## Download Axiomatic Set Theory by Gaisi Takeuti, Wilson M. Zaring (auth.) PDF By Gaisi Takeuti, Wilson M. Zaring (auth.)

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Extra resources for Axiomatic Set Theory

Sample text

Since S is dense in P, Sn G "# O. On the other hand, since G u {p} is compatible G contains no elements incompatible with p. e. (3q ~ p)[q E G]. Since G is P-generic it follows that p E G. This is a contradiction. 19. If P = E M, if B is the Boolean algebra of regular open subsets of P, if G is P-generic over M, and if F = {b E IBI nM I b = b- o G' = {p I [P] - 0 E F} /\ b n G"# O} then G = G'. , G s; G'. 18 that G = G'. 20. If B is a natural Boolean algebra, if F is a proper Mcomplete ultrafilter for BM, and if G = {p I [p] - 0 E F} P = {b E IBI nM I b = b- o 1\ bn G =j: O} then F = P.

Vp, q E G)(:Jr E G)[r :::; p /\ r :::; q]. Remark. 2 the topology is that induced on P by the partial ordering :::;. Consequently, condition 3 asserts that every element of A that is a dense subset of P, in the order topology, has a nonempty intersection with G. 3. If P = is a partial order structure, if p is a minimal element of P and if G = {q E PI p :::; q} then G is P-generic over A (in the strong sense). 25 Prao! 11' ql' q2 E G then p ::s; ql and p ::s; q2' Therefore Gis compatible.

3a E F)[N(a) s; G] ---7FEG. FE GM -;. I 0, GM = G. 15. Let GI and G 2 be open sets 01' a topological space I f for euch regular open set H . is regular open. 2. If G is a regular open subset of P then G* is regular open.