvious right-ended sequence ...101, ...110, ...111. Are you claiming > (perchance!) you have specified a "sequence" that includes all of the > 2-adics? In which case, which of ...1010101 and ...0101010 comes > first? Those are both right-ended, if you insist, though they both have unending strings of zeros to the right of the binary point. Which comes first, 01 or 10? I think I know. Which is greater, 0.10101010... or 0.010101...? > >> Tony Orlow >> http://realitorium.net > > My browser can't find that domain... > It's probably your ISP's fault..... > Brian Chandler > (really) http://imaginatorium.org >
From: Tony Orlow on 31 Mar 2007 18:22 Brian Chandler wrote: > Tony Orlow wrote: >> 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: > > <oh grief, I expect he did> > Oh Good Grief, it's Chandler Brown. >>>>> Under what definition of sequence? > >>>> 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. > > Unlike Lester, I think you really do have enough brains to understand > simple mathematics if you tried. Why oh why do you not read a book, so > you wouldn't need to spew out confused babble like the above. (Pure > poetry though it may be in your own private language.) > > Brian Chandler > http://imaginatorium.org > Forsooth and what ho! Whyforever do I not? Tony Orlow http://Realitorium.net
From: Tony Orlow on 31 Mar 2007 18:27 stephen(a)nomail.com wrote: > In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >> 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. > > So in other words >>>>>>> An actually infinite sequence is one where there exist two elements, one >>>>>>> of which is an infinite number of elements beyond the other. > is not your "correct" definition of an "actually infinite sequence", > which was my point. You are so sloppy in your word usage that you > constantly contradict yourself. > > If all you mean by "actually infinite" is "uncountable", then > just say "uncountable". Of course an "uncountable sequence" > is a contradiction, so you still have to define what you mean > by a "sequence". > > Please do expliculate what the contradiction is in an uncountable sequence. What is true and false as a result of that concept? >>>> 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? > > Aleph_0 is not a finite number. Care to try again? > It's also not the size of the set. Wake up. >>> 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 > > But the question is not about the number of elements up and including > any finite element of N. I asked how many elements are between 1 and w > in the set {1, 2, 3, ..., w }. w-2 are between w and 1. x-2 are between 1 and x. w is not an element of N, nor is it finite. Oh, then why mention it? > I know you are incapable of actually thinking about all the elements of N, > but that is your problem. In any case, N is not an element of N. > Citing Ross as support is practically an admission that you are wrong. > > Stephen > Sure, of course, agreeing with someone who disagrees with you makes me wrong. I'll keep that in mind. Thanks.. Tony
From: Tony Orlow on 31 Mar 2007 18:30 cbrown(a)cbrownsystems.com wrote: > On Mar 31, 5:30 am, Tony Orlow <t...(a)lightlink.com> wrote: >> Virgil wrote: > >>> In standard mathematics, an infinite sequence is o more than a function >>> whose domain is the set of naturals, no two of which are more that >>> finitely different. The codmain of such a function need not have any >>> particular structure at all. >> That's a countably infinite sequence. Standard mathematics doesn't allow >> for uncountable sequences like the adics or T-riffics, because it's been >> politically agreed upon that we skirt that issue and leave it to the >> clerics. > > That's false; Please elucidate on the untruth of the statement. It should be easy to disprove an untrue statement. people have examined all sorts of orderings, partial, > total, and other. The fact that you prefer to remain ignorant of this > does not mean the issue has been skirted by anyone other than > yourself. > There have always been religious and political pressures on this area of exploration. >> However, where every element of a set has a well defined >> successor and predecessor, it's a sequence of some sort. >> > > Let S = {0, a, 1, b, 2, c}. > > Let succ() be defined on S as: > > succ(0) = 1 > succ(1) = 2 > succ(2) = 0 > succ(a) = b > succ(b) = c > succ(c) = a Okay you have two sequences. > > Every element of S has a well-defined successor and predecessor. What > "sort of sequence" have I defined? Or have you left out some parts of > the /explicit/ definition of whatever you were trying to say? > > Cheers - Chas > Yes, I left out some details.
From: Tony Orlow on 31 Mar 2007 18:31
Lester Zick wrote: > On 30 Mar 2007 10:23:51 -0700, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > >> On Mar 30, 9:39 am, Tony Orlow <t...(a)lightlink.com> wrote: >> >>> They >>> introduce the von Neumann ordinals defined solely by set inclusion, >> By membership, not inclusion. >> >>> and >>> yet, surreptitiously introduce the notion of order by means of this set. >> "Surreptitiously". You don't know an effing thing you're talking >> about. Look at a set theory textbook (such as Suppes's 'Axiomatic Set >> Theory') to see the explicit definitions. > > Kinda like Moe(x) huh. > > ~v~~ Welcome back to your mother-effing thread. :) E R. 01oo |