Galileo's Paradox
in [Math]
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From: Michael Stemper on 15 Dec 2006 09:57 In article <458191a1(a)news2.lightlink.com>, Tony Orlow writes: >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, Why not? Suppose that I start with all integers, and then discard the ones that have prime factors other than two or three. I'll also get rid of all of the numbers that don't have at least one factor of three. Finally, I'll get rid of zero because I don't like it. That would leave me with: {... -12, -9, -6, -3, 3, 6, 9, 12, ...} Then, I write each of these in the form (2^m)*(3^n) for positive and (-1)(2^m)*(3^n) for for negative. Each of these corresponds to a rational number: either m/n or -m/n. Examples: -18 = (-1)(2^1)(3^2) --> -1/2 -3 = (-1)(2^0)(3^1) --> -0/1 6 = (2^1)(3^1) --> 1/1 54 = (2^1)(3^3) --> 1/3 We've gotten rid of a lot of integers (infinitely many, in fact), and can turn what's left into all of the rationals. Some of the rationals get hit more than once (maybe all of them). That doesn't change the fact that we've created *all* rationals from a subset of the integers. -- Michael F. Stemper #include <Standard_Disclaimer> Reunite Gondwanaland!
From: Michael Stemper on 15 Dec 2006 12:47 In article <200612151457.kBFEvrL4143386(a)walkabout.empros.com>, Michael Stemper writes: > Suppose that I start with all integers, and then discard the ones that > have prime factors other than two or three. I'll also get rid of all > of the numbers that don't have at least one factor of three. Finally, > I'll get rid of zero because I don't like it. > > Then, I write each of these in the form (2^m)*(3^n) for positive and > (-1)(2^m)*(3^n) for for negative. Each of these corresponds to a > rational number: either m/n or -m/n. > > We've gotten rid of a lot of integers (infinitely many, in fact), and > can turn what's left into all of the rationals. Some of the rationals > get hit more than once (maybe all of them). Well, I must be slow this morning. Every rational gets generated by infinitely many different integers, because if z_1 generates m/n, then so does (z_1)^p, where p in Z+. For instance: 18 = (2^1)(3^2) --> 1/2 324 = (2^2)(3^4) --> 2/4 = 1/2 Which just makes my original point stronger. We can discard an infinite subset of the integers, and then generate each rational once from each member of an infinite subset of the remaining integers. I predict that, in response to this, somebody will stick their fingers in their ears and go "La, la, la, I can't hear you." -- Michael F. Stemper #include <Standard_Disclaimer> A preposition is something that you should never end a sentence with.
From: Six on 15 Dec 2006 23:51 On 14 Dec 2006 11:00:21 -0800, "MoeBlee" <jazzmobe(a)hotmail.com> wrote: >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. Right. I accept just about all your points there. >> 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. That's certainly a cogent point of view. And it's difficult for me to dispute it with my limited mathematical knowledge. I do still strongly suspect that you are underestimating the paradox. I wish I was sure you were alive to the shallowness of creating different senses of a word in effect for the sole purpose of 'resolving' a paradox. This is something I was trying to get across to Stephen. I see paradoxes as engines of progress, and think it is important to be strenuous in resolving them. Admittedly the case of mathematics is special, or different from some others, because the mathematics has to be constructed, and mathematicians are entitled to so work as they see fit. I recognize the force of your 'pre-systematic'. >> >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. > Thanks. The point indeed is that it is a simple bijection, and on that basis might be thought to be comparable to the original one -- one having a 2:1 flavour, the other a 1:1 flavour. The drift being that perhaps bijection as such isn't the important thing. >> 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. OK > >> >> 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. OK >> 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. OK. I certainly take your point, in your opening remarks that the notion of size of infinite sets may make little difference to the practice and development of set theory. I have just begun to wonder if my idea has any bearing on the theory of ordinals, but anyway that's a thought for another time, if at all. No reply is particularly called for. I've made a reply to Imaginatorium, if you're interested. Thanks for some cogent and helpful comments. Six Letters
From: David R Tribble on 16 Dec 2006 14:36 cbrown wrote: >> 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". > Tony Orlow wrote: > I am using it in the same way - "to the right along the real line," but > farther than any finite value. There is no real number that matches this definition, because all reals in the real number line are finite values. There is no non-real number that matches this definition, because the real number line contains only real numbers (obviously). You continually persist in inventing "numbers" outside the realm of the reals (and naturals), and then shoehorn them into the reals as if they were there all along, just hidden until you came along and found them.
