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> We produce a succession of nested intervals. For a given algebraic, eventually every interval will not include it. That means that no algebraic is in more tha

> We produce a succession of nested intervals. For a given algebraic, eventually every interval will not include it. That means that no algebraic is in more than finitely many intervals Going back over the post -- this number only "exists" because of the least upper bound axiom. You're right that I probably didn't read it carefully enough but there's no particular reason why the intervals need share any elements (e.g. [0,0.5) [0.5,1]) or contain their boundary points -- we got division and < from the rationals without violating anyone's intuition. Personally, I've always found these arguments unsatisfying. The proof is of the "existence" of something but in the end we have no idea what it looks like -- is it >0.5? No clue. And the proof relies on an existence axiom that seems contrived. (I guess at heart I'm a constructivist.) The desire for "completeness" flows from a false intuition of the nature of the world -- the surface of this table is a "continuum" right? But it turns out not to be. In contrast, I really liked the discussion of the Reals linked yesterday which sidetracked into a robust description of the Reals with infinitesimals (and infinites) and no, I repeat no, least upper bound axiom.

gjm11 13y ago
> in the end we have no idea what it looks like -- is it >0.5? No clue. No, that isn't true. The proof Colin gives is quite explicit. There's an explicit listing of the algebraic numbers (except that he leaves the order of various things unstated; for present purposes I'll just work with the order of appearance in his writeup). Then there's a procedure that considers these one by one and chooses intervals of exponentially decreasing size to concentrate on. You can get as many decimal places as you like of the transcendental number Colin's constructing, just by doing enough steps. Let's be more explicit. The first algebraic numbers in Colin's list are 0, 0, -1, 1, 0. (I assume we aren't counting repeated roots of a single polynomial, but also aren't eliminating duplicates between different polynomials in the list.) So, first we take the intervals [0,1/3], [1/3,2/3], [2/3,1] and the algebraic number 0. We take the leftmost interval not containing 0, namely [1/3,2/3]. Now we subdivide into three again: [1/3,4/9], [4/9,5/9], [5/9,2/3]. And again our algebraic number is 0. The leftmost interval not containing 0 is [1/3,4/9]. (So we've already established that the number is < 0.5.) In the next three steps we never see an algebraic number inside our interval so we'll just take the leftmost third each time; so we get [1/3,1/3+1/243]. We've now found the first two decimal places of the transcendental number we're constructing, namely 0.33. It would (at least in principle) be easy to computerize the process and find many more decimal places.
ColinWright 13y ago
> this number only "exists" because of the > least upper bound axiom. The whole point of this article is to show that any countable list (because although it does do it specifically in the context of the algabraics, it works for any list) does not contain all the reals. In doing so we have to have some definition of the reals, and this article is, in an unspoken manner, using the definition to be "The Completion of the Rationals." In this context, "completion" means "closure under the taking of Cauchy Sequences", which in turn is equivalent (in the context of the rationals) to be the existence of the least upper bound for every bounded, non-decreasing sequence. In this sense, the existence of the least upper bound isn't an axiom, it's a definition of the closure. > ... there's no particular reason why the intervals need > share any elements (e.g. [0,0.5) [0.5,1]) or contain > their boundary points ... We avoid all concerns about whether the intervals are open, closed, contain their end-points, etc., by using three sub-intervals. It side-steps all the necessary detail of who has the end-points. Other choices are equally valid and can be made to work. I don't see why any of them are easier or better than the one made here. You may not share my opinion on that. > ... we got division and < from the rationals without > violating anyone's intuition. ? > Personally, I've always found these arguments unsatisfying. > The proof is of the "existence" of something but in the end > we have no idea what it looks like -- is it >0.5? No clue. We have. I gave an explicit method of enumerating the algebraics, and we can use that to construct this non-algebraic explicitly. Not that that's the point - the real point is that nearly every (real) number is non-algebraic. Indeed, nearly every real number is non-constructable, non-computable, and, in a very real sense, non-visualisable. > And the proof relies on an existence axiom that seems > contrived. (I guess at heart I'm a constructivist.) The > desire for "completeness" flows from a false intuition > of the nature of the world -- the surface of this table > is a "continuum" right? But it turns out not to be. My experience is that most people think of the line as being "complete," and in that sense this seems to match the intuition of most people. However, maybe this doesn't match your intuition, or the way your think. That's fair enough. More, though, if you have the rationals, you don't have limits of Cauchy Sequences. That alone would be enough to talk about the "missing numbers," and that doesn't really use the line at all. > In contrast, I really liked the discussion of the Reals > linked yesterday which sidetracked into a robust description > of the Reals with infinitesimals (and infinites) and no, > I repeat no, least upper bound axiom. It's easy enough to construct the reals as an abstract thing, without reference to the line. Most people have trouble with such abstract concepts and derivations. I suspect this article just isn't for you. You might prefer the one I'm writing now which starts with naive finite sets and constructs, completely in abstract, Natural Numbers, Integers, Rationals, Reals, Complex Numbers, and Quaternions. Your comment has made me think about the deliberate and explicit separation of the construction from the visualisation of the reals as a model of the line. I'll think about that.

