Galileo's Paradox
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From: Six on 24 Nov 2006 06:35 GALILEO'S PARADOX 1 2 3 4 5 ................. 1 4 9 16 25 ............... There is a paradox because the 1:1 Correspondence suggests the sets are equal in size, by extension from the finite case, and yet clearly the second set is contained in the first set. That an infinite set can be put into 1:1 C with a proper subset is not by itself paradoxical. That is only the beginning, the facts of the case. The paradox is that the squares seem to be both smaller than N and the same size as N. I want to suggest there are only two sensible ways to resolve the paradox: 1) So- called denumerable sets may be of different size. 2) It makes no sense to compare infinite sets for size, neither to say one is bigger than the other, nor to say one is the same size as another. The infinite is just infinite. My line of thought is that the 1:1C is a sacred cow. That there is no extension from the finite case. If we want to compare the two sets for size we would write, not the above, but: 1 2 3 4 5 6 7 8 9 ............... 1 2 3 4 5 6 7 8 9................ ^ ^ ^ (The intention here is to highlight the squares in the second row of integers.) Then we would notice that the relative size of the squares set becomes ever smaller as n increases, that increasingly large numbers of integers are missed out. In fact, if we wanted to find a plausible candidate for a set eual in size to N, then we would choose not the squares, but the non-squares. The contrived nature of the 1:1C becomes more obvious when we compare with N, sets that appear to be larger than N. The clearest example is Z. We have: 1 2 3 4 5 6 7 ......... 1 -1 2 -2 3 -3 4 ......... which is mildly clever, but again if we wanted to compare the two sets for size we would write: 0 1 2 3 4.... ....-4 -3 -2 -1 0 1 2 3 4.... with a perfect 1:2 Correspondence. Here one would like to say, since not only is there 1:1C between Z and a proper subset, but an identity (1 2 3 4 ......), that however you define infinity there has got to be more in Z than in N. Not, of course, if you make all countable sets equal in size by definition. But for me, that doesn't relieve the paradox at all. On the contrary it builds it into the foundation of the mathematics. I would like to suggest that the existence of 1:1C between the two sets is a CONSEQUENCE of the fact that they are both infinite. The infinities are what gives one room to manoeuvre, to manufacture a 1:1C. It has no bearing on their relative size. Can one make sense of Z = 2N, of Q = N^2, etc.? (Incidentally the number of squares would be sq.rt. of N, since after n^2 integers there are n squares.) Maybe it's complete rubbish, but my argument is that the alternative is the ineffable infinity. If it does make sense, there is no place for a diagonal argument, or a power set argument, since it would already be conceded that 10^N > N, that 2^N > N, or in general that k^N > N, just as Z > N and Q > N. There remains of course Cantor's proof that R cannot be put into a 1:1C with N, which is very interesting. But what does it mean? Maybe something like this: So-called denumerable sets can be represented on a finite-dimensional lattice, so that a self-avoiding walk can be shown to systematically cover the entire line, are, volume or hyper-volume. For R understood as a set of decimals (to choose that -- perfectly good -- representation), by contrast, every decimal place can be construed as an axis. In any case what I don't understand is how this affects the simple paradox with which we began. However, it may very well be that my insufficiently tutored brain has flown its coop again, in which case I would be very grateful for any illumination. Six Letters 24/11/06
From: Peter van Liesdonk on 24 Nov 2006 07:01 > 2) It makes no sense to compare infinite sets for size, neither to say one > is bigger than the other, nor to say one is the same size as another. The > infinite is just infinite. This is exactly the case. All infinite sets have the same size: infinite. Actually, talking about size is a bit vague in this case. You give the example: 1 2 3 4 5 1 4 9 16 25 It is obvious that in this finite example (N=5), the second row is not contained in the first one. The same is true for every finite N, no matter how large. It is only when N actually is infinity that the second row is contained in the first. Because Philosophize about these questions to get an understanding: * If you have two sets of infinite size, is the union of these sets than larger than infinity? What would larger than infinity mean? * If you have an infinite size set and you remove a finite amount of elements, how large will the result be? Regards, Peter
From: Richard Tobin on 24 Nov 2006 07:20 In article <1164369693.301566.8460(a)l12g2000cwl.googlegroups.com>, Peter van Liesdonk <peter(a)liesdonk.nl> wrote: >> 2) It makes no sense to compare infinite sets for size, neither to say one >> is bigger than the other, nor to say one is the same size as another. The >> infinite is just infinite. >This is exactly the case. All infinite sets have the same size: >infinite. Actually, talking about size is a bit vague in this case. We don't usually go as far as that. Not all infinite sets are the same size; there are different infinite sizes. But some infinite sets that you might like to have different sizes don't. This is because - as I think the original poster realises - there just isn't a consistent extension of size to infinite sets that satisfies all our intuitions about size. If you want to have both (a) sets in 1-1 correspondence are the same size and (b) proper subsets are smaller than their supersets you just can't do it, because (as the squares example demonstrates) you can put infinite sets into 1-1 correspondence with certain of their proper subsets. On the other hand, we don't have to throw everything out. It's not always the case that an infinite set can be put into 1-1 correspondence with a particular proper subset, as Cantor showed. It turns out that we can have a number of different infinite sizes without running into contradiction. >* If you have two sets of infinite size, is the union of these sets >than larger than infinity? What would larger than infinity mean? That this is not a stumbling block should be clear of you replace "infinite" with "big". The union of two big sets can be bigger than either of the original big sets. Like "big", "infinity" is not a number; it's a description of certain numbers. -- Richard -- "Consideration shall be given to the need for as many as 32 characters in some alphabets" - X3.4, 1963.
From: Bob Kolker on 24 Nov 2006 08:03 Six Letters wrote: > > I want to suggest there are only two sensible ways to resolve the > paradox: > > 1) So- called denumerable sets may be of different size. Same size (i.e. same cardinality) means there is a one to one correspondence between the sets. That is the definition of "same size". > > 2) It makes no sense to compare infinite sets for size, neither to say one > is bigger than the other, nor to say one is the same size as another. The > infinite is just infinite. False. Where a correspondence can be established it makes perfectly good sense. > > > My line of thought is that the 1:1C is a sacred cow. That there is > no extension from the finite case. Dead wrong. You have been wrong for over a hundred years. Bob Kolker
From: Six on 24 Nov 2006 09:41
On 24 Nov 2006 04:01:33 -0800, "Peter van Liesdonk" <peter(a)liesdonk.nl> wrote: >> 2) It makes no sense to compare infinite sets for size, neither to say one >> is bigger than the other, nor to say one is the same size as another. The >> infinite is just infinite. > >This is exactly the case. All infinite sets have the same size: >infinite. Actually, talking about size is a bit vague in this case. > >You give the example: >1 2 3 4 5 >1 4 9 16 25 Actually this was meant as 2 infinite sequences, but it doesn't affect your point. >It is obvious that in this finite example (N=5), the second row is not >contained in the first one. The same is true for every finite N, no >matter how large. It is only when N actually is infinity that the >second row is contained in the first. Because This is an interesting way of putting it, but I am not convinced it does more than re-state the paradox. >Philosophize about these questions to get an understanding: >* If you have two sets of infinite size, is the union of these sets >than larger than infinity? What would larger than infinity mean? The short anwer is that maybe you can have two infinite sets, one larger than the other. I mean for denumerable sets, but as Tobin points out this is already accepted for sets in general because of Cantor's proofs. >* If you have an infinite size set and you remove a finite amount of >elements, how large will the result be? At a guess, the size will be unaffected. >Regards, >Peter Thankyou, Six Letters. |