By Andrzej Mostowski
ISBN-10: 0444534210
ISBN-13: 9780444534217
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E A , then CHAPTER I1 GENERAL PRINCIPLES OF CONSTRUCTION 1. Sufficient conditions for a class to be a model We state several simple lemmas which express sufficient conditions for a class to be a model of ZF. Throughout this section A is a transitive class. 1. Axiom (I) is always valid in A. 2. Axiom (11) is valid in A if and only i f A is closed under the operation of forming pairs. 3. , a E A + a E A. 4. Axiom (IV) is valid in A if and only A contains a non-void set a such that whenever b is in a, there is a set c which properly contains b and is an element of a: All these lemmas are proved in a straightforward way.
Zf 1 < k < r, then plk(a) = pll(a)+k-1. PROOF. It follows from the definition of the antilexicographic ordering that if 1 < i < r, then the element (i+ 1, a , p , 7 ) is an immediate successor of the element (i, a , p, 7 ) . Hence if to, t1, ... , 5@,... , t Q + k - l , ... is the increasing transfinite sequence of all ordinals 7 satisfying Zq = k. Hence plk(a) = E,+k-1 = pll(a)+k-l. 2. CONSTRUCTIBLE SETS 39 2. Constructible sets Let S be the class of all transfinite sequences. Thus a E S if and only if there is an ordinal 5 E On, called the type or the length of a, such that a is a function and Dom(a) = 5.
6 is an increasing and continuousfunction. It follows from the definition that the set X , = { a : 6, < a < 6 ~ + 1 } has the order type rdj. ,y < 6,. We denote the quadruple which corresponds to a by ( l a , K a , L a , Ma). 3. ZfS, < a < 6,+,, then 1 < la < r andKa, La, M a < 6,. It is not very important how exactly we fix the mapping of r * 6: onto X,; but for the sake of definiteness we may agree to order (antilexicographically) the quadruples ( i , p , Y , e ) where 1 < i < r, p, Y , c < 6, and correlate with each quadruple the order type of the set of all quadruples which precede it.
Constructible Sets with Applications by Andrzej Mostowski
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