From: Six on 16 Dec 2006 18:39
On 16 Dec 2006 09:21:17 -0800, imaginatorium(a)despammed.com wrote: >Thanks for reading what I wrote... And to you too. snip >> >> It took me some time and a little study to understand what you >> meant about partial ordering and so on. The thing is that I am not, indeed, >> going to try and construct an alternative notion of size, to compete, as it >> were, with the cardinality one. I know that is futile, and I do not require >> it. I will come back to the business of strings at the end. > >"Compete" is an odd term, but I don't think there's anything at all >wrong with contemplating alternative notions for infinite sets that >correspond in some way or another to size for finite sets. OK snip >> >> >The "paradox" comes from assuming that since all finite sets have a >> >"size", it must be possible to define a "size" for infinite sets that >> >preserves the properties of "size" that we are used to with finite sets >> >- the two most salient of which being that: (a) If you remove some >> >elements from a set you get a different set that is smaller, and (b) If >> >you can count two sets together such that one does not end before the >> >other then the sets are of the same size. There is no definition of >> >"size" for infinite sets such that (a) and (b) are both preserved. >> >> Agreed. That's a good explication of the paradox. snip >> >> >> >> Now what difference do Cantor's discoveries make to all this? I >> >> would say none, as far as the reasoning goes, but everything in terms of >> >> what is of interest. Everthing said about the infinity considered (the >> >> denumerably infinite) is still valid. It is as if infinity was a piece of >> >> elastic, which would stretch or shrink to cover all cases. But it turns out >> >> there are hyper-infinities which exceed the elastic's breaking point. Does >> >> this mean our first infinity has a fixed size after all? No. But there is >> >> now a comparison to be made. It is still fallacious to think that a >> >> bijection demonstrates directly equality of size. >> > >> >Sorry to jump in the middle. IF you have defined "size" to mean "equal >> >cardinality", it is most certainly not fallacious. IF you have defined >> >"size" to mean [something relating to subsethood, in which case "equal >> >size" actually means "the same set"... if you can make a rigorous >> >definition like this], then yes, it is fallacious. IF you have not >> >really defined what you mean by "size", but hope that by talking enough >> >your audience will know what you mean, it is not really fallacious, >> >just meaningless, going on philosophy. >> >> This is I believe an unfair trichotomy. Obviously I do not mean the >> first or second options. But no I am not going to define size, that is to >> say equality of number. This is a primitive notion. > >Sorry, what is a primitive notion? You really think the trichotomy >unfair? But you don't want to define what "size" means? How on earth >can you usefully say anything mathematical about something you refuse >to define. This _really_ is not what mathematics is about. I don't think you have taken my point. I am not refusing to define anything. I am refusing to go along with the conventional viewpoint that different notions of size are involved in the paradox. In just the way that bijections suggest equality of (infinite) size, so does removing elements from an (infinite) set suggest inequality of number. Consider the possibility that I am not overtly or covertly appealing to different notions of size. If you have made your mind up that this is the way to resolve the paradox, you will not agree with me, but you can at least acknowledge the, if you like, logical possibility that both intuitions in the paradox are wrong. Then what precise definitions are needed, if any, will come out of resolving the paradox in this way. If I am right, it is not I who has a problem with defintions, it is you. If you are right, then it is indeed a question of clarifying notions of size. But you can hardly make a failure to define (in)equality of (infinite) number a critical test of my position. >Oh, wait a minute, you say "size" is *equality* of *number*. Well, >'equality' is OK, but then the obvious question is: what is "number" in >the context of an infinite set? There _really_ is no obvious answer. If >you define the "number" of elements in a set to be the cardinality of >the set, then you get one answer; if you define the "number" of >elements in the evens (for example) to be the density of the evens in >the naturals, followed by an indexing symbol (I'll write m for >omega-upside-down), you get a different answer m/2. One of the crank >threads not so long ago went on and on and on, literally for months, >with everyone trying to find out what the crank meant by "size", and >the crank repeated over and over again that it was "just the number of >elements in the set". No-one (including the crank, actually) has the >faintest idea what this could possibly mean. The naive notion of >"number" in this sense is that you point at the elements in turn, while >singing a ditty ("one-two-three-and-so-on..."), then shout out the >number you stop at. Try this on an infinite set, and the ditty never >ends. So what number do you stop at? There isn't one. I like your ditty. Here is a game to go with it. A paradox about blah? No problem. Two senses of blah. Blah blah blah blah blah .............. >> It clearly is not >> adequate as it stands for dealing with infinite sets, but that does not >> mean there is any pre-systematic (to borrow MoeBlee's nice phrase) or 17th >> Century confusion about it. What I would like to do is go back 400 or so >> years and build the mathematics in a way that does justice to it. >> You paraphrased the paradox yourself very well, although it is >> slightly inaccurate to talk about intuitions about finite sets. It is >> intuitions about infinite sets which leads to the paradox, though >> presumably