Logic

Constructible Sets with Applications by Andrzej Mostowski

By Andrzej Mostowski

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Example text

If 1 is a limit number > 0, then c d ( l ) is predicatively closed; also the class K = K(a) of sets constructible in a is predicatively closed. 4. In order to prove the second it is sufficient to note that K = Cd(=). 6. 4) and each transitive sequence a the class KB(a)of sets constructible from a by means of B is a model of ZF. 13. It is then obvious that the A, form an increasing sequence and that KB(a)is their union. Assumption (ii) is satisfied because d(A) = sup(d(5): E < A} for each limit ordinal A.

Hence x u y = {x, y } is an element of Ca(a+4) according to the cases already discussed. It follows that A S @ ,b) c Ca(a+4) and the theorem follows as in the previous cases. u i = 6. Again all we have to show is A&, b) c Cd(a+l). Let x = C,, y = C,, . Hence x-y = A&, y ) nx = A,(C,, C,,)n C, and we obtain x--y = C, for any y such that Zy = 8, K y = 5, Ly = q, My = E. Hence x-y E Ca(a+l)since a y with these properties exists among the numbers < d(a+ 1). i = 7. An element x o y of A&, b) consists of pairs (p, q) such that (p ,r ) E x and (r: q) ~y for a set r.

P n ) . 4. I f R ( A , p , , ... , pn> is a strongly definable relation and f ( A , q ! , ... , qm)a strongly dejnable functor of the second kind, then the relation R(A ,f ( A ,q1, ... ,p n ) is strongly definable. 5. The ternary relations expressed by the formulae ( P E A ) A ( 4 E A)A ( P E d, ( P E A ) A (4 E A ) A ( P = 4 ) are strongly dejnable. 54 IV. The characteristic functors En and Id of these relations have as their definitions the formulae = (XI (xo = xo) = (XI = X J . 6. I f f , g are strongly definable functors of the second kind, then the relations f ( A , p1, ...

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