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Topic:
About the existance of infinite products of integers
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About the existance of infinite products of integers
Posted:
Jul 24, 1998 9:58 AM


About the existance of infinite products of integers
16 October, 1995
Dear Mr Ben Jebara,
Thank you for your fax letter of 15 October. The model you proposed will violate Cn not only for even n as you intented but also for odd n >1, which you wanted to avoid. I had told you that I'd try to get a second opinion; that succeeded better than I had expected. The person I asked for the opinion pointed out that, in every FraenkelMostowski model, if the product (n +1)*(n +1)*(n +1)*... is well defined, then so is n*n*n ....So the type of FraenkelMostowski model needed for your argument, where the product n*n*n ... is welldefined for odd n but not for even n, cannot exist. Here's a sketch of the proof. The main point (which I had overlooked when I thought about the question earlier)is that in FraenkelMostowski models the continuum is wellorderable (because it is a pure set). That means that, if we take Ai={1,2,...,n+1}for every natural number i, then the product IIiAi is well orderable. The assumption that (n +1)*(n +1)*(n +1)*... is welldefined therefore makes IIiBi wellorderable for any sequence of (n+1)element sets Bi. Now let Ci be any sequence of nelement sets; I'll show that the cardinality of IIiCi doesn't depend on the particular sequence of Ci's; in other words, n*n*n*...is welldefined. For this purpose, fix some object q that belongs to none of the Ci, and let Bi =CiU{q}. By the preceding paragraph, IIiBi is well orderable; fix a wellordering < of it. Now we can define (uniformly with respect to i) a wellordering <i of each Ci as follows. (Note that I'm not saying merely that each Ci has a wellordering, which is trivial since its cardinality is n, but that there is a sequence of wellorderings, <i, one for each Ci.) If a,b belong to Ci, let fa be the first, with respect to <, element of IIiBi whose ith component is a , and define fb similarly with its component b. Then let a<ib if and only if fa < fb. Page 2 Now using the sequence of wellordering <i, we define a sequence of bijections gi:Ci>{1,2,...,n} as the unique bijections taking the order <i on Ci to the standard order on {1,2,...n}. The existence of this sequence of bijections implies that IIiCi has the same cardinality as IIi{1,2,...,n}, namely n power the power of N, independent of the particular sequence.
Sincerly yours, Andreas Blass Attention of Mr Andreas BLASS
Dear Mr Andreas BLASS
At the end of the 16 October 1995 letter's first page you write if a,b belong to Ci you should write (a0,a1,...,an,...) belong to IIiCi (b0,b1,...,bn,...) belong to IIiCi which is assuming IIiCi # 0 (empty set) Regards Adib Benjebara
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