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Topic: About the existance of infinite products of integers
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Posts: 10
Registered: 12/8/04
About the existance of infinite products of integers
Posted: Jul 24, 1998 9:58 AM
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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 Fraenkel-Mostowski model, if the product (n +1)*(n
+1)*(n +1)*... is well- defined, then so is n*n*n ....So the type of
Fraenkel-Mostowski model needed for your argument, where the product n*n*n
... is well-defined 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 Fraenkel-Mostowski models the continuum
is well-orderable (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 well-defined
therefore makes IIiBi well-orderable for any sequence of (n+1)-element sets
Bi. Now let Ci be any sequence of n-element 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* well-defined. 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 well-ordering < of it. Now we can
define (uniformly with respect to i) a well-ordering <i of each Ci as
follows. (Note that I'm not saying merely that each Ci has a well-ordering,
which is trivial since its cardinality is n, but that there is a sequence of
well-orderings, <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 well-ordering <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)
Adib Benjebara

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