The Definition of Points
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From: Lester Zick on 19 Apr 2007 12:02 On Wed, 18 Apr 2007 14:49:36 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Tue, 17 Apr 2007 13:39:45 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>>> Sure. Happens all the time. However if you're asking whether a >>>> statement must be one or the other the answer is no. There are >>>> problematic exceptions to the so called excluded middle. >> >>> Please eloborate. >> >> "Black is crows" is ambiguous in general terms and neither true nor >> false since "crows are black". Hence we find that "crows are black" is >> true but "black is not crows" is true too in general scientific terms. >> >> ~v~~ > >Okay, consider a universe where the ONLY black things are crows. In that >universe, would not "black is crows" be true? What else could black be, >besides some number of crows? > >I think you are assuming that all crows are black, and noting the proper >subset relation. C is a proper subset of B, because all members of C are >members of B, but not all members of B are members of C. So, indeed, >black(crow) might be equal to 1, meaning 100% of crows are black, while >crow(black) might only be equal to say 0.05, because only one out of >twenty black things are crows. It's a 5% probability that if something >is black, it has the property of crowness, while if it is a crow, it has >100% probability of being black. Make sense? No. ~v~~
From: Lester Zick on 19 Apr 2007 12:02 On Wed, 18 Apr 2007 14:52:06 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Tue, 17 Apr 2007 13:39:45 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>>> Well your phrase "exploring the meaning of truth" is ambiguous, Tony, >>>> because what you're really doing is exploring consequences of truth or >>>> falsity given assumptions of truth or falsity to begin with, which is >>>> an almost completely trivial exercise in comparison with the actual >>>> determination of truth in mechanically exhaustive terms initially. >>>> >>> I am exploring the mechanics of truth, and its pursuit, which you are >>> not, really, as far as I can tell. >> >> You're exploring the mechanics of using truth once established but I >> see no indication you're exploring the mechanics of truth otherwise. >> >> ~v~~ > >I am trying to get there, but you're struggling against even defining in >any mechanical terms what your statements mean, which is obviously >deliberate. The mechanics of deduction are pretty straightforward. The >mechanics of induction are a little harder to ascertain, because we >doing it unconsciously by nature, but can be developed by looking at the >deductive mechanics. That's nice. ~v~~
From: Lester Zick on 19 Apr 2007 12:02 On Wed, 18 Apr 2007 14:39:33 -0400, Tony Orlow <tony(a)lightlink.com> wrote: >Lester Zick wrote: >> On Tue, 17 Apr 2007 12:20:59 -0400, Tony Orlow <tony(a)lightlink.com> >> wrote: >> >>> Lester Zick wrote: >>>> On Fri, 13 Apr 2007 14:33:20 -0400, Tony Orlow <tony(a)lightlink.com> >>>> wrote: >>>> >>>>> Lester Zick wrote: >>>>>> On Thu, 12 Apr 2007 14:35:36 -0400, Tony Orlow <tony(a)lightlink.com> >>>>>> wrote: >>>>>> >>>>>>> Lester Zick wrote: >>>>>>>> On Sat, 31 Mar 2007 20:58:31 -0500, Tony Orlow <tony(a)lightlink.com> >>>>>>>> wrote: >>>>>>>> >>>>>>>>> How many arguments do true() and false() take? Zero? (sigh) >>>>>>>>> Well, there they are. Zero-place operators for your dining pleasure. >>>>>>>> Or negative place operators, or imaginary place operators, or maybe >>>>>>>> even infinite and infinitesimal operators. I'd say the field's pretty >>>>>>>> wide open when all you're doing is guessing and making assumptions of >>>>>>>> truth. Pretty much whatever you'd want I expect.Don't let me stop you. >>>>>>>> >>>>>>>> ~v~~ >>>>>>> Okay, so if there are no parameters to the function, you would like to >>>>>>> say there's an imaginary, or real, or natural, or whatever kind of >>>>>>> parameter, that doesn't matter? Oy! It doesn't matter. true() and >>>>>>> false() take no parameters at all, and return a logical truth value. >>>>>>> They are logical functions, like not(x), or or(x,y) and and(x,y). Not >>>>>>> like not(). That requires a logical parameter to the function. >>>>>> Tony, you might just as well be making all this up as you go along >>>>>> according to what seems reasonable to you. My point was that you have >>>>>> no demonstration any of these characteristics in terms of one another >>>>>> which proves or disproves any of these properties in