Galileo's Paradox
in [Math]
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From: Virgil on 14 Dec 2006 15:08 In article <458191a1(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > Eckard Blumschein wrote: > > On 12/11/2006 7:10 PM, Tony Orlow wrote: > >> Eckard Blumschein wrote: > > > >>> More is just inappropriate as to describe something infinite. There are > >>> not more naturals than rationals. There are not equaly many of them, > >>> there are not less naturals than rationals. > >>> > >>> > >> So, you wouldn't agree that every natural is a rational, > > > > Every natural is a rational, yes. > > > >> and that there are "more" rationals besides those? > > > > No. In this even the dullest Cantorians agree with me. > > They say: There is a bijection between both. > > I add: Galilei: incomparable > > > > > > Bijections aside, if you have the set of naturals, and you wish to turn > it into the set of rationals, you do not remove any elements, but > between every two elements you had originally, you have to add > infinitely many "more", to get the desired set. The bijection approach > alone is hundreds of years old. Galileo was brilliant, but he was not > the "last prophet", any more than Georg Cantor. There is no such animal. > It's time to work measure into count, and distinguish what are obviously > different size sets. If TO is suggesting that he is some sort of heir to Galileo Galilei and Georg Cantor. Not bloody likely!
From: imaginatorium on 14 Dec 2006 15:14 MoeBlee wrote: > Tony Orlow wrote: > > MoeBlee wrote: > > > Tony Orlow wrote: > > >> You cannot have a smallest infinity, and also not have it. <snip> > > In pressing > > this point, I see set theorists backing away from the notion that omega > > is even a "number" It's an "ordinal" with no relation to "real" numbers. > > Call it a 'number' or a 'skalumber' or whatever you want. There is a > least infinite ordinal. And it exists in Robinson's work with > non-standard naturals and non-standard reals. Of course omega is not a > member of the same FIELD as the non-standard reals, but omega exists as > an object IN THE SAME THEORY with the non-standard reals. > > > I'm satisfied with that admission. Omega is really not a number. > > It's not a matter whether it's called a 'number' or not. As far as I > know, set theory doesn't have a defined predicate 'is a number'. And if > we ever want to define such a predicate, then we can decide how to > define as that will determine whether ordinals satisfy that definition > or not. > > Anyway, your pressing some informal question as to whether ordinals are > numbers or not is a red herring as far as what we were talking about: It may be a red herring to you, but I think it's central to Tony's thinking - speaking from considerable experience of reading his brain dumps. You see, to you or me, {0} is a set, and {0, 1, 2} is a set, and {0, 1, 2, 3, 4, ... } is a set, and we can easily talk about these things, since we know there is a strict formal basis to doing so. But to Tony there's just the "number line". We might think of a "number line" as modelling the integers - extending without limit in two directions. But in Orlovia, _all_ numbers are somehow on this line. Our line merely extends without limit, but Tony's line goes further than that, into zones including "infinite naturals" and all sorts of strange beasts. Obviously, therefore, since if "omega" were a number, it'd be somewhere on this line, then to the left of it would be another number, which would have to be different from omega, couldn't be a "finite number" (P), since then P+1 would equal omega, and even Tony can see this is wrong, so it must be a different "infinite number", and we might as well call it omega-1. Thus the notion of a "least infinite" (point on this "number line") is contradictory. I think that within the notion of this "line that extends further than indefinitely", Tony's reasoning is, um, well, persuasive. The trouble is, of course, that you can't actually discuss things like a correlate of the set {0, 1, 2, ...} using mental pictures of lines, but Tony can't handle set theoretic notation, so you're stuck. It is slightly amusing to wonder quite how he is reading Robinson. I wonder, if he read Conway's ONAG, would he be able to see any difference? Well, good luck with this. Don't wear out the CAPS key, eh? Brian Chandler http://imaginatorium.org
From: Virgil on 14 Dec 2006 15:14 In article <458193cc(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> You cannot have a smallest infinity, and also not have it. > > > > You are actually TRYING NOT to listen. > > > > 'Smallest infinity' means two DIFFERENT things in two DIFFERENT > > contexts. I've explained too many times already now but you just keep > > going right past the point to make your same error over and over and > > over. > > > > MoeBlee > > > > The infinite numbers in set theory have no relation to the infinite > numbers in NSA. They exist within two different theories. In pressing > this point, I see set theorists backing away from the notion that omega > is even a "number" It's an "ordinal" with no relation to "real" numbers. > I'm satisfied with that admission. Omega is really not a number. "Number" is used in a variety of ways in mathematics, so many that unless one has a context limiting the meaning, it can mean almost anything its user wants it to mean. So that in some contexts every ordinal, including omega, IS a number, So when you say that omega is NaN, you must specify what are and what are not numbers in the context in which you make that statement or it is meaningless.
