The Definition of Points
in [Math]
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From: Tony Orlow on 1 Apr 2007 11:34 Virgil wrote: > In article <460f22e6(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >>> The obvious question is why haven't /you/ studied them; instead of >>> making vague and uninformed statements about them (regardless of what >>> you choose to call these ordered sets). >>> >> The question is, "is there an acceptable term with which to refer to >> such uncountable linearly ordered sets?" > > The set of real numbers, whether with or without infinitesimals is an > uncountable linearly ordered set, but of course not discretely ordered. > > And I cannot believe that TO, who is usually quite inventive, if not > always accurate, cannot create one. I did, the H-riffics. > >>>>>> However, where every element of a set has a well defined >>>>>> successor and predecessor, it's a sequence of some sort. > > Not necessarily. If a set is partitioned into two or more subsets each > with such an order on it, but with no order between partitions, then the > set itself is not even an ordered set even though every member has a > well defined predecessor and successor. It's certainly easy enough to order the partitions, though not without infinite descending sequences either within or between the partitions, in an uncountable set, as far as I can see.
From: Mike Kelly on 1 Apr 2007 11:37 On 1 Apr, 16:07, Tony Orlow <t...(a)lightlink.com> wrote: > cbr...(a)cbrownsystems.com wrote: > > On Mar 31, 7:27 pm, Tony Orlow <t...(a)lightlink.com> wrote: > >> step...(a)nomail.com wrote: > >>> 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. > > No, you were told that to define "an uncountable sequence" as "a > > sequence which is uncountable" makes about as much sense as defining > > "a quadriangle" as a "a triangle that has four sides". That is /not/ > > the same as being told "there is no such thing as a polygon with four > > sides". > > I defined it as a sequence where there exist elements infinitely beyond > other elements. As you're incapable of providing your definition of "sequence" that doesn't elucidate anything. > >>>>>>>>>> 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. > > > Great! All that's left then is for you to define "quantity", > > "manipulated", and "quantitative results" without using the words > > number, quantity, manipulated, and quantitative results. > > Don't be a boor. "Quantitative results" are quantities indicated by the > resulting symbolic expression. "Manipulate" means "produce a new string > from an existing one according to rules". A quantity is a point on the > real line. A number is a string that indicates a point. Arithmetic is > the manipulation of strings. But, you know all that. Do you think people are *lying* when they say they can't understand you? > >> I > >> might be wrong, but I'm sure you can apprise me of the official meaning > >> of "number", mathematically. ;) > > > There really isn't one. Honest. Sure, there's a definition of natural > > number, rational number, algebraic number, adic number, complex > > number, Stirling number, number field, and so on. But by itself, the > > word "number" is too vague to have a useful mathematical definition; > > just like the word "size". > > > Cheers - Chas > > That was tongue-in-cheek. I know there's no definition of "number",and > mathematicians seem quite satisfied with that for themselves, but insist > that I produce a definition of a word they use every day without knowing > what they even mean themselves. Mathematicians refer to "natural numbers" and "real numbers" and "ordinal numbers" and "cardinal numbers" and so on. They don't refer to plain old "numbers", because "number" isn't defined. You don't seem to comprehend this. -- mike.
From: Tony Orlow on 1 Apr 2007 11:48 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. > > If the size of N is Q, then Q is the last element of N. It doesn't exist. >>>> 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. > > 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). Actually, let's tweak the example, noting that incrementing ...111 results in an infinite carry and the string ...000. So, here's the 2-adics in a single order without gap, both right and left endless: { ..., ...100, ...101, ...110, ...111, ...000, ...001, ...010, ...011, ...} > > 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 > Well, I addressed this with the T-riffics. I thought there was something about that in the post you just responded to. With the tweak above, such numbers will end up infinitely far to the right or left. It can't be determined which, without additional information such as what's used by the T-riffics. Tony Tony
From: Tony Orlow on 1 Apr 2007 11:49 Virgil wrote: > In article <460f2439(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > > >> On the finite scale, it takes an infinite number of infinitesimals to >> achieve measure. > > Are infinitesimal units ever equal to their reciprocals, as are all > finitesimal units in the reals? No, infinitesimal units have infinite units as reciprocals.
From: Mike Kelly on 1 Apr 2007 11:54
On 1 Apr, 16:48, Tony Orlow <t...(a)lightlink.com> wrote: > 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. > > If the size of N is Q, then Q is the last element of N. It doesn't exist. Irrespective of what notion of "size" is being used? -- mike. |