The Definition of Points
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From: stephen on 1 Apr 2007 01:21 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 Tony Orlow <tony(a)lightlink.com> wrote: <snip> >> >>> Oh. What word shall I use? Supersequence? Is that related to a >>> subsequence or consequence? >> >> As long as you define your terms it does not matter to much what you >> call it. You could just call it an uncountably infinite sequence, but you >> need to define what that is if you want anyone to know what you are >> talking about. Why are you so reluctant to define your terms? >> > I did that, and was told no such thing exists. Gee, then, don't talk > about unicorns or alephs. Where did you define "uncountably infinite sequence"? Just naming it is not defining it. >>>>>>>>> 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 is the cardinality of a set. >> >>> Is that a number? >> >> What is your definition of "number"? aleph_0 is called a transfinite >> number, but definitions, not names, are the important thing. >> > A number is a symbolic representation of quantity which can be > manipulated to produce quantitative results in the form of symbols. I > might be wrong, but I'm sure you can apprise me of the official meaning > of "number", mathematically. ;) And how does aleph_0 not fit that definition? Stephen
From: stephen on 1 Apr 2007 01:22 In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> >> None of the options mention "size" Tony. What does "size" have >> to do with a, b or c? >> > Ugh. Me already tell you, nth one is n, then there are n of them. So > easy, even a caveman can do it. Size is difference between. Brilliant Tony. Act like an idiot when backed into a corner. Did you learn that trick from Lester? You are truly pathetic. Stephen
From: Brian Chandler on 1 Apr 2007 01:36 stephen(a)nomail.com wrote: > In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > > stephen(a)nomail.com wrote: > >> > >> None of the options mention "size" Tony. What does "size" have > >> to do with a, b or c? > >> > > > Ugh. Me already tell you, nth one is n, then there are n of them. So > > easy, even a caveman can do it. Size is difference between. > > Brilliant Tony. Act like an idiot when backed into a corner. > Did you learn that trick from Lester? Don't think so. You think Lester is acting? Brian Chandler http://imaginatorium.org
From: Brian Chandler on 1 Apr 2007 01:43 ** Sorry, some confusion here *** Brian Chandler wrote: > Tony Orlow wrote: > > Brian Chandler wrote: > > > Tony Orlow wrote: > > > >> 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.... > > Indeed. So your "characterization" of Q isn't much use, because it > doesn't distinguish Q from 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. > > No: the elements (2-adics) are right-ended bitstrings, but I was > referring to the sequences you have included in your "Try" expression. > > ...000, ..001, ...010, ... > > is a left-ended sequence. At the left end is ...000, and I can > reasonably assume that the three dots on the right mean that after ... > 010 the sequence continues with ...011 then ...100 then ...101 and so > on, but on a good day even you can see that this sequence has no right > end. Of course, every element in this sequence has the property that > if I look sufficiently to the left in the bitstring I find that I have > reached the leftmost 1, and the remainder of the left-end-less > bitstring consists only of zeros. > > The remainder of the content of your "Try" is: > > ..., ...101, ...110, ...111 > > and this is a right-ended sequence. Again, starting from ...111 on the > right, I can see how to generate the next value on the left, and do > this indefinitely. But again, every element in this sequence has the > property that if I look sufficiently to the left in the bitstring I > find that I have reached the leftmost 0, and the remainder of the left- > end-less bitstring consists only of 1s. > > So if a mathematician wrote something resembling your "Try", it would > include all 2-adics that have an endless string of 0s or and endless > string of 1s to the left. > > But I surmise you claim to have included all of the 2-adics somehow, > so I'm asking you to explain how. I notice that the elements you have > named explicitly are all in "conventional numerical" order, or what we > might call "reverse lexicographical" order. *** No we mightn't. Since all of the elements have either an unending sequence of 0s or of 1s to the left, they are in normal "lexicographic" order extended in the obvious way; we don't need to look at the "left end" of the bitstring, since there isn't one - we look at the part to the left where the digits don't vary any more (i.e. ...0000xxx or ...1111xxx) put the ...000 ones before the ...111 ones, then sort the two-ended remainders of the bitstrings in the normal way. Anyway this does not give us a way to order ...0101010 and ...1010101. > > Note that for reverse-lexicographical order we are using the right end > of the bitstring, so for any element (except the last one ...111!) we > can find the successor by (the obvious extension of) normal binary > addition (to left-end-less strings). > > However, if you wish to claim that ...1010101 and ...0101010 are > somehow both included in this thing - in some sort of hiatus in the > middle of the central five dots, at least you need to say which one > comes first. > > > Which comes > > first, 01 or 10? I think I know. Which is greater, 0.10101010... or > > 0.010101...? > > Uh, yeah? The binary fraction 0.101010... is greater than the binary > fraction 0.0101010... but so what? > The binary fraction 0.100... is greater than the binary fraction > 0.0101111... but ...1110101 comes after ...0001 in (the comprehensible > parts of) your Try above. > > We have been through this before: to provide a *sequence* of two-ended > bitstrings is easy: you use the left end to start the lexicographic > orderings, and you use the right end to generate successors. Although > this is no proof that a set of one-ended strings cannot form a > sequence, it means you cannot rely on hand-waving to assure that it > does. > > Do you want to try again? > > Brian Chandler > http://imaginatorium.org
