Galileo's Paradox
in [Math]
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From: Tony Orlow on 14 Dec 2006 13:24 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 = ">". > > 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. > > 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. > > 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? > > But we /can/ deduce things like "if infinite(x) and infinite(y), then > exactly one of x > y, x = y, or x < y is true" or "if infinite(x) then > x > 1.72". Which I think is the type of thing you want to prove. > > Cheers - Chas > Yeah, sure. Whether x and/or y are infinite, if they are both elements of an ordered set, then trichotomy holds. That's order for you. Tally Ho! Tony
From: MoeBlee on 14 Dec 2006 14:00 Six wrote: > On 11 Dec 2006 12:48:49 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: > 1) Cardinality (based on bijection) is a kind of count; it is understood to > be a measure (of how many). > I take it that is the conventional point of view. Not measure, since that is a another concept. But, yes, I think most people take cardinality to to capture the notion of size insofar as the notion of size can be understood also in the infinite. However, whatever the pros or cons for that notion, it is still an informal notion and is not itself required for formal set theory. Personally, for the finite, I do look at 'same count' as bijection. Then, insofar as (I'm stressing 'insofar as') as we'd like to be able to regard infinite sets has having comparative "size", the notion of bijection seems as suitable to me as any other proposal I've seen. But again, my understanding of set theory and mathematics does not at all depend on bijection serving to represent the notion of "size" in the infinite, but rather my primary interest is in the proofs of theorems from axioms and definitions. So, as far as I'm concerned, you could call it 'bjectabitlity' or 'schmardinality' or 'kurperfluginality' or whatever you want. > 2) Subsethood has also an implication for measure, for how many. That seems to me a remnant of a pre-systematic way of thinking. Given certain reasonable axioms (and I consider the axiom of infinity and the power set axiom to be reasonable) we see things differently upon a closer look. > >Because it is indeed 'bi'; it goes both ways. (Ha! I didn't even intend > >that to be a pun on sexuality.) > > Is it really that simple? Isn't that just the bias towards > bijection of which zuhair complained (though ultimately he realized his > solutions were futile), and which others perceive or think they perceive? I suppose you could describe just about any tendency as a bias. Bijection, though, is pretty basically intuitive, I think. > Consider the binary relation P (to be thought of as 'pairs with') > which is to operate on sets of natural numbers with one or two elements. > We will say A P B iff |A| not = |B|, |> This, I hope, expresses the mapping between: > > 1.2 -> 1 > 3,4 -> 3 > 5,6 -> 5 > etc. I don't think you need all that P stuff. We could easliy formalize your mapping, and it's clear enough what it is, even unformalized. > It is a bijective mapping (if I've done it right). It's a bijection. Let X = the set of unordered pairs of natural numbers such that the least member is odd and the other member is the least number plus one. Let Y = the set of odd natural numbers. Then your mapping is a bijection from X onto Y. > We can construct an analogous mapping, P*, to express the > correspondence between: > > 1 -> 1 > 2 -> 3 > 3 -> 5 > 4 -> 7 > etc. Okay. > but straightaway we know it is not going to be a bijective mapping as we > want {3} P* {5}, but we also want {3} P* {2}. Whatever is going on with P, the above is still a bijection. It's a bijection from omega onto Y. > >> I would draw the conclusion that it makes no sense to think of N or > >> any denumerable set as having a fixed size. N is a piece of elastic, or > >> more accurately a piece of elastic with no resting state. We know it's > >> being stretched out to the rationals, and shrunk again when it goes down to > >> the even numbers, or squares. But there is no way of measuring these > >> stretchings. N is the ruler. > > > >Okay. You might find a way to formalize that in set theory or some > >other theory of your invention. > > If you are not kidding me, I would love to know how to even begin to do > that. No, I'm not kidding. It is not out of the question that you could devise a formal theory that captures your notion of elasticity. Though I know something about formal theories, I don't know how to formalize your notion, but that doesn't entail that you or someone else couldn't figure out how to do it. > Even IF (ie big if) I was right, as I say I don't think it makes > any difference to anything. Except possibly to arrive at a better picture > of infinity. In very elementary treatments one often sees the sentiment > that it is wrong to think of infinity as a number, usually for all the > wrong reasons. The thrust of my argument is that is wrong to think of > denumerable infinity as any kind of count, EXCEPT, as it turns out, > relative to non-denuerable infinities. It is only the existence of those > (bigger pieces of elastic) which gives meaning to aleph_0 being a count. > Does that make sense? I think I get the general idea you have, though I pretty much don't buy it. MoeBlee
