From - Tue Sep 15 21:12:02 1998 Date: Tue, 15 Sep 1998 05:12:28 -0400 (EDT) From: Mark Adkins cc: Mark Adkins Subject: A Mathematical Mystery MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Newsgroups: sci.math THE MYSTERY OF THE MISSING INTERVAL (A Mathematical Mystery Inspired by Brian Scott and Ilias Kastanas) It had been a long, hot summer. A succession of tawdry adultry investigations had left me tired and apathetic, but the money was good and I had bills to pay. As I sat feet upon my desk, fedora over my eyes, pondering the futility of modern romance, my secretary Marianne announced over the intercom: "There's someone here to see you." "Send 'em in" I replied without enthusiasm. Into my office slinked the sexiest dame I had seen for months. She had legs up to her armpits and a kisser to match. (Whatever that means.) "What can I do for you?" I asked with a polite hat removal and flourish. "I'd like you to find my cousin, Elmer Interval. He ran off to avoid some gambling debts, with two goons hot on his heels, and I haven't seen him since." "Any idea where he might have gone?" I asked. "Yes, he told me about a plan he had to lose the goons. He went down to the railroad, the Real Line, and hopped a local-service train. He raved about a sure scheme to get them off his trail." "The only sure things in life are death and taxes," I said sententiously, because I knew this was the sort of hardboiled private-eye crap the customers expected to hear. The leggy dame went right on without batting an eye. "He said he'd paid the conductor a bundle to zip back and forth along the line at ever greater speeds. He was going to spend the first hour in the left half of the (local) Line, then spend the next 1/2 hour in the right quarter of the Line, then the next 1/4 hour in the left eighth of the Line, then the next 1/8 hour in the right sixteenth of the Line, and so on. He said anybody trying to predict where he would end up after two hours would have to choose between the terminal stations No. 0 and No. 1, but that even though he would get arbitrarily close to each, he wouldn't end up at either. I thought to myself: these gamblers. Always a racket, always some surefire, complicated, and usually wacky scheme. What I actually said to the dame was, "I'm paid $200 a day plus expenses. Plus a $500 retainer. Take it or leave it." She took it. The first order of business was to visit the railyard and talk to the supervisor, a Mr. Brian M. Scott. I told him about Mr. I's scheme and asked him where he would have to be after two hours. I could track him from there. Scott checked his maps and timetables, and agreed that Mr. I would never stop at either terminal station. But he also said that the train would run out of coal after two hours, so I asked him where along the line Mr. I would be then. He checked his maps and timetables again and shrugged. "Maybe one of the N's killed him," he said, plucking a day old food crumb from his bushy, unkempt mustache and popping the morsel into his mouth. He explained that there was a very large family named N, which lived along the tracks next to the Real Line between terminal stations Nos. 0 and 1. It was run by a polygamist named Finite N, who had named all of his offspring (and their offspring) after him. There was Finite N the First, the patriarch, and then came his son Finite N (II), and so on, without end. I thought it was a bit soon to think about murder, but I agreed that the N's might make good witnesses and so I left the railyard and went down to the tracks to interview them. It was a big job, but somebody had to do it. Old Finite N (I) said that when he last saw Mr. Interval, he had been alive and well. His son, Finite N (II) said, "When I saw him, he was only half the man he had been, but he was still alive and well." I interviewed each member of the N family in sequence, all aleph_0 of them. They all said the same thing: Mr. Interval had been alive and well at every place along the Line he had been seen, though he seemed to be shrinking from sight, as if he wanted to avoid notice. I thanked them all and left. I wondered if perhaps there had been other witnesses, perhaps stepsons. As a stranger to the locale, I decided it would be best to check with the local Sheriff, Mr. Ilias Kastanas. Sheriff Kastanas said that he knew everyone living along the tracks. He told me that there are no other "steps" and that everything that happened to Mr. I along the Line would have been seen by one of the members of the Finite N clan. I left the Sheriff and stopped by Anthony's Bar & Grill for a drink and to consider the evidence so far collected. First, I knew from Scott the supervisor that the train ran out of fuel after two hours, but he also said that he couldn't determine any point along the Real Line where the train would come to rest at that time. Next, I knew from my interviews with the N clan, vouched for by the Sheriff, that Mr. Interval had been alive and well at every second during those two hours. I thought about interviewing the train conductor, but he had disappeared along with Mr. Interval. Could there be a *black hole* in the Real Line? Was I going crazy, or was somebody lying or mistaken? I decided to have another drink... From - Sat Sep 19 21:29:04 1998 Date: Sat, 19 Sep 1998 01:54:26 -0400 (EDT) From: Mark Adkins Reply-To: Mark Adkins Subject: Re: A Mathematical Mystery MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Newsgroups: sci.math Here is the solution to "The Mystery of the Missing Interval". In "The Mystery of the Missing Interval" the question was asked, what happened to Elmer Interval after aleph_0 steps. The answer is that he is still alive and well and has entered an interval of infinitesimal (e.g., surreal or hyperreal) length. Because the maps and timetables consulted by the railyard superviser do not acknowledge or contain intervals of infinitesimal length, because the sequence is divergent, and because there are only aleph_0 real intervals, Mr. Interval seemed to have mysteriously vanished from the train track called The Real Line. Note that the scenario was different from a Grandi series, which oscillates between 0 and 1 taking one of these values at every point in the series, whereas Mr. Interval oscillates ever closer to 0 and 1 but never reaches either point nor does he have a potential location in any real interval (or among any pair of them) after aleph_0 steps, since there are only aleph_0 real intervals. Specifically, he resides within one of two hyperreal intervals: 1/omega, or 1-1/omega. We can't say which, but at least this solves the disappearance of Mr. Interval and determines his location on the geometric line within the narrow confines of two specified intervals. Also see my entries in the thread titled "Dense Sets In R: A Logical Paradox?" in this newsgroup. For a rigorous formulation of infinitesimals as hyperreal numbers, see Abraham Robinson, _Non-Standard Analysis_ (Amsterdam: North Holland, 1974).