An uncountable countable set
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From: MoeBlee on 7 Sep 2006 13:57 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> My point was that that axiom only applied to sets, but I went further to > >> say that even urelements can be considered to be equal to the sets of > >> all 1--place predicates which apply to them, > > > > Then they are not urelements. An urelement has no members. If the set > > of 1-place predicates that hold of an object has members, then it can't > > be an urelement, since an urelement has no members. > > Yes, I was saying that there may be no such thing as a urelement, You were? I don't see that in what you wrote. My point is not to play another round of "said it - no you didn't - yes I did..." but rather, hopefully, to foreclose on another of your complaints that I and a whole bunch of other people are not reading what you said. > if > every object is considered to be the set of all of its attributes. > However, what we have here are two different kinds of sets: sets of > objects, and sets of attributes. If a urobject, which has no object > elements, represents a set of attributes, then this indicates a need to > distinguish between sets of objects and sets of attributes. I suppose > this is the basis for type theory, which I'll try to read up on. But > Virgil says it's an anachronism. We'll see. You're only vaguely on point regarding type theory. But you say you'll read up on it. Why not organize your reading so that first you read material that will allow you to properly understand what you will read about type theory? > > One more time: an urelement has no members, so an urelement cannot > > equal a set that has members, and the set of 1-place predicates that > > hold for an object is a set with members, namely the 1-place predicates > > that hold for the object. Focus, Tony, focus. > > If you reread what I wrote before, and focus, you'll see I was drawing a > distinction between two perspectives on the matter. It might be best to > call the set of attributes of an object its "nature" or "character", > rather than a set, and reserve "set" for objects. No, it's not MY READING that is the problem. It is that NOW you are ADDING something that wasn't in what you wrote, wiseass. Fine, add explanation. Of course you are welcome to do that. But what a jerk you come off as for faulting me for not having read your mind. > > You are conflating the fact that properties define sets with your own > > notion that the sets ARE the set of properties that define the set. > > That is exactly the point of the video image analogy I made, which you > > dismissed. Just because a property (actually, in Z set theories, since > > Skolem, for precision, we use formulas rather than properties, but that > > is a technical point here) defines a set does not entail that the set > > IS the set of properties that define it. > > When you say you use formulas, that is for sets of real numbers > (including rational and naturals). Now, please distinguish the curve > defined by a formula and the set of points in that curve. I am cnflating > the curve and the formula. They are one. The curve is the set of points. The formula that defines that set of points is not that set of points. Go ahead and try to build a system in which the curve IS the formula. Fine. But that is not the case in set theory. > >> Yes, at times you've been very generous, and I appreciate your > >> contributions. You just seem rather cranky lately. > > > > Because it is frustrating talking with you, as I prefer not to consider > > you stupid, but your willful ignorance (not just in refusing to a read > > a single book on the subject but also in ignoring so many crucial > > points made in posts made to you) and arrogance cause you to say so > > many stupid things. > > My arrogance is driven by conviction and success in formulating an > alternative. Your frustration comes from trying to prove me wrong in that. It seems that it is in your NATURE to be hopelessly lost. > > The thust of what I said is that the idea is inconsistent with Z set > > theories, so you need to devise some other system if you want to > > implement the idea; also, I mentioned that even in some system of your > > own, I suspect (suspect, not proven) that your idea entails a vicious > > circle that will cause a contradiction within its own system. > > I haven't disagreed with that, but I do believe there is a way out of > the vicious cycles you mention, even when trying to incorporate the > fundamental notion of discernibility by property into set theory. So > far, there seems to be some perceptions of contradiction, but as far as > I can tell, no actual contradictions between any ideas I've put forth. All of that would be meaningful if you just gave us a system. > >>> The set is an object in some domain of discourse. The definition is a > >>> syntactical object, which is a member of the theory but almost never > >>> (if ever) a member itself of the domain of discourse. > >> So, there are members of the theory that transcend the domain of > >> discourse? > > > > No, that's nonsense what you just said, and