An uncountable countable set
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From: Virgil on 2 Jul 2006 12:40 In article <1151844660.650801.186480(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Daryl McCullough schrieb: > > >> And in the set of all mappings from N to P(N), there are injections but > > >> no surjections. > > > > > >The latter is an empty assertion. > > > > It is a provable fact. Yes, you don't accept the proof, because > > you don't accept the basic principles of mathematics, but the > > proof certainly follows from those principles. > > "I don't know what predominates in Cantor's theory - philosophy or > theology, but I am sure that there is no mathematics there" (L. > Kronecker). The general view is that Kronecker's mathematics belied his philosophical ukases. > Your principle is that the complete set N including the largest element > does exist. Ours is that a complete N -> without<- any largest exists. It is ->yours<- that needs a largest. > But infinity obeys laws different from finite numbers (Galilei). As I recall, Galileo Galilei was a great scientist, but not a mathematician. Besides which, he was dead for about 3 centuries before any of the modern notions of infinity were under serious consideration.
From: Virgil on 2 Jul 2006 12:47 In article <1151844816.810459.198700(a)75g2000cwc.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > The difference being that Cantor's sequence of non-diagoal digits need > > not be created seqeuentially, but can be done independently and > > simultaneously, whereas yours cannot. > > Poor arguing! Is that all you have? (Euler thought a function must be > defined by a single expression.) > > We have infinitely much time. Further I can supply you with a > single-step creation of all transpositions required. But that is > unimportant. You need an infinite sequence of such infinite sequences of transpositions, and two successive such sequences of transposition must be applied in a particular order to achieve a particular result. They do not commute and they cannot be applied simultaneously. > > > Ask Cantor, how he can be sure that his diagonal is different from > > > every number of the list in case of infinitely many numbers. > > > > One does not have to ask Cantor to explain what is transparently clear. > > All Cantor needs do is specify a rule which can be applied > > simultaneously to ever number in a given list to produce something that > > cannot be a member of the list. > > > And the last number is included with certainty? There is no such "last" in ZF or NBG. What system are you messing up in?
From: Virgil on 2 Jul 2006 12:55 In article <1151845027.264084.242050(a)p79g2000cwp.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > > If one sets conditions on the mappings allowed, perhaps, but what if > > there are no such restrictions, other than having domain R and codomain > > P(N). In that case nothing prohibits surjection, and, in fact, > > surjections have been constructed. I have, myself, even constructed a > > bijection. > > Note that Hessenberg's trick is inavlidated by introducing one > unnatural number like (-1). The mapping from N and (-1) onto R is not > disprovable if (-1) is mapped on the set of non-generators. Then the > mapping can be surjective. Would you deny that (|N and (-1)) and |N > have same cardinality? So, what only can destroy it in one case while > leaving it in the other? AS the mapping at issue is from R to P(N), there is no mapping "from N" involved here. "Mueckenh" really should learn to read what he is responding to before shooting off such irrelevant responses. But in any case of a mapping from any countable set (which can be bijected with N) to either P(N) or R, one can produce, respectively, an analog of K or of Cantor's anti-diagonal rule to prove that the function cannot be a surjection.
