The Definition of Points
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From: Virgil on 1 Apr 2007 00:57 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?
From: Virgil on 1 Apr 2007 00:59 In article <460f246a(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <460f1a41(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > >> > >> You have seen two apples, and three? > >> > > Are apples numbers? > > Naturals apply to objects. That is not an answer to what I asked. And is irrelevant to that question.
From: Virgil on 1 Apr 2007 01:02 In article <460f24eb(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Virgil wrote: > > In article <460f1b3e(a)news2.lightlink.com>, > > Tony Orlow <tony(a)lightlink.com> wrote: > > > >> Virgil wrote: > >>> In article <460ef839(a)news2.lightlink.com>, > >>> Tony Orlow <tony(a)lightlink.com> wrote: > >>> > >>>> Virgil wrote: > >>>>> In article <460ee056(a)news2.lightlink.com>, > >>>>> Tony Orlow <tony(a)lightlink.com> wrote: > >>>>>> Please do expliculate what the contradiction is in an uncountable > >>>>>> sequence. What is true and false as a result of that concept? > >>>>> A mathematical sequence is a function with the naturals as domain. > >>>>> If TO wishes to refer to something which is not such a function, he > >>>>> should not refer to it as a sequence if he wishes to be understood in > >>>>> sci.math. > >>>>> > >>>>> > >>>> Pray tell, what term shall I use???? > >>> TO is so inventive in so many useless ways that I cannot believe that > >>> his imagination will fail him in such a trivially useful way. > >>>>>>> I know you are incapable of actually thinking about all the elements > >>>>>>> of > >>>>>>> N, > >>>>>>> but that is your problem. In any case, N is not an element of N. > >>>>>>> Citing Ross as support is practically an admission that you are > >>>>>>> wrong. > >>>>>>> > >>>>>>> Stephen > >>>>>>> > >>>>>> Sure, of course, agreeing with someone who disagrees with you makes me > >>>>>> wrong. I'll keep that in mind. Thanks.. > >>>>> > >>>>> It is not so much that Ross disagrees with one person, it is that he > >>>>> disagrees with everyone, frequently including himself. > >>>> Ross has a vision, even if not axiomatically expressed. In fact, he's > >>>> entirely honest about that, expounding an axiom free system. I like > >>>> Ross. So do you. Admit it. :) > >>>> > >>> > >>> Like Russell? > >>> > >>> What is there about him to like? > >> You don't like Russell? > > > > I don't know him well enough to like or dislike. I dislike his > > anti-mathematical idiocies. > > Define "mathematics" before you accuse anyone of being > "anti-mathematical". I doubt I agree with everything Russell said, but, > whatever. Never mind. Be as crotchety as you like. One definition might be the study of axiom systems and their consequences. Since Russell rejects all axiom systems, at least those of interest to mathematicians, he is anti-mathematical.
From: Virgil on 1 Apr 2007 01:11 In article <460f294b(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > cbrown(a)cbrownsystems.com wrote: > >>> What does it mean for an ordering to be "discrete" or "linear"? What > >>> does it mean for something to "occur in" an ordering? > >> Linear means x<y ^ y<z ->x<z > > > > Funny; everyone else calls that "a transitive relation". > > Yes, is that unrelated? Since equality is a "linear relation" according to TO's definition, , TO must be requiring that equality be an ordering. > > > > > Let S = {a,b,c,f,g, h} > > > > Impose the following ordering: > > b < a > > c < b > > c < a > > f < a > > g < f > > g < a > > if x <> h, then h < x. > > > > The equivalent Hasse diagram: > > > > a > > / \ > > b f > > / \ > > c g > > \ / > > \ / > > \ / > > h > > > > This ordering satisfies, for all x,y,z in S: if x<y and y<z, then x<z. > > > > This is not what most people mean when they say "a linear ordering". > > Instead, it's an example of what people usually call a partial order. > > Okay, but that's an ordering that is based on some finite set of rules > regaring some finite set of points, which doesn't suffice to specify the > relationships between every pair of points. We can't say, from the > specified relationships, whether f<b or f<c or b<g or c<g. That's why > there's a parallel route, and so the diagram is "nonlinear". It could be > all on a line, but there would be several possible ordering given the > stated relationships. Nothing in TO's definition requires any more than "if x<y and y<z, then x<z", so that anything satisfying TO's definition must be, by his own standards, a linear ordering. On the other hand, TO could retract his insufficient definition and replace it with a valid one. > >>> So when you say "sequence" you're using an undefined term. As such, > >>> it's rather hard to your evaluate claims such as "There are actually > >>> infinite sequences". I have literally no idea what you are even trying > >>> to say. > >>> -- > >>> mike. > >> Oh gee, there has to be some word for it... > >> > > > > There almost certainly is; but as usual, it depends on what the /heck/ > > you're talking about. Perhaps the words "well-order" or "total order" > > actually already satisfy your requirements; or some particular proper > > subset of all non-isomorphic total orders satisfy your requirements. > > Or not. But how will you ever know if you refuse to /learn/ what these > > words refer to? > > > > Cheers - Chas > > > > I guess not by asking around here. Geeze. Those things have all been properly definied here so that TO's failure to learn them here is strictly his own fault.
From: Virgil on 1 Apr 2007 01:18
In article <460f2ac1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Let's examine what '<' means. x<y ^ y<z -> x<z. True for real > quantities, and true if '<' is taken to mean "is a proper subset of". > The proper subset is less than the whole, and the evens are half the > naturals. That's a very primitive result. This does not explain what "<" means. Perhaps TO needs to do a little work: http://en.wikipedia.org/wiki/Order_relation#Special_types_of_orders > > > > What I don't understand is what name you would like to give to the set > > {n : n e N and n <> N}. M? > > > > Cheers - Chas > > > > N-1? Why do I need to define that uselessness? Because if you can't do that, it is you that are useless. |