nothing I said deserves > > such a nonsensical reply. > > You said a syntactical object may be a member of the theory, but not of > the domain of discourse. It's right above. No. I said those syntactical objects that are DEFINITIONS are almost never, if at all, themselves members of the domain of discourse. One may manage to devise a model that has such syntactical objects as members of the domain of discourse, but it is not usually done, and certainly not REQUIRED as you think it shoud be. Moreover, some syntactical objects, such as 1-place operation symbols, are members of certain domains of discourse, such as in Henkin's proof of completeness. A theory is a set of sentences closed under entailment. Those sentences talk ABOUT objects (the objects in the domain of discourse) but those objects do not have to be the sentences themselves (and usually are not the sentences themselves). Again, the distinction: The members of the theory are sentences. The members of the domain of disourse are the objects that the sentences talk ABOUT. > If the set is a member of the domain of discourse, rather than a member > of the theory itself, why is the definition of the set not a member of > the domain of discourse? As I said, I don't know that it is impossible for definitions to b
From: Virgil on 7 Sep 2006 14:07 In article <4500181d(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <44fe1466(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> A better way to view the universe of sets is as a relation between the > >> objects within it and 1-place predicates concerning objects, where each > >> object has a truth value associated with each predicate, between 0 and 1 > >> inclusive. Thus, these predicates define subsets of the universe, eh? > > > > TO appears to be suggesting that there are truth values strictly between > > 0 and 1, as in "fuzzy logic". > > I am actually referring to probabilistic logic, which doesn't have the > kludgy aspects of "fuzzy" logic. > > > > > That doesn't work well in mathematics, TO. > > Probability doesn't work well in mathematics? Numerical truth values other than 0 or 1 do not work well in mathematics. > > > > > And allowing unrestricted predicates leads to all sorts of anomalies, > > like sets which are not members of themselves. > > Yes, predicates like "is a set" cause self-referencing problems. Not in ZF, ZFC and NBG. > > > > > Besides which mathematics has no need to include the entire universe, > > and those why try to make it do so get themselves in over their heads. > > In which case they must swim. Or, like TO, be fishy.
From: stephen on 7 Sep 2006 14:15 Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: >> Tony Orlow <tony(a)lightlink.com> wrote: >>> stephen(a)nomail.com wrote: >>>> Mike Kelly <mk4284(a)bris.ac.uk> wrote: >>>> >>>>> I was actually thinking of that problem in the car a week ago. Surely >>>>> even if there are "infinite integers" to go on the balls, those balls >>>>> get removed infinitesimally before Noon? The nth ball gets removed at >>>>> the nth iteration, at time -(1/2) ^ n. Surely Tony would argue that >>>>> this is valid in the infinite case. Oh well. >>>> Tony's objection was that because you are always adding balls, >>>> you can never get 0. But apparently his own math says that >>>> if you keep adding 1, you eventually get, ..111111111111, and >>>> if you add 1 to that, you get 0. So in Tony's world, if >>>> you just add 1 ball at a time, and never remove any balls, >>>> you can end up with 0 balls at noon. >>>> >>>> Stephen >>>> >> >>> It's nice to see you are enjoying making fun of the number circle. >>> People used to make fun of the Round Earth Theory too. What shape is our >>> universe, and why? Have you ever asked yourself that question? I thought >>> not. >> >> I'm making fun of you, not the number circle. Once again >> you are ignoring an argument and responding with adolescent >> philosophical maunderings. >> >> Here is a simple question for you: if you keep adding balls >> to a pile, and never remove any, and you do this an "infinite >> number of times", is it possible to end up with zero balls? >> >> Your math seems to say "yes". Do you agree with your math? >> Or does your answer depend on what shape of the Universe >> happens to be that day? >> >> Stephen > The universe is always expanding at the speed of light. At any given > moment, it wraps back upon itself, but given that we cannot travel as > fast as it expands, we cannot circumnavigate it. We cannot make the full > circle, and never can, given the relationship between space and time. > While a snapshot of the continuum may lead us to the conclusion that > adding forever gets us nothing, it only ever gets us farther away from > our origin, as that continuum expands. > Tony Was that supposed to be an answer? How do you know that the universe is always exanding at the speed of light, and what does that have to do with the question I asked? Once again, according to you ...11111111 = 1+1+1+... and ...11111111 + 1 = 0 So if we add ball i to the pile at time 12:00-(1/2^i), is it possible that there will be 0 balls at time 0? According to your math this is possible. Are you willing to accept the logical consequences of your mathematics or are you going to go on some tangent about the speed of light? Stephen