From: Virgil on 2 Jul 2006 13:16 In article <1151845298.388602.107330(a)a14g2000cwb.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de wrote: > Virgil schrieb: > > If you claim that any member of any list is not differentiated from the > > number created for that list by any of uncountably many variations of > > Cantor's constructions, then you must be declaring there there exists a > > first number in some list which is not differentiated according to some > > version of Cantor's rules. > > > > So Give us an example of this alleged situation. If you can. > > Nothing easier than that. Exchange 0 by 1 in: > > 0.0 > 0.1 > 0.11 > 0.111 > ... > > The diagonal up to any line number n is contained in line number n+1. > > If you think 0.111... is not in the list, remember that any number the > digits of which can be indexd by finite numbers is in the list. Since every number in your list terminates, butyour "diagonal" 0.111... does not, it differes from each list member in infinitely many digit positions, and it is clearly not in your list. So YOU have provided a nonmember to prove your own error. If > 0.111.. is not in the list, then it must have more digits than can be > indexed (and hence, can exist). They are satisfactorily indexed by the infinite set of finite natural numbers, N. > > As the list is supposed to be endlessly long and the decimal (or other > > base) representation of each member is supposed to be endlessly "wide", > > why should either endlessness be greater than the other? > > Why should it not? Remember, there must be an exact equality between > width and length (better than +/- 1). If "mueckenh" insists on exact equality and "mueckenh" simultaneously insists on a difference, "mueckenh" is in trouble. Is that precision achievable? > Isn't it a fact, that already a list of all rationals is by far longer > than wide, even in binary representation? > 0.0 > 0.1 > 0.01 > 0.11 > ... > for instance. Except that "mueckenh" has omitted the endless string of 0's which each list member requires for its completion. > > I would rather believe in the fairys in your garden and even under your > bed than in the Cantor-list to be a precise square. You seem compelled to embed your list in geometry. I do not. The nearest I would come is to regard a list of reals between 0 and 1 in decimal notation as representable by a mapping f:N^2 --> {0,1,2,...,9}, where N^2 is the Cartesian product of N with itself, and with f(m,n) representing the n'th digit of the m'th number in the list. In this representation, one can construct a rule generating uncountably many decimal expansions g:N --> {0,1,2,...,9} such that 0 < g(n) < 9 for all n in N and g(n) <> f(n,n) for all n in N.
From: David Hartley on 2 Jul 2006 13:56
In message <1151844027.410356.228200(a)m79g2000cwm.googlegroups.com>, mueckenh(a)rz.fh-augsburg.de writes > >David Hartley schrieb: > > >> >The set of rationals is taken to be well-ordered. The following >> >transpositions operate on that set simultaneously (given are the >> >indices): >> >(1,2), (3,4), (5,6), ... >> >where Elements q_2n-1 and q_2n are interchanged, if they deviate from >> >order by size. >> >In the next step the pairs >> >(2,3), (4,5), (6,7), ... >> >are ordered by size, in the next step the pairs (1,2), (3,4), (5,6), >> >... are ordered by size, and so on. There are exactly as many steps as >> >are required to define the diagonal of a Cantor list. And there is the >> >same definition of "infinitely" as in Cantor's diagonal proof. >> >> You have not proved - or even given any justification at all - that this >> process has any meaningful limit, nor that if it does, it is ordered as >> you wish. > >This is exactly the same with Cantor's diagonal proof which not valid >for an infinite list. He has not proved - or even given any >justification at all - that this process has any meaningful limit. In >fact, it has not a meaningful limit, because all digits of an unending >number like 0.111... (to take a very simple case) which can be >identified are contained in the list of numbers: >0.1 >0.11 >0.111 >... >0.111...1 >... > >But in this list the number 0.111... is not contained. Hence not all of >its digits can be identified. > >> To see that this is not straightforward, consider the negative naturals, >> indexed by the corresponding positives, > >-1, -2, -3, ... indexed by > 1, 2, 3, ... > >> and apply your process. The >> first few steps give > >the indices after the first step are >> >> 2 1 4 3 6 5 8 7 ... > >This is correct, but the next four steps would supply >2 4 1 6 3 8 5 10 7... >4 2 6 1 8 3 10 5 ... >4 6 2 8 1 10 3 ... >6 4 8 2 10 1... >because we must always alternate between pairs (2n-1, 2n) in one step >and (2n, 2n+1) in the next (for all n e |N). > I see we have different ideas of what these transpositions mean. I took (1 2) to mean swap the pair originally indexed by 1 and 2, wherever they now are. You seem to mean swap the current occupants of positions 1 and 2 in the list. This is certainly more likely to have the effect you desire. Indeed, I agree that it is possible to apply a sequence of transpositions and change a well-ordering of the rationals to the usual ordering. (I posted my own example last night.) However, I take this to imply simply that such transformations do not preserve well-ordering, not that there is a contradiction in standard set theory. -- David Hartley |