# MTH 132 Calculus 1 by Stewart

By Stewart

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K). Note that for every x , , . ,xr E w , (i) U:= 1Mi( { x 1, .. ,x,})'s are disjoint for distinct i's. Let 9 be a non-principal ultrafilter on w. , x , } ) 9 ~ holds for exactly o n e j (1 C j C K). , of w as follows; if x,,, We define a sequence of distinct elements x o , x I ... ,x,,,}) I 1 C i C K, m,< - - . < m, < n and since this intersection belongs to 9 ,it is infinite, and we can also choose < n. , K. Then [N']'= C;u * * * u CL and so there exist. an infinite X C N' and a j such that [ X ] r q.

3 The following result shows that we need only consider the relation rn+(n): when r is finite. 5 (Erdos-Rado). (n),”o. Let M be a set of power m. We well-order [M]’o by a relation < . Elements of [Ml’o will be denoted by A’, Y , .... We place X in C, if X has a <-smaller subset Y , and in C, ’ If 1Y’l’ C Ci(iE J ) then n’ < r, so rn’+(n’); holds vacuously. We shall frequently use the device of ordering the base set X. , x n } for which xI < . . < xn. (x,, Ch. 0,$ 51 45 PARTITION CALCULUS otherwise.

0 is the (unique) isomorphism + r ’ Moreover, if R is a well-ordering relation of ON for which every initial section is a set, then (ON,R)=(ON, E ) (cf. GODEL[l]). Ch. 0, 3 21 between "CONSTRUCTIBLY" DEFINEDINACCESSIBLE CARDINALS ( T ~ E) , 25 and (ON,R)). 13. M'R,A'(X)= U M'R3c'( X ) if A is a limit ordinal, h > 0. 14. The preceding definition should be thought as a generalization of M"); likewise, MR is a generalization of M'"). The class M('vv) is increase with 5. E)'s O,, 1 is subtracted, as in the definition of Pq), to insure that the condition a EMR(X) depends only on the relation R restricted to ordinals less than a.