Axiomatic Set Theory, Volume 2 (Symposium in Pure by Jech, T.J; Scott, D. (editors)

By Jech, T.J; Scott, D. (editors)

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I}, {(i, i + 1) I i < I}). ) Moreover show that the dass of finite acyclic digraphs can be axiomatized by a sentence "IP1jJ, where P is unary and 1jJ is a first-order sentence over {E,P}. 8. Show that there is a formula 1jJ(x, y) ofthe form 3Pep, where P is unary and ep is a first-order formula over {E,P}, expressing in finite graphs that x and y are in the same connected component. In fact, as 1jJ(x, y) one can take a formula expressing that x = y or that there is a subset P containing x and y such that x and y have an edge to exactly one member of P and every other member of P has an edge to precisely two members of P.

Ar . We restrict ourselves to the case ai < a < ai+1 for some i. Then, dj(ai,a) = 00 and dj (a,ai+1) = 00; hence, dj+t{ai,aHd = 00 and therefore, dj+1(P(ai),p(aHl)) = 00. Thus there is absuch that p(ai) < b < p(ai+1), dj(p(ai), b) = 00, and dj(b,p(aHd) = 00. One easily verifies that q := pU {(a,b)} is a 0 partial isomorphism in Ij. 6. Let T = { <, min, max} be as in the preceding example and a = TU{E} with a binary relation symbol E. For n 2: 3 let An be the ordered T-structure with An = {O, ...

Suppose that for the sentence (*) and any finite graph 9 is connected Hf for some Pi, ... , Pr ~ 9, we have: G: (Q, Pi, ... , Pr) F 1/J. 2. Since 910 is connected, there are Pl, ... ,Pr such that (Qlo'Pl, ... ,Pr ) F 1/J. Then, (Q;o'Pl, ... 4, but 9~0 is not connected, a contradiction. 6. The dass of finite and connected graphs can be axiomatized by a formula of the form 3R 1/J, where R is binary and 1/J is a first-order sentence over the vocabulary {E,R}. Proof. Let 1/J be a sentence expressing that R is an irreflexive and transitive relation with a minimal element, and that Exy holds for any immediate Rsuccessor y of X; that is, 1/J is the conjunction of Vx-,Rxx /\ VxVyVz((Rxy /\ Ryz) -+ Rxz), 3xVy(x = y V Rxy), VxVy((Rxy /\ Vz-,(Rxz /\ Rzy)) -+ Exy).

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