By F. William Lawvere, Stephen H. Schanuel

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Y X˛ , then deﬁne g˛ to be this function. Otherwise, deﬁne g˛ to be any function X˛ ! Y X˛ whatsoever. Now pick any function g W X ! Y X, and let S denote its graph viewed as a subset of X Y X. From the Diamond Principle we obtain that the set Eg of ˛’s with g X˛ D g˛ is stationary in Ä, as desired. Another result of Jensen’s which will be needed is as follows. 5 (V = L). Let Ä be a regular cardinal which is not weakly compact. There exists a stationary subset E of Ä which consists of limit ordinals coﬁnal with !

The elements of an inﬁnite cyclic group C generated by c are nc (all distinct) with n running over the additive group Z of integers. C is isomorphic to Z, an isomorphism is given by the correspondence nc 7! n 2 Z. Thus all inﬁnite cyclic groups are isomorphic. Along with c, also c can be a generator of C, but no other element alone can generate C. m 1/c. Because of mc D 0, we compute in C just as with the integers mod m. Consequently, C is isomorphic to the additive group of residue classes of the integers mod m; this group is Z=mZ.

Given a countably inﬁnite set S, there exist two families, † D fS j < †0 D fS0 j g and < g; of almost disjoint subsets of S such that S S 0 (i) < S D S and < S D S; (ii) for all ; < , the intersection S \ S0 is inﬁnite. f from the set nN D Proof. , let Fn denote the set of all functions S f0; 1; : : : ; n 1g to the set f0; 1g. Evidently, the set S D n