based on extensions from the finite, so we won't quibble. >> People say you can't trust intuitions, you can't expect intuitions >> about the finite case to carry over to the infinite. Precisely. Exactly. >> But what do they do? They throw one of them out and seem surprised when >> people claim it's not been done justice. >> Solution? Kill them both. They are both inadmissible extensions >> from the finite case. > >Can you state what *exactly* you want to kill? With respect to >cardinality, two things come to mind. >(1) Don't say, suggest, or hint that cardinality is "size". Don't use >the word "size" in the context of cardinality. ** This one is no >problem at all. Cardinality is exactly unchanged, we just have to be >careful with words. > >(2) (the tricky one) Abolish the concept of cardinality. How could we >do this? Is there a false theorem? If not what would it mean to >"abolish" a concept? (This can be done, but only in a totalitarian >state.) "Kill" is explicated by the sentence that followed it. They (the two intuitions that constitute the paradox) are both inadmissible extensions from the finite case. Exactly what the consequences of this are I am not sure, beyond what I have written already about elasticity. I do not think they are necessarily radical. >I think the following paragraph is somewhat confused, but excuse me >picking a bit out... > >> In the infinite case, it is neither true that a bijection >> establishes an equality of size nor is it true that removing members from >> such a set necessarily reduces its size. This is the radical and satisfying >> way of resolving the paradox. There is not a solution which preserves the >> intuitions -- we would not really have expected infinity to be so >> reducible. It does justice to both intuitions perhaps only in the >> convoluted sense that the opposite of each is denied. But this does seem >> right to me, not out of some abstract principle of dialectical democracy, >> but just by looking at the mathematical objects. Yes, bijection has force, >> but look how convoluted (not necessarily hard to find) these bijections are >> between for eg N and Q, or N and Z. In the latter case, for example, one >> can make some sense of the idea that the infinity at the left hand end of Z >> is being rolled up and shoved over at the right hand end of the bijectable >> ordering of N -- there's room for it, because that right hand end is >> infinite. > >The "right hand end is infinite"??? Um, no. The right hand end isn't >there. There is no right hand end. Endless means there _is_ no end, not >that the end is somehow in a different sort of universe. Sorry. 'end' here was meant synonymously with 'side'. Poor choice of word. There is certainly some working out to do with the thoughts in this paragraph, but I think the basic idea of rejecting both intuitions in Galileo's paradox is reasonably clear. > I can't give a precise meaning to those thoughts, perhaps because >> the goal of any such precision would be to reify the >> intuition-about-removing -elements to the sole arbiter of size, or perhaps >> a precise meaning could be given to those intuitions about rolling up and >> shoving. On the other hand, the intuitiion that removing elements from an >> infinite set reduces it has force -- whatever infinity is, one would like >> to say, surely it is reduced by precisely the elements one has removed from >> it. Yet one can begin to understand that the concept of infinity is >> flexible enough or peculiar enough to deny that that is true. >> >> >It is sort of interesting to speculate that suppose Cantor (or anyone >> >else) had never investigated transfinite set theory, and never noticed >> >that the power set of an infinite set can't be put in 1-1 >> >correspondence with the set, we might have chosen to use a word like >> >"unsized" to mean what "infinite" means now. Would the cranks (by which >> >I don't of course include you, since you appear to have a genuine wish >> >to find out what's going on here) still argue about [not Cantor's?!] >> >theory??? Who knows. >> > >> >> Yes, and I would add that literally 'unsized' makes sense. There >> are some superficially surprising consequences, but given the elastic >> nature of infinity we've arrived at they should be trivial. For example, we >> cannot say that |N| is base-invariant. > >Obviously it depends what "| |" means. But you will have to reconstruct >the totality of mathematics if somehow there are alternative versions >of the natural numbers with different properties. I don't see that as a consequence. Just about everything that could be said about N before can still be said. > It will not be true either (make >> sense to say) that there are more decimal strings than birnary stirings >> representing the same natural numbers >> Not really surprising since we cannot say anything about |N|. Except, that >> is, in relation to sets which, as it turns out, are bigger (R). Does that >> mean |N| turns out to have a range after all? Maybe there is a real problem >> with this concept of elasticity here. How can N be indefinitely elastic if >> it is smaller than another set? Perhaps bijection has something to do with >> an upper bound on N, whereas removing elements of N has to do with a lower >> bound on N. I don't know. > >Hmm. Carry on with the introductory reading, I think. > When I began this thread I thought I had just about enough mathematics to keep my head above water, but at various times through it I have found myself gasping for breath. In particular I ought to do some homework on set density (actually myriad things). While fairly sure that I am not appealing to any such notions in my argument it would be better of my convictions were based on knowledge rather than faith. I have some rough ideas about how to extend the thoughts in the last paragraph, but indeed this is probably a time for catching up on more basics. It's very kind of you to help my groping, fumbling mind. Six Letters. |