From: MoeBlee on 14 Dec 2006 14:31 Tony Orlow 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. Please stop spouting misinformation that comes from your not having even read the material. You're COMPLETELY missing the point of Robinson. In set theory (whether formal or informal), we can define certain languages and theories. Then we show (usually through ultraproducts) the existence of certain models, of which we take a certain one to be our non-standard model. The universe of this non-standard model is thus proven to be a set - proven in set theory. Whether this is formally axiomatized set theory or whether it is informal intuitive set theory, we still use all the principles - extensionality, separation, pairing, union, power set, infinity, and choice (or possibly a weaker principle than full choice, but still independent of Z). (Maybe replacement too, depending on the treatment?) Meanwhile, IST is a different approach, which gets the nonstandard field not by languages, models, and ultraproducts, but rather by EXTENDING ZFC with ADDITIONAL axioms. Whatever extra objects are proven to exist in IST, we also have the existence of ALL the objects proven to exist in ordinary set theory, including infinite ordinals, including the least infinite ordinal. All in the same theory. You are FLAT OUT INCORRECT about this. > 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: You are STILL FLAT OUT INCORRECT about this "two different theories" business. Ordinary set theory does have the existence of omega and non-standard naturals and non-standard reals, and so does IST. >There > is no establish language for it that supplies anything like a > calculation. Meanwhile, omega is the least infinite ordinal and that is NOT inconsistent with the existence of non-standard naturals and non-standard reals. > Real numbers? They exist for me. Sorry, Eckard, they're > real. "Limit ordinals" and "transfinite cardinals"? Those are fictions, You have shown no theory in which real numbers, or natural numbers, or non-standard real numbers, or non-standard natural numbers exist and such that we can perform operations such as taking subsets, intersections, and the like but without having the existecnce of a limit ordinal nor the existence of a transfinite cardinal. You're a pathetic joke. You think non-standard real numbers are so great, but you don't even read the mathematics that goes into proving their existence and properties, which includes the mathematics that proves the existence of limit ordinals and transfinite cardinals. MoeBlee
From: MoeBlee on 14 Dec 2006 14:48 Tony Orlow wrote: > MoeBlee wrote: > > Tony Orlow wrote: > >> Can you or anyone please cite where Robinson mentions omega in > >> Nonstandard Analysis? I'm not saying it isn't there, but I haven't seen > >> it. Granted, I'm not very far through it, but so far I see no need for it. > > > > Page 52. Not only does he mention omega, but he explicitly uses it in a > > formula of ordinal arithmetic to state the order type of the > > nonstandard *N. > > > > Now will you PLEASE stop claiming that Robinson's nonstandard analysis > > is incompatible with having a least infinite ordinal? > > > > Maybe, just MAYBE, you'll take ONE MINUTE to think toward understanding > > what I've said to you so many times already about your confusing two > > different senses of 'infinite'. > > > > MoeBlee > > > > Yes, I stand corrected. > He mentions it in the context of countable > neighborhoods surrounding each infinite value, in each direction. He mentions omega explicity as part of the description of the order structure of the non-standard system. > That's > exactly like m T-riffic countable neighborhoods around each declared > infinite limit point (digit), with rational "repeating" digit strings > connecting them. He doesn't refer to omega as being "infinite" in the > sense that he's using for his theory, however, does he? OF COURSE HE DOES! He's in ordinary set theory and mathematical logic. He talks about countable infinity in several places. He talks about the set of natural numbers in several places. His MAJOR POINT is that non-standard analysis can be derived from ordinary classical mathematics. Omega is the least infinte ordinal. That is part of the set theory that Robinson mentions as "RUDIMENTARY" [all caps mine] for his work. As far as I know, he uses no principles that are not expessible and derivable from ordinary set theory, and I do know that he uses ordinary set theoretic principles - extensionality, separation, power set, union, pairing, infinity, and choice (Zorn's lemma specifically). Omega as he refers to it is exactly omega as it is in ordinary set theory, and it is a limit ordinal. > He's referring > to a countable set in the way most easily understood by his readers. He HIMSELF says that mathematical logic and abstract set theory are the rudiments of his work. He's not using 'countable' in a way that is different from ordinary set theory. You were COMPLETELY INCORRECT about all of this, as I've been trying to get through your thick skull since at least a couple of months ago. It would help if you stopped spewing misinformation about that which you are not even prepared to speak on account of your not even having taken the time to learn the RUDIMENTS of the subject. MoeBlee
From: MoeBlee on 14 Dec 2006 14:53
Tony Orlow wrote: > Yeah, sure. Whether x and/or y are infinite, if they are both elements > of an ordered set, then trichotomy holds. That's order for you. WRONG. There are orderings that do not have trichotomy. MoeBlee |