From: Virgil on 14 Dec 2006 15:31 In article <458196bf(a)news2.lightlink.com>, Tony Orlow <tony(a)lightlink.com> wrote: > cbrown(a)cbrownsystems.com wrote: > > Tony Orlow wrote: > > > >> This still makes no sense. x and y are reals, or at least reside along > >> the same metric from comparison. It's like asking whether a baseball is > >> more or less than a washcloth. triangles are not quantities, but > >> geometrical objects. > > > > I agree. So when you say > > > > infinite(x) <-> A y in R, x > y > > > > I can only assume you mean x is a real number being compared to y in > > "the usual way"; because that is the usual meaning of "x > y" when y is > > a real number. And there is no such real number x such that > > infinite(x). > > Did I say xeR? xe*R. > > > > > That's my complaint about your definition - ">" is defined for real > > numbers already, but you are "secretly" using the same symbol (">") to > > mean a /different thing/ - to compare real numbers with "a quantity". > > I am using it in the same way - "to the right along the real line," but > farther than any finite value. > > > > > Which is fine and a common thing to do, /if/ you have defined what you > > mean by ">" and "a quantity"; otherwise it's just something floating > > around in your head that you haven't stated explicitly. We're not mind > > readers! > > A quantity is a point on the line which represents a value equal to its > distance to the right of the origin. Right = ">". Nonsense. ">" is an order relation on R, meaning that it is a subset of RxR such that for all x,y and z in R (1) not (x < x) (2) if (x < y and y < z) then x < z (3) if x != y then (x < y xor y < x) > > > > > Your verbal definition of "a quantity" seems to be limited to: > > something that you claim can be compared to a real number using the > > symbol ">". For example, if a triangle can be compared to a real number > > using the symbol ">", then a triangle is a "quantity"; otherwise it's > > not. > > ">" implies linear order, such that the elements can be represented by a > point in a linear space. ">" implies a subset of RxR as defined above. "Linear space" has a quite different meaning. > > > > > That does not include a host of other things that I know you want, but > > have not said. I assume you will later claim that a "quantity" is not > > simply a thing that can be compared to a real number using the symbol > > ">"; but also has the property that it can be compared to every other > > "quantity" in such a way that the symbol ">" is a total order on > > "quantities" and real numbers, that preserves the usual meaning of "x > > > y" when both x and y are real numbers. > > > > But until you actually /state/ your definitions, axioms and rules, this > > is only guessing. > > > > For example, here's a sketch of what I think you want: > > > > "Let the set of quantities be some superset of R, with a total order > > ">=" which extends the usual ordering ">=" of R. Then if x is a > > quantity (is in the set of quantities), then infinite(x) <-> for all y > > in R, x >= y". > > Right. I never said xeR. Since "<" is an order relation on R, "x>y" only makes sense for x and y both in R, or at least in some explicit set, which has not been provided here. > > > > > This of course assumes that we mean the usual things by "R", "set", > > "superset", "extends", "is in the set", and "total order". > > > > Note that we still /cannot/ prove from this definition that the > > triangle T is a quantity; nor can we prove that it is /not/ a quantity. > > For all we know from the above definition, the set of quantities could > > possibly be exactly the same as the set R union {T}; with ">=" defined > > so that x >= T for all x in R. > > One would have to define how to compared a 2D object with a point on a > line. I didn't bring up that idea. So, what's your solution? To do as you have done and merely declare it so.
From: cbrown on 15 Dec 2006 02:52
Tony Orlow wrote: > Since it is a statement of implication, it's a rule, and an axiom. That > it serves as the definition of infinite doesn't make it not a rule for > determining what values are infinite, once finites are defined. > But we've already established that these rules don't apply to /everything/; for example you apparently don't want to imply that these rules apply to triangles. Triangles are neither "finites", nor are they "infinites"; because they are not the kind of thing you mean by "quantities". So let's agree to call these things "quantities"; with the understanding that not everything is a quantity: such a thing must obey certain rules before we can say it's a quantity. The down side of this for you is that it requires you /not/ to go haring off saying "therefore (omega/2)^2 is infinite" before we have /agreed/ that omega is, indeed, a quantity; unlike the triangle T, which is not a quantity, and for which the statement "(T/2)^2" has no agreed upon meaning at all. > >> A whole array of theorems pop forth from infinite-case induction > >> and IFR, such as that the size of the even naturals is half that of > >> the naturals. > > > > Not unless you declare them as axioms, as neither holds without being > > assumed. > > If n>k -> f(n) (inductively proven), and infinite(n) -> n>k, then > infinite(n) -> f(n), and the property can be said to hold in the > infinite case. I suppose what you want is an exact statement of IFR so > that we can determine which inductive proofs are allowable? Or, what? Obviously, if X -> Y, and Y -> Z, then X -> Z is valid logic. So as infinite(x) implies x > y for all y in R; and you claim to have a proof that for some k in R, n>k implies that f(n) then of course f(x) follows. The question remains: what constitutes a valid proof (in the world of quantities) of "for some k in R, n>k implies f(n)"? For example: You have previously frequently said you believe that: if f(0) and for all naturals n, f(n) -> f(n+1), then infinite(x) -> f(x). This is the metaphor regarding "induction" which you attempt to extend to "infinite induction". Except that you /also/ have said that you realize that (not infinite(x)) -> (not infinite(x+1)); so that by this logic, for the function f(n) = not infinite(n), infinite(x) -> f(x) -> not infinite(x). Which you don't want! So, your axiom of (truth about) infinite induction must be carefully phrased in such a way that it does not allow us to think that you mean that the statement (f(0) and f(n) -> f(n+1)) /always/ implies that infinite(x) -> f(x) for /all/ functions f of quantities. But how are we to know what /other/ exceptions are lurking out there? Is lim (n->oo) f(g(n)) /always/ equal to f(lim (n->oo) g(n)), when we extend the range of lim to include quantities as well as reals? Or is there perhaps some exception? A theory that says "this is mathematically true; unless of course it isn't: anyway I /feel/ it is true" is what Stephen Colbert might call "mathiness". You don't think it with your head, you /feel/ it in your /heart/. http://en.wikipedia.org/wiki/Truthiness Cheers - Chas |