The Definition of Points
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From: Tony Orlow on 31 Mar 2007 19:04 Virgil wrote: > In article <460e56a5(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Virgil wrote: > >>> But all other mathematical objects are equally fantastic, having no >>> physical reality, but existing only in the imagination. So any statement >>> of mathematical existence is always relative to something like a system >>> of axioms. >> Sure, but the question is whether any such assumption of existence >> introduces nonsense into your system. > > It has in each of TO's suggested systems so far. > If thou so sayest, Sire. >> With the very basic assumption >> that subtracting a positive amount from anything makes it less > > That presumes at least a definition of "positive" and a definition of > "amount" and a definition of "subtraction" and a definition of "less" > before it makes any sense at all. Yes, it does. I think we all know basically what those terms mean. But, let's say we don't. We have a system where x<y and y<z implies x<z. We define 0, and say if x<0, then y+x<y, and if x>0 then y+x>y, and of course, if x=0, then y+x=y. y+x=z <-> z-y=x and z-x=y. x<y means y is in the "positive" direction from x. x is the distance from 0, positive if to the right of 0, negative if to the left, and we move that distance in the opposite direction from any point to subtract x from it, and determine the poit indicating the result. If we start at one point, and move a nonzero distance, do we not end up on another point, different from the first? Tony (I notice you never sign your posts, so, while I kind of like to mirror the signing styles of various posters in my responses, with you, I got nuthin' to work with. So, I'll just sign, Tony, and then maybe, you'll start signing, Evrett, or whatever your name "actually" is. :))
From: Tony Orlow on 31 Mar 2007 19:05 Lester Zick wrote: > On Fri, 30 Mar 2007 12:06:42 -0500, Tony Orlow <tony(a)lightlink.com> > wrote: > >> Lester Zick wrote: >>> On Thu, 29 Mar 2007 09:37:21 -0500, Tony Orlow <tony(a)lightlink.com> >>> wrote: >>> >>>>>> You might be surprised at how it relates to science. Where does mass >>>>>> come from, anyway? >>>>> Not from number rings and real number lines that's for sure. >>>>> >>>> Are you sure? >>> Yes. >>> >>>> What thoughts have you given to cyclical processes? >>> Plenty. Everything in physical nature represents cyclical processes. >>> So what? What difference does that make? We can describe cyclical >>> processes quite adequately without assuming there is a real number >>> line or number rings. In fact we can describe cyclical processes even >>> if there is no real number line and number ring. They're irrelevant. >>> >>> ~v~~ >> Oh. What causes them? > > Constant linear velocity in combination with transverse acceleration. > > ~v~~ Constant transverse acceleration? 01oo
From: Virgil on 31 Mar 2007 19:05 In article <460edc26(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Bob Kolker wrote: > > Tony Orlow wrote: > >> > >> As I said to Brian, it's provably the size of the set of finite > >> natural numbers greater than or equal to 1. No, there is no last > >> finite natural, and no, there is no "size" for N. Aleph_0 is a phantom. > > > > No. It is the cardinality of the set of integers. > > No, Bob, that's a Muslim lie, perpetrated by the Jews as a joke on the > xtians. And does TO pretend to have a mathematically valid proof of that claim?
From: Tony Orlow on 31 Mar 2007 19:12 Virgil wrote: > In article <460e571f(a)news2.lightlink.com>, > Tony Orlow <tony(a)lightlink.com> wrote: > >> Mike Kelly wrote: >>> On 30 Mar, 18:25, Tony Orlow <t...(a)lightlink.com> wrote: >> ~v~~ >>>> An actually infinite sequence is one where there exist two elements, one >>>> of which is an infinite number of elements beyond the other. >>>> >>>> 01oo >>> Under what definition of sequence? >>> >>> -- >>> mike. >>> >> A set where each element has a well defined unique successor within the >> set. Good enough? > > No! if we define the successor of x as x + 1, as we do for the ntaurals, > then the set of rationals and the set of reals, with their usual > arithmetic, both satisfy your definition of sequence. > > A sequence should at least be well ordered and have only one member, its > first, without a predecessor. I agree that what are considered normal sequences have first elements, but I don't see that the integers, or the adics, aren't a sequence in a broader sense, if we choose any arbitrary starting point. We can say a sequence has some single element without predecessor, or some element without successor, or both, so as to limit the line to a ray or segment. But the most general rule is that it may go one forever in both directions, as y -> Ex Ez : x<y<z, ala Archimedes. Yesno? :D Tony
From: Virgil on 31 Mar 2007 19:14
In article <460ee056(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > stephen(a)nomail.com wrote: > > In sci.math Tony Orlow <tony(a)lightlink.com> wrote: > > If all you mean by "actually infinite" is "uncountable", then > > just say "uncountable". Of course an "uncountable sequence" > > is a contradiction, so you still have to define what you mean > > by a "sequence". > > > > > > 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. > > > 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. |