gjm11 13y ago

> in the end we have no idea what it looks like -- is it >0.5? No clue. No, that isn't true. The proof Colin gives is quite explicit. There's an explicit listing of the algebraic numbers (except that he leaves the order of various things unstated; for present purposes I'll just work with the order of appearance in his writeup). Then there's a procedure that considers these one by one and chooses intervals of exponentially decreasing size to concentrate on. You can get as many decimal places as you like of the transcendental number Colin's constructing, just by doing enough steps. Let's be more explicit. The first algebraic numbers in Colin's list are 0, 0, -1, 1, 0. (I assume we aren't counting repeated roots of a single polynomial, but also aren't eliminating duplicates between different polynomials in the list.) So, first we take the intervals [0,1/3], [1/3,2/3], [2/3,1] and the algebraic number 0. We take the leftmost interval not containing 0, namely [1/3,2/3]. Now we subdivide into three again: [1/3,4/9], [4/9,5/9], [5/9,2/3]. And again our algebraic number is 0. The leftmost interval not containing 0 is [1/3,4/9]. (So we've already established that the number is < 0.5.) In the next three steps we never see an algebraic number inside our interval so we'll just take the leftmost third each time; so we get [1/3,1/3+1/243]. We've now found the first two decimal places of the transcendental number we're constructing, namely 0.33. It would (at least in principle) be easy to computerize the process and find many more decimal places.

ColinWright 13y ago

> this number only "exists" because of the > least upper bound axiom. The whole point of this article is to show that any countable list (because although it does do it specifically in the context of the algabraics, it works for any list) does not contain all the reals. In doing so we have to have some definition of the reals, and this article is, in an unspoken manner, using the definition to be "The Completion of the Rationals." In this context, "completion" means "closure under the taking of Cauchy Sequences", which in turn is equivalent (in the context of the rationals) to be the existence of the least upper bound for every bounded, non-decreasing sequence. In this sense, the existence of the least upper bound isn't an axiom, it's a definition of the closure. > ... there's no particular reason why the intervals need > share any elements (e.g. [0,0.5) [0.5,1]) or contain > their boundary points ... We avoid all concerns about whether the intervals are open, closed, contain their end-points, etc., by using three sub-intervals. It side-steps all the necessary detail of who has the end-points. Other choices are equally valid and can be made to work. I don't see why any of them are easier or better than the one made here. You may not share my opinion on that. > ... we got division and < from the rationals without > violating anyone's intuition. ? > Personally, I've always found these arguments unsatisfying. > The proof is of the "existence" of something but in the end > we have no idea what it looks like -- is it >0.5? No clue. We have. I gave an explicit method of enumerating the algebraics, and we can use that to construct this non-algebraic explicitly. Not that that's the point - the real point is that nearly every (real) number is non-algebraic. Indeed, nearly every real number is non-constructable, non-computable, and, in a very real sense, non-visualisable. > And the proof relies on an existence axiom that seems > contrived. (I guess at heart I'm a constructivist.) The > desire for "completeness" flows from a false intuition > of the nature of the world -- the surface of this table > is a "continuum" right? But it turns out not to be. My experience is that most people think of the line as being "complete," and in that sense this seems to match the intuition of most people. However, maybe this doesn't match your intuition, or the way your think. That's fair enough. More, though, if you have the rationals, you don't have limits of Cauchy Sequences. That alone would be enough to talk about the "missing numbers," and that doesn't really use the line at all. > In contrast, I really liked the discussion of the Reals > linked yesterday which sidetracked into a robust description > of the Reals with infinitesimals (and infinites) and no, > I repeat no, least upper bound axiom. It's easy enough to construct the reals as an abstract thing, without reference to the line. Most people have trouble with such abstract concepts and derivations. I suspect this article just isn't for you. You might prefer the one I'm writing now which starts with naive finite sets and constructs, completely in abstract, Natural Numbers, Integers, Rationals, Reals, Complex Numbers, and Quaternions. Your comment has made me think about the deliberate and explicit separation of the construction from the visualisation of the reals as a model of the line. I'll think about that.