The Definition of Points
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From: Michael Press on 20 Apr 2007 05:22 In article <23659382.1174324080949.JavaMail.jakarta(a)nitrogen.mathforum.org>, "G.E. Ivey" <george.ivey(a)gallaudet.edu> wrote: > I believe it was Hilbert who said that "If you replace > points and lines by beer steins and tables, every statement should still be > true." Ever pair of tables has a beer stein in common? No wonder he had so many problems. -- Michael Press
From: Tony Orlow on 20 Apr 2007 11:41 Mike Kelly wrote: > On 19 Apr, 15:01, Tony Orlow <t...(a)lightlink.com> wrote: >> Mike Kelly wrote: >>> On Apr 18, 6:55 pm, Tony Orlow <t...(a)lightlink.com> wrote: >>>> Virgil wrote: >>>>> In article <4625a...(a)news2.lightlink.com>, >>>>> Tony Orlow <t...(a)lightlink.com> wrote: >>>>>> Mike Kelly wrote: >>>>>> < snippery > >>>>>>> You've lost me again. A bad analogy is like a diagonal frog. >>>>>> Transfinite cardinality makes very nice equivalence classes based almost >>>>>> solely on 'e', but in my opinion doesn't produce believable results. >>>>> What's not to believe? >>>>> Cardinality defines an equivalence relation based on whether two sets >>>>> can be bijected, and a partial order based on injection of one set into >>>>> another. >>>>> Both the equivalence relation and the partial order behave as >>>>> equivalence relatins and partial orders are expected to behave in >>>>> mathematics, so what's not to believe? >>>> What I have trouble with is applying the results to infinite sets and >>>> considering it a workable definition of "size". Mike's right. If you >>>> don't insist it's the "size" of the set, you are free to do with >>>> transfinite cardinalities whatever your heart desires. What I object to >>>> are statements like, "there are AS MANY reals in [0,1] as in [0,2]", >>>> and, "the naturals are EQUINUMEROUS with the even naturals." If you say >>>> they are both members of an equivalence class defined by bijection, then >>>> I have absolutely no objection. >>> Then you have absolutely no objection. Good that you recognise it. >>>> 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? > > No interval in the ordering on the real line. But there are orderings > where there are infinitely many naturals between each pair of > rationals. You seem hung up on the idea of orderings on reals/ > rationals/naturals that aren't the standard one. Maybe because you > think about everything "geometrically"? > I think about quantity in terms of measure, yes. Where we have one atom of space we have a point, without form. Where we have two, we have a measurable difference, a line. Half of mathematics arose from geometrical concerns, largely for dividing land and building buildings. The other half arose from financial concerns, keeping books regarding accounts. The one starts from the notion of points and lines, "real" objects, and the other from the notion of a tally or count, in units. That modern mathematics considers the second to supersede the first is, I think, a mistake. Any theory that integrates mathematics should satisfy both notions, no? >>>> without feeling a twinge of >>>> inconsistency there, then that can only be the result of education which >>>> has overridden natural intuition. That's my feeling. >>> 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. > > I imagine most people have never heard of it. From what I can remember > of freshman analysis some people had problems with it, some people > didn't. I certainly don't think there was a clear majority who did. > And I don't remember anyone taking 2 years to understand that all > cardinality is really about is bijections. > I never imagined it would take anyone two years to still not understand, as I've repeatedly stated, that bijections alone are not sufficient for proper relative measure of infinite sets. When I say that, doesn't that sort of imply that I understand that cardinality is based on bijection alone? The mapping functions describe the relative size, as long as both sets follow quantitative order. Otherwise it's slightly more complicated. >>>> I'd rather acknowledge that omega is a phantom quantity, >>> By which you mean "does not behave like those finite numbers I am used >>> to dealing with". >> Or, does not fit into what I understand quantities to be. > > OK, whatever. You don't think Omega is a "quantity". You haven't > defined "quantity" in any unambiguous way but I'm perfectly happy to > believe you don't think the set of all finite ordinals is a > "quantity". So what? I don't think omega sounds much like a > "quantity", either. So? This doesn't change anything about my > understanding of set theory. It doesn't make me say silly things like > "aleph_0/omega is a phantom". Who cares if they're called "numbers"? > A raw quantity is a point on the real line. Points to the right represent greater quantities than points to the left. When we add a negative quantity, we indicate a point to the left of our starting point, which is a different point on the real line, representing a lesser quantity. If omega, or aleph_0, lies anywhere on this infinite line containing all quantities, then aleph_0-1<aleph_0. If a set size is defined as "an integral quantity" of elements, then all set sizes lie on this line, and obey that principle. x-1<x, but aleph_0-1=aleph_0. So, whatever kind of "number" one may consider omega, or aleph_0, they appear to have no relationship to actual set sizes, naturals, or reals. >>>> and preserve basic notions like x>0 <-> x+y>y, and extend measure to the infinite scale. >>> 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. > > Great. Don't let me discourage you from your noble quest. I'll > continue to glance at your posts when I see them and see if there's > ever going to be anything of yours that actually interests me. And, > yes, when you make what I consider amusingly egregious errors about > set theory or other areas of mathematics I will "defend the status > quo" by pointing out that you're being silly. > As you wish. >>>>>> I'm working on a better theory, bit by bit. I think trying to base >>>>>> everything on 'e' is a mistake, since no infinite set can be defined >>>>>> without some form of '<'. I think the two need to be introduced together. >>>>> Since any set theory definition of '<' is ultimately defined in terms of >>>>> 'e', why multiply root causes? >>>> It's based on the subset relation, which is a form of '<'. >>> 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. > > I do say so. Does this mean you're going to stop claiming that > relations in set theory aren't based solely on 'e'? > > -- > mike. > No, not when sequences are defined using a recursively defined successor function, which is a relation between two elements, as opposed 'e', a relation between an element and a set. The combination of the two is what produces an infinite set, no? tony.
