Give me ONE digit position of any real that isn't computable up to that digit position!
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From: Marshall on 6 Jun 2010 09:13 On Jun 5, 10:54 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > Yet one will be hard-pressed to find an uncountable > object in the real world. You'll never find a three in the real world either, but that doesn't mean that three is therefore a somehow problematic concept. > Indeed, some physicists > wonder whether length, time, etc., can be quantized > (Planck units), which would imply that there's no > real-world example of an _infinite_ object, much less > an uncountable object. Sure; so what? The point of math isn't to exactly faithfully model the universe. There is exactly and precisely only one *complete* model of the universe; that is the universe itself. Any other model is necessarily leaving something out. (Probably quite a lot in fact.) > > Higher infinities and incompleteness and most of your uncomputable claims are just > > platonic drivel. > > I agree (except for the word "drivel"). And yet there is no bijection between the natural numbers and the real numbers. You might be able to make up a theory in which you can prove that there is, but you won't be able to demonstrate it, because it doesn't exist. (And from there a trivial application of the pigeonhole principle shows the existence of uncomputable reals.) Marshall
From: George Greene on 6 Jun 2010 12:36 On Jun 5, 9:07 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: > You could always state whether you agree with the proposition or not, and THEN > argue WHY it holds or it doesn't. DAMN, you're stupid. THAT is what YOU have to do! WE have already stated everything! For a CENTURY now! > > (NoBoxOfIndexes -> NoBijectionToPowerset -> HigherInfinity) > -> > (NoBoxOfIndexes -> HigherInfinity) > > You explicitly agree with what's inside the first set of parenthesis NO, I DON'T, dumbass. THE FIRST question is about what YOU believe, NOT what I believe! And it is whether you do or don't BELIEVE the proof of Cantor's Theorem! Do YOU believe there is a bijection??
From: George Greene on 6 Jun 2010 12:58 On Jun 5, 11:00 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > Of course, this doesn't necessarily mean that Herc has > actually disproved Turing's Halting Theorem in _ZFC_. He hasn't disproved it PERIOD. Theorems follow FROM AXIOMS. You have to start with some sort of axiomatic framework. The halting problem is just one of many diagonalization problems, from Russell's Paradox on down. This is ONE problem. In order to say proved (or disproved) this or that VARIANT of it, you have to START INSIDE the FRAMEWORK in which that variant is EXPRESSIBLE. In other words, if he hasn't disproved it in ZFC, he hasn't disproved it PERIOD. ZFC is sort of the default overarching framework around here.
From: FredJeffries on 6 Jun 2010 13:10 On Jun 5, 10:54 pm, Transfer Principle <lwal...(a)lausd.net> wrote: > > Yet one will be hard-pressed to find an uncountable > object in the real world. Indeed, some physicists > wonder whether length, time, etc., can be quantized > (Planck units), which would imply that there's no > real-world example of an _infinite_ object, much less > an uncountable object. You would be hard pressed to find a 7-dimensional object in your real world either. But the mathematics of more-than-three dimensional spaces is used all the time in economics, programming, meteorology, agriculture, war, advertising, ... and give accurate useful answers and predictions. Are all of these answers and predictions invalidated because the scientists and engineers used illegitimate mathematics? There's no such thing as a seventh dimension in the real world. So all the bridges and buildings start falling down? You're taking set theory too rigidly by insisting that the only possible real world application is to collections of tangible objects. If hyper-mahlo demi-measureable non-inaccessible sub-cardinalities are ever found in the "real world" it will be in some area hitherto not well modeled mathematically like psychology, ethics, linguistics, swearing, deodorants, ... > > It is precisely for this reason that I am open to > reading about alternate theories, as long as those > theories don't contradict empirical evidence. Thus, > even I won't defend a theory which seeks to prove > that 2+2 = 5, since one can prove in the real world > that 2+2 = 4, not 5. But there's a great yet-to-be-discovered cryptographic system based on a system where 2 + 2 = 5. Not to mention rabbit breeding or bowling (I get 6 pins with my first ball, 4 with my second, 8 with my third and roll a gutter ball with my fourth. What's my score?). > But since the real world can't > prove anything about infinity, Your real world doesn't prove anything about anything. It is human minds who sift through their sense impressions, highlighting some, discarding others, who decide what is to be proved, what constitutes a proof and whether a proposed proof really is a proof of what it alleges to prove. > I'm open to many > different theories about infinity, including NFU, > PST (Pocket Set Theory), intuitionism, as well as > finitist set theories like ZF-Infinity in which one > can't necessarily prove that infinite sets exist. I > am even somewhat open to _ultrafinitist_ theories > where there's an upper bound on the largest natural > number that exists, as long as that upper bound is > larger than any number that can describe objects in > the real, physical world. Theories don't describe objects in the real, physical world. People describe. And one of the tools people use in making descriptions is mathematics. Someday a brilliant scientist uses an ultrafinitist theory with an upper bound on the largest natural number that exists (lets call it W) to give a unified theory-of-everything which is able to be used to predict everything about the universe, solve all of our problems and establish world peace. Except flightless waterfowl. To explain penguins he needs to use W + 274 but the ultrafinitist theory with an upper bound of W + 274 doesn't work for the rest of the universe. So you would not be open to that theory? Not even somewhat?