mechanical terms >>>>>> starting right at the beginning with the ideas of true and false. >>>>>> >>>>>> ~v~~ >>>>> Sorry, Lester, but that's an outright lie. I clearly laid it out for >>>>> you, starting with only true and false, demonstrating how not(x) is the >>>>> only 1-place operator besides x, true and false, and how the 2-place >>>>> operators follow. For someone who claims to want mechanical ground-up >>>>> derivations of truth, you certainly seem unappreciative. >>>> Only because you're not doing a ground up mechanical derivation of >>>> true or false. You're just telling me how you employ the terms true >>>> and false in particular contexts whereas what I'm interested in is how >>>> true and false are defined in mechanically reduced exhaustive terms. >>>> What you clearly laid out are the uses of true and false with respect >>>> to one another once established. But you haven't done anything to >>>> establish true and false themselves in mechanically exhaustive terms. >>>> >>>> ~v~~ >>> Again, define "mechanics". >> >> Tony, time for you to do a little work for yourself. I've already gone >> through this. You describe for me the mechanics of using binary truth >> values and I explain to you I'm interested in truth not binary truth >> values and how to ascertain truth in mechanical terms initially and >> not how to work with truth values mechanically once ascertained. > >So, you are of the opinion that science can be performed without >collecting any data, doing experiments or studies, and ascertaining >truth from fact? Then you are in as much a religious limbo as the >Cantorian ball replicators. How far has this Ivory Tower approach gotten >you so far? If you get down the mechanics of deduction, then you can >consider more readily what underlying principles may be causing whatever >phenomena you're investigating. > >> >> By the way what is the truth value of "square triangles" and how does >> that differ from the truth value of "blue squares" and how do you know >> the difference? >> >> ~v~~ > >"The triangle is square" is false because the definitions of "triangle" >and "square" are mutually self-contradictory, because 3<4. > >"The square is blue" is not inherently self-contradictory, because there >is no contradiction between "triangle" and "blue". So, the truth value >depends on the particular triangle, and perhaps what one considers >"blue" as opposed to "green" or "purple". Yeah, Tony, look I really don't have any further interest in your philosophy of truth values. ~v~~
From: MoeBlee on 19 Apr 2007 12:49 On Apr 19, 7:06 am, Tony Orlow <t...(a)lightlink.com> wrote: > MoeBlee wrote: > > You just need to learn it theorem by theorem from the axioms. All it > > takes is an introductory set theory textbook, and all of them will > > prove the well ordering theorem for you. > But, a well order cannot have infinite descending chains of any sort, so > if there are an uncountable number of limit ordinals within the order, > then there will exist an infinite descending chain within that sequence > of limit ordinals itself. How does one get around that? Again, you're reasoning from vague ideas you have of what the terminology means. If you were to pin down these terms you use such as 'infinite descending chain' with the precise definitions, then tried to derive a formal contradiction, you wouldn't succeed to prove any such contradiction nor evidence of anything that we must "get around" here. Ironically, your response is right after I reiterated that you need to study this in a systematic and precise way, definition by definition, theorem by theorem. It is just more effort down the drain for me to untangle your confusions over terms of which you've not bothered to learn their precise definitions from primitives and axioms. A well ordering doesn't preclude that a member of the field of the ordering may have infinitely many predecessors. What a well ordering does demand is that in any nonempty set of predeccessors, there is a least one, which doesn't preclude that there may be infinitely many "on top" (as we're "working downwards") of that least one. You don't even need to go to the uncountable; all you need is just one limit ordinal in the field of the well ordering to see the point: Just take w +1. The well ordering on w+1 by the membership relation has w itself as member of the field of the ordering. But to "travel from" w "to" the least member of w requires "travelling through" an infinte number of members of the domain of the field of the ordering. I'm using