From: imaginatorium on 7 Sep 2006 14:24 Tony Orlow wrote: > stephen(a)nomail.com wrote: > > Tony Orlow <tony(a)lightlink.com> wrote: > >> stephen(a)nomail.com wrote: > >>> Mike Kelly <mk4284(a)bris.ac.uk> wrote: > >>> > >>>> stephen(a)nomail.com wrote: > >>>>> imaginatorium(a)despammed.com wrote: Ooh look! I must've said something... > >>>>>> Tony Orlow wrote: <snip> > > Here is a simple question for you: if you keep adding balls > > to a pile, and never remove any, and you do this an "infinite > > number of times", is it possible to end up with zero balls? > > > > Your math seems to say "yes". Do you agree with your math? > > Or does your answer depend on what shape of the Universe > > happens to be that day? > > > > Stephen > > The universe is always expanding at the speed of light. At any given > moment, it wraps back upon itself, but given that we cannot travel as > fast as it expands, we cannot circumnavigate it. We cannot make the full > circle, and never can, given the relationship between space and time. > > While a snapshot of the continuum may lead us to the conclusion that > adding forever gets us nothing, it only ever gets us farther away from > our origin, as that continuum expands. Oh, right. So you mean like Big'un is bigger now than when you invented it - was it a few months ago? How much bigger? I mean just roughly? 20%? 1-Little'unth-of-somepercent? Wow! Dynamic infinity. You know, I have suggested this to Lester, but I think you too should work something about the stockmarket into your book title. "Investments: Turning Little'un into Big'un with IFR". Something like that... Brian Chandler http://imaginatorium.org
From: Mike Kelly on 7 Sep 2006 14:42
Tony Orlow wrote: > Mike Kelly wrote: > > Tony Orlow wrote: > >> Virgil wrote: > >>> In article <44fe2642(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Virgil wrote: > >>>>> In article <44fd9eba(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>> > >>>>>> Dik T. Winter wrote: > >>>>>>> In article <44ef3da9(a)news2.lightlink.com> Tony Orlow <tony(a)lightlink.com> > >>>>>>> writes: > >>>>>>> Your axiom uses things that are not defined. What is the *meaning* of > >>>>>>> "x<z"? > >>>>>> Geometrically it means that x is left of z on the number line. > >>>>> And for someone standing on the other side of the number line would x be > >>>>> on the right of z? > >>>>> > >>>>> And does the line stay horizontal as one moves around earth? Which way > >>>>> is larger if the line ever goes vertical. And how does the "larger" work > >>>>> at antipodes? > >>>>> > >>>> Silly questions. > >>> In response to a silly definition. > >>>>>> It means > >>>>>> A y (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y). That says about all it > >>>>>> needs to, wouldn't you say? > >>>>> Not hardly. > >>>>> A y (y = x) v (y = z) v (y<x ^ y<z) v (x<y ^ y<z) v (x<y ^ z<y) > >>>>> is a bit better but still insufficient. > >>>> True, I should have specified y<>x and y<>z. I guess it's usually done > >>>> using <= for this reason, eh? > >>>> > >>>>>>> > > That is not a definition, because it makes no sense. "The set of > >>>>>>> > > naturals > >>>>>>> > > is as large as every natural"? > >>>>>>> > > >>>>>>> > It is not larger than all naturals > >>>>>>> > >>>>>>> That is something completely different again. > >>>>>> It's not LARGER than every finite. > >>>>> Which natural(s) is it "not larger" than", in the sense of not being a > >>>>> proper superset of that natural or having that natural as a member? > >>>> ....11111 binary (all bit positions finite) > >>> Unless that string has only finitely many bit positions as well as only > >>> finite bit positions, it is not a natural at all, as it is then neither > >>> the first natural nor the successor of any natural, and every natural > >>> has to be one or the other. > >> It is the successor to ....11110. Duh. I've already proven that this is > >> a finite value, given that all bit positions are finite, and that > >> therefore no place in that string can achieve an infinite value, and > >> that any such number has predecessor and successor. The cute thing is > >> that the successor to ...1111 is 0, and that ...1111 is essentially -1. :) > > > > Does it not bother you that nobody else agrees with, or even > > understands, your proof? > > > > I find it disappointing, but not surprising, that you don't understand > such a simple proof, since it's contradictory to your education. I do > find it annoying that you feel the right to disagree with it without > understanding it. If you feel there is a problem with the proof, please > state the logical error I made. If the string is all finite bits, and > none of them ever can possibly achieve an infinite value, then how can > the string have an infinite value? There's nowhere in the string where > that can occur. It's that simple. Grok it. 1) A finite string of 1s represents a (finite) natural number. 2) An infinite string of 1s represents a (finite) natural number. 1) doesn't imply 2). -- mike. |