An uncountable countable set
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From: Virgil on 28 Oct 2006 16:57 In article <454394e9(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > imaginatorium(a)despammed.com wrote: > > Here's another question - I' ve asked before, so I don't hold out much > > hope of an answer, but anyway: Suppose you could ask Abraham Robinson > > what he thought of your ideas: what do you suppose he would say? Do you > > think he might just latch onto the IFR, N^L=S (whaddeveritwas), your > > T'rrible numbers, the twilight zone, etc., or do you suppose he would > > dismiss it as total nonsense? > > I suppose that depends on whether he liked me. I don't think hed find my > ideas objectionable. He was obviously an original thinker. In order to get his own thoughts in order, Robinson would have had to be an extremely critical thinker, who would have found TO's sloppy thinking and nonsensical derivations anathema. The only way he could have "liked" TO would have been if TO kept mute. > > Do you actually think you might ever find > > a real mathematician who thought there was anything at all of merit in > > what you have to say? > > Why don't you tell me? Did Boole? Did Cantor? They were fortunate enough never to have had to deal with anything TO wrote. > > Here's a suggestion: Robinson died in 1974, at a > > rather early age, but Conway is very much still alive - he's a very > > helpful person (I've seen him laboriously explaining something rather > > elementary on a geometry list I think it was), and he also created > > another non-standard collection of numbers, with what's more, > > constructions like omega/2 in it. > > Yes, the surreals. > > So send him an email, of not more > > than say 200 words, setting out your most basic ideas; ask him if he > > thinks you're wasting your time? Don't forget to mention that you are > > quite sure the set of natural numbers is not infinite. > > > > That's worth a try. Since Conway has a Puckish sense of humor, and does not always suffer fools gladly, be prepared to be held up to the ridicule you deserve, TO.
From: Virgil on 28 Oct 2006 17:11 In article <454396a8(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > David Marcus wrote: > > Stephen suggested that this problem would "not cause any fuss at all", > > i.e., everyone would agree what the answer is. In reply, you wrote, "It > > would still be inductively provable in my system that IN=OUT*10." We all > > took this to mean that you disagreed that |IN\OUT| = 0. Now, you seem to > > be saying that you agree that |IN\OUT| = 0. > > > > Care to clear up this confusion? > > > > No, I don't care, but I'll do it anyway. :) Just kidding. Of course I > care, or I wouldn't waste my time. > > I am beginning to realize just how much trouble the axiom of > extensionality is causing here. That is what you're using, here, no? The > sets are "equal" because they contain the same elements. That gives no > measure of how the sets compare at any given point in their production. TO seems to be under the mistaken impression that sets vary with time. That is on the same order of idiocy as presuming that the value of a number varies with time. Functions can vary as their arguments change, but constants do not vary, or they would not be called constants. Set functions of time (functions whose domain is time and whose codomain is some family of sets) can vary over time, but the value of such a function at any fixed time in its domain is a fixed set. > Sets as sets are considered static and complete. Quite correct. > However, when talking about processes of adding and removing > elements, the sets are not static, but changing with each event. Then one has a different set every time an element is added or removed. > When speaking about what is in the > set at time t, use a function for that sum on t, assume t is continuous, > and check the limit as t->0. One has then a function of time whose values are sets, but there is no possibility of continuity as there is no metric or topology on the codomain family of sets in question. So there is also no question of limits being definable. > Then you won't run into silly paradoxes and > unicorns. It is TO who keeps bringing in "unicorns". Like his infinitely numbered balls.
From: Virgil on 28 Oct 2006 17:24 In article <45439743(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > The purpose being to try to obscure details of the stated problem. That does seem to be TO's purpose. The stated problem was [ with one modification]: Given an initially empty vase. Given the infinite set of finite natural numbers, staring with 1, and a ball with each number marked on it. At times in minutes before noon: at t = 1/n balls numbered 10*(n -1) +1 to 10*n are inserted into the vase and then in that same instant ball n is removed. [At noon, a cube is placed in the vase.] What is the state of the vase at noon ?
From: Virgil on 28 Oct 2006 17:28 In article <454398cd(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > I think that the concept of a least infinite number in any real sense > violates the fact that one can always remove 1 from it and it will still > be infinite. But the issue is not "in any real" sense, it is "in any ordinal" sense, which issue TO avoids. >Robinson directly uses this idea. Robinson does not such thing, as he avoids ordinals entirely, at least in non-standard analysis.
From: Virgil on 28 Oct 2006 17:36
In article <4543a593(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > How do I account for it? I don't TO being unaccountable again. |