By James E. Baumgartner (ed.)

ISBN-10: 0821850261

ISBN-13: 9780821850268

ISBN-10: 1719792712

ISBN-13: 9781719792714

ISBN-10: 7719717317

ISBN-13: 9787719717313

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A, are free and the vectors 67 LINEAB DEPENDENOE 4,.. ,,,b are free, then at least one vector b, is free from +, . ,a,, b, are h). %, . . , a , (that is, Proof. The matrix of the a, has rank r ; we may suppose that the determinant d formed out of its first r columns is # 0. ,Y ; a= 1, . , r 1) , (8= 1,. + r + 1) As the b, are free, there is a determinant, formed out of their components with subscripts jl, .. ,it+,, which is # 0. The corresponding determinant formed out of the c, is 0, so we find at leaat one pair of subscripts t , u with b , # c,.

1) After xn has been chosen, at least one and a t most three values are admissible for x,,+~,so S is finitary. I shall show that S coincides with [a, b ] . (1) shows that every element, of S coincides with an element of [d, c]. Conversely, let x be any element of [d, c], and let { ~ , , 3 - ~be) a canonical ips coinciding with x. As before, we may suppose that d n s x n ~ c , , , so that x coincides with an element of AS. 4. The fan theorem T h e o r e m . If an integer-valued function ~ ( 6 ) is defined for every element 6 of a finitary spread 8,then a natural number N can be computed from the definition of T, such that ~ ( 6is) determined by the first N components of 6 ; that is, if 6, and 6, are such elements of S that the first N components of S, are equal t o the first N components of 6,, then q1(6,)=p(6,) [L.

Let Q be a bounded infinite species of real numbers and let m, n be natural numbers. Then there exists an interval of length 2-n which contains at least m elements of Q. P r o o f . Let h, k be integers such that Q is contained in the interval (h,k). Set r = ( k - h + 1)2n+1. Let R be a subspecies of Q with rm elements. For every element x of R we determine an integer z % +so ~ that (z,,+~-1)2-n-1

### Axiomatic Set Theory by James E. Baumgartner (ed.)

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