From: Tony Orlow on 20 Apr 2007 11:58 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: >>>> stephen(a)nomail.com wrote: >>>>> In sci.math Tony Orlow <tony(a)lightlink.com> wrote: >>>>>> Mike Kelly wrote: >>>>>>> The point that it DOESN'T MATTER whther you take cardinality to mean >>>>>>> "size". It's ludicrous to respond to that point with "but I don't take >>>>>>> cardinality to mean 'size'"! >>>>>>> >>>>>>> -- >>>>>>> mike. >>>>>>> >>>>>> You may laugh as you like, but numbers represent measure, and measure is >>>>>> built on "size" or "count". >>>>> What "measure", "size" or "count" does the imaginary number i represent? Is i a number? >>>>> The word "number" is used to describe things that do not represent any sort of "size". >>>>> >>>>> Stephen >>>> Start with zero: E 0 >>>> Define the naturals: Ex -> Ex+1 >>>> Define the integers: Ex -> Ex-1 >>>> Define imaginary integers: Ex -> sqrt(x) >>>> i=sqrt(0-(0+1)), so it's built from 0 and 1, using three operators. It's >>>> compounded from the naturals. >>> That does not answer the question of what "measure", "size" or "count" i represents. >>> And it is wrong on other levels as well. You just pulled "sqrt" out of the >>> air. You did not define it. Claiming that it is a primitive operator seems >>> a bit like cheating. And if I understand your odd notation, the sqrt(2) >>> is an imaginary integer according to you? And sqrt(4) is also an imaginary integer? > >> No, but sqrt on the negatives produces imaginary numbers. Besides, sqrt >> can be defined, like + or -, geometrically, through construction. > > You cannot define the sqrt(-1) geometrically. You are never going to draw a line > with a length of i. > >>> You also have to be careful about about claiming that i=sqrt(-1). It is much safer >>> to say that i*i=-1. If you do not see the difference, maybe you should explore the >>> implications of i=sqrt(-1). >>> >>> So what is wrong with >>> Start with zero: E 0 >>> Define the naturals: Ex -> Ex+1 >>> Define omega: Ax >>> I did that using only one operator. >>> >>> > >> Ax? You mean, Ax x<w? That's fine, but it doesn't mean that w-1<w is >> incorrect. > > No, I meant Ax. All of the natural numbers. I know you are incapable of > actually imagining "all", but others do not have that limitation. > > Of course, who knows what your notation is really supposed to mean. > What is (Ax)-1 supposed to be? How do you subtract one from all the naturals? > >>>> A nice picture of i is the length of the leg of a triangle with a >>>> hypotenuse of 1 and a leg of sqrt(2), if that makes any sense. It's kind >>>> of like the difference between a duck. :) >>> That does not make any sense. There is no point in giving a nonsensical >>> answer, unless you are aiming to emulate Lester. >>> >>> Stephen >>> >>> > >> It's not nonsensical, and may even apply to uses of imaginary numbers in >> practice, but you can ignore it as I knew you would. That's okay. > >> Tony > > It is nonsense. Such a triangle does not exist. > > A > # > # > # > # > # > C############B > > Are you claiming this is a triangle? Are you claiming the distance from A to C > is i? How exactly is this supposed to be a picture of i? What is i measuring > in this picture? > > If this is not what you meant, please draw your picture of i. > > And you still have not answered what "size", "count" or "measure" i represents. > Is i a number, or not? > > Stephen i is a number produced by extending completeness to the multiplicative field, after introducing completeness to the additive field to produce -1, like I said above. Consider that you have a right triangle so A^2+B^2=C^2. Now, if you swap the values between the hypotenuse and one of the legs, that formula will produce a second leg of the original length times i. So, in a way it is like picturing an impossible triangle. You have to "imagine" the other leg of length i. :) A |\ | \ | \ 1 sqrt(2) | \ | \ | \ B i C Tony
From: Tony Orlow on 20 Apr 2007 12:00 Lester Zick wrote: > On Wed, 18 Apr 2007 14:57:50 -0400, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Tue, 17 Apr 2007 13:29:44 -0400, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>>> Let me ask you something, Tony. When you send off for some truth value >>>>> according to "true(x)" and it returns a 1 or 0 or whatever, how is the >>>>> determination of that "truth value" made? >>>> From the truth values of the posited assumptions, of course, just like >>>> yours. >>> So you just posit truth values and wing it whereas I'm more inclined >>> to demonstrate the truth of what I posit instead? >>> >>> ~v~~ >> What truth have you demonstrated without positing first? > > And what truth have you demonstrated at all? > > ~v~~ Assuming two truth values as 0-place operators, I demonstrated that not(x) is the only functional 1-place operator, and then developed the 2-place operators mechanically from there. That was truth about truth. 01oo
From: Tony Orlow on 20 Apr 2007 12:01
Lester Zick wrote: > 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~~ That's kinda sad, Lester. You couldn't follow any of that? Have a cup of coffee or something. 01oo |