The Definition of Points
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From: Tony Orlow on 19 Apr 2007 10:22 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 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. >> 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
From: Mike Kelly on 19 Apr 2007 11:04 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"? > >> 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'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"? > >> 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. > >>>> 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.
From: stephen on 19 Apr 2007 11:48 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
From: Lester Zick on 19 Apr 2007 12:00 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~~
From: Lester Zick on 19 Apr 2007 12:01
On Wed, 18 Apr 2007 15:25:30 -0600, Virgil <virgil(a)comcast.net> wrote: >In article <462669b3(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Lester Zick wrote: >> > On Tue, 17 Apr 2007 13:33:39 -0400, Tony Orlow <tony(a)lightlink.com> >> > wrote: >> > >> >>>> >> >>>> Is there a set >> >>>> of statements S such that forall seS s=true? >> >>> No idea, Tony. There looks to be a typo above so I'm not sure exactly >> >>> what you're asking. >> > >> >> I am asking, in English, whether there is a set of all true statements. >> > >> > No. There are predicates to which all true statements and all false >> > statements are subject respectively but no otherwise exhaustively >> > definable set of all true or false statements because the difference >> > between predicates and predicate combinations in true or false >> > statements is subject to indefinite subdivision. >> > >> > ~v~~ >> >> Uh, what? > >Just Zickism at its most opaque. As opposed to a Virgilism at its most transparent? ~v~~ |