From: |-|ercules on 6 Jun 2010 13:39
"Marshall" <marshall.spight(a)gmail.com> wrote > On Jun 5, 8:45 pm, "|-|ercules" <radgray...(a)yahoo.com> wrote: >> "Marshall" <marshall.spi...(a)gmail.com> wrote.. >> >> > On Jun 5, 8:12 pm, Transfer Principle <lwal...(a)lausd.net> wrote: >> >> >> But this all goes back to the question that I've been >> >> asking this past fortnight or so, ever since Herc >> >> started this recent posting spree. Is Herc really >> >> trying to introduce a new theory (or "theories"), or is >> >> he trying to prove that classical ZFC is "wrong"? >> >> > Neither one. >> >> > He's using the word "theory" to mean "theorem." >> > What he's trying to do is prove some established >> > theorems false. >> >> Right! >> >> 1/ I designed the simplest computer fetch cycle >> >> 2/ I proved Godel's proof places no bounds on knowledge >> >> 3/ I proved the possible existence of an effective halt algorithm >> >> 4/ I showed that higher infinities are thought to be implied by the non existence >> of a box that contains the numbers of the boxes that don't contain their own number >> >> No-one has agreed or disagreed with 1 - 3, a couple have disagreed with 4 without >> substantiating why. > > For what it's worth: > > 1) Is uninteresting to me; neither agree nor disagree. The simplest computer model is uninteresting? > 2) The way you phrased it in this post is sorta weird, but in > other posts you've said this one says that a computer and > a human are on equal footing with regards to Incompleteness; > I agree. Does "this statement has no proof" limit the boundaries of your comprehension? > 3) "disagree" insofar as this one is provably false. Do you think the following is flawed? --------------------------------------------------------- Once there was no polynomial time algorithm to determine if a number is prime. However Rabin found a probabilistic solution based on witnesses to compositeness. Over half of numbers from 1 to any composite number are witnesses so they are easy to find. If you run 100 iterations of the algorithm and no witness is found, the number is prime with probability 1 - (1/2)^100. P(x is prime) = 99.9999999999999999999999999999999999%. Such an attack could be used for a halting algorithm, add a parameter to the probabilistic Halt function which has 3 outcomes. pHalt(program, input, p) = {Halts | NotHalts | DontKnow} If half of values of p produce the correct result Halts or NotHalts, then performing numerous iterations would give an effective procedure to determine if a program halts or not. --------------------------------------------- > 4) Never was clear on what this was claiming. If it's claiming > that the real numbers are countable, then disagree. So you think the nonexistence of a box that contains the numbers of all the boxes that don't contain their own number implies higher infinities? > >> Aatu said they were all wrong then disappeared to work on >> his *informal proof* of the incontrovertible fact that all informal proofs can be formalized. > > Disagree. So you agree with Aatu his "demonstration of incontrovertible fact" was of a non-mathematical nature and not in the scope of it's own demonstrated assertion? Herc |