picture language such as "travelling through" to get this across to you, even though I think it is reliance on picture language that is a large part of your problem in failing to come to grips with the precise mathematics. The alternative is for me to explicate such terms as you misuse, but, as I said, that is effort down the drain if you continue to refuse to do the minimal work yourself of learing the definitions from the primitives. One thing I should say is that I was too glib by saying that one might easily see the intuitive basis of well ordering from choice. While it is true that choice from well ordering is a piece of cake, but getting well ordering from choice is usually done with Zorn's lemma and/or transfinite recursion, and those are not simple matters. But to at least appease your curiosity about this (which you continue to refuse to satisfy properly by actually studying the material), when you think about limit ordinals, you're thinking about unions, and that's the main connection with Zorn's lemma and transfinite recursion too. The proofs in Stoll run about three pages and even then you have to fill in some of the sub-steps mentally or on paper for yourself. The proof in Enderton is about half a page, but it is based on a couple of chapters worth of development of transfinite recursion and theorems about ordinals. I can't properly compress all of that into a post. Especially when such proofs are so complex, there comes a point at which you just have to study a textbook and cannot expect to have it explained to you in casual postings. (Or perhaps take a look at the relevant parts of that paper by Kanamori I linked you too.) P.S. In another post I mentioned 'weak' versus 'strong' versions of well ordering. Actually, that should be 'weak' versus 'strict'. MoeBlee
From: MoeBlee on 19 Apr 2007 13:27
On Apr 19, 7:01 am, Tony Orlow <t...(a)lightlink.com> wrote: > Mike Kelly wrote: > >> If you say in the same breath, "there > >> are infinitely many rationals for each natural and there are as many > >> naturals overall as there are rationals", > > > And infinitely many naturals for each rational. > > How do you figure? In each 1-unit real interval, there is exactly one > natural, and an infinite number of rationals. Which interval has one > rational and an infinite number of naturals? That there are denumerably many rationals but only one natural in a unit interval defined by the standard ordering on rationals doesn't refute that there exists the kind of correspondence Mike Kelly mentioned. Several months ago I defined for you a dense linear ordering on the set of natural numbers. You ignored it. > > Or, maybe, other people don't share the same intuitions as you!? Do > > you really find this so hard to believe? > > Most people find transfinite cardinality "counterintuitive". Surely, you > don't dispute that. Probably most people who are not familiar with the theorem by theorem proofs of set theory would find uncountability unintuitive. But I know of no evidence that most people in general can't grasp the idea of an infinite set such as the set of natural numbers if the matter is presented to them in a clear way. Then, if one accepts that there are infinite sets, and one grasps the notion of a power set and some other basic concepts about sets, uncountability follows. > > Well, maybe you'd like to do that. But you have made no progress > > whatsoever in two years. Mainly, I think, because you have devoted > > rather too much time to very silly critiques of current stuff and > > rather too little to humbling yourself and actually learning > > something. > > That may be your assessment, but you really don't pay attention to my > points anyway, except to defend the status quo, so I don't take that too > seriously. You're ridiculous. Certain people have even OVER-indulged you by showing in detail, ad nauseam, exactly what's whack in your various proposals. And it doesn't even matter WHO is telling you - that you need to study this stuff from a good textbook is just plain, basic, good adive that you do need to take seriously. Or continue to be a nonsense spouting crank. Your choice. > > Get this through your head : every relation between objects in set > > theory is based on 'e'. It's really pathetic to keep mindlessly > > denying this. Set theory doesn't just "try to base everything on 'e'". > > It succeeds. > If you say so. No, not just "if he says so". He says so and he's RIGHT. And you can VERIFY for yourself just by reading a textbook already. MoeBlee |