From: cbrown on 1 Apr 2007 02:56
On Mar 31, 8:44 pm, Tony Orlow <t...(a)lightlink.com> wrote: > cbr...(a)cbrownsystems.com wrote: > > On Mar 31, 5:45 pm, Tony Orlow <t...(a)lightlink.com> wrote: > >> Mike Kelly wrote: > > >>> When we say that a set has cardinality Aleph_0 we are saying it is > >>> bijectible with N. Are you saying it's impossible for a set to be > >>> bijectible with N? Or are you saying N does not exist as a set? > >>> Something else? > >> I have been saying that bijection alone is not sufficient for measuring > >> infinite sets relative to each other. > > > Since it is certainly sufficient for comparing sets by their > > cardinality, I can only ask: what do you mean by "measuring infinite > > sets relative to each other"? There are many, many ways of "measuring" > > one set against another; which do you have in mind? > > Let's examine what '<' means. By all means; that's the correct place to start. Can you provide a definition? > x<y ^ y<z -> x<z. So "<" is a transitive relation. Ok; we established that earlier. But you're being coy as usual: you haven't mentioned /all/ of the properties that you're /actually/ /thinking/ of. Consider me a total blank slate; an alien unfamiliar with your method of writing mathematics. If x < y, can it also be true that y < x? If x < y, then /must/ y > x? Can it be true that x < x? Given distinct x and y, /must/ it be the case that either x < y or y < x? Can I make it simpler for you? Suppose S is a set. Then "<" is a PARTIAL order on S means: (1) For all x, y in S, if x = y then not (x < y). (2) For all x, y in S, if a < b then not (b < a) (3) For all x, y, z in S, if a < b and b < c then a < c To make it a TOTAL order, we add the additional constraint: (4) For all x, y in S, x = y, or x < y, or y < x. Note: There is /nothing whatsoever/ in the above definitions to do with "+", "-", "subset of" and so on; or that S must be a subset of some well-known set like the naturals or the reals, or any other notion. "Is a (blood) descendant of" is a PARTIAL order on all people living and dead. If g is my father's father, and f is my father, and m is me, and I write m < f and f < g for "I am a descendant of my father" and "my father is a descendant of my grandfather", then it follows that m < g: "I am a descendant of my grandfather". But it would be bizzare in the extreme to then claim (as you sometimes do) that therefore m - g < 0 without first very clearly saying what "me - my grandfather is a descendant of 0" is supposed to mean. What person, living or dead, is m - g supposed to be? What person, living or dead, is "0"? Ordering and arithmetic functions like "+" and "-" /don't/ / automatically/ follow one from the other. > True for real > quantities, and true if '<' is taken to mean "is a proper subset of". Yes, but -1.5 is not a proper subset of pi, nor are the sets {a,b} and {a,b,c,d,e} real quantities. You have simply given examples of how the same symbol, "<" can have 2 /different/ meanings; in one case describing a total order, and in the other, a partial order. It is NOT the case that for any subsets A, B of S that either A = B, A < B, or B < A. That's why, in that case "<" is not a total order: it doesn't satisfy constraint (4); but it /does/ obey the other constraints. You need to have ALL FOUR constraints apply to get a TOTAL or linear order. Not just one, or two of them; ALL FOUR of them. When you say something like: x<y ^ y<z -> x<z. you are only mentioning /ONE/ constraint (3); and making us "guess" what other constraints you have in mind. > The proper subset is less than the whole, and the evens are half the > naturals. That's a very primitive result. > If by "primitive" you mean "naive", I agree. It is also a "primitive result" that if a truly, honestly, really, fair coin has come up heads 20 times in a row, it is more likely to come up tails on the next flip than heads. That doesn't make this primitive intuition actually / true/. 20 heads in a row comes up about once in a million trials of 20 flips. There are currently over 6 billion people living on the planet; so one can imagine that /thousands/ of actual /living/ human beings have been confronted with this disturbing situation. Honestly, even I'd be inclined to bet against logic; even believing that the coin was really, truly, cross-my-heart-and-hope-to-die fair. "The evens are half the naturals" does not follow /solely/ from the observation that the evens are a /proper subset/ of the naturals. You need a lot /more/ than that observation to make sense of the statement "the evens are half the naturals". Thus the need to be very /specific/ about what you mean when you say things in a mathematical sense. > >> Yes, NeN, as Ross says. I understand what he means, but you don't. > > > What I don't understand is what name you would like to give to the set > > {n : n e N and n <> N}. M? > > > Cheers - Chas > > N-1? Why do I need to define that uselessness? I find the set of naturals to be a useful concept, whatever you wish it were called. For example, I can say "if n is a member of (the set of naturals), then n has a unique factorization as a product of primes". Doesn't that seem at all useful to you? It's called the Fundamental Theorem of Arithmetic: http://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic > I don't want to give a > size to the set of finite naturals... Fine. Then don't give it a size if you don't want to. But it seems rather self-defeating: wasn't your idea to put a consistent "measure" of "size" on /all/ sets? Cheers - Chas |