The Definition of Points
in [Logic]
|
Prev: On Ultrafinitism
Next: Modal logic example
From: Brian Chandler on 31 Mar 2007 13:02 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! 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!! > 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? > <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. > > 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? > Tony Orlow > http://realitorium.net My browser can't find that domain... Brian Chandler (really) http://imaginatorium.org
From: Brian Chandler on 31 Mar 2007 13:05 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> > >>> 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
From: stephen on 31 Mar 2007 13:05 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". >>> 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 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 is not an element of N, nor is it finite. 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
From: cbrown on 31 Mar 2007 14:40 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; 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. > 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 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
From: Lester Zick on 31 Mar 2007 14:43
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~~ |