By James E. Baumgartner (ed.)

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Additional info for Axiomatic Set Theory

Sample text

Since rx is positive, we replace some of the + by -,at the same time requiring that in each rank there appears exactly one value +. Next, we reject all systems whose elements are all - (if such occur). With each system we associate the set of ranks for which it takes the value +, and substitute for these the same atom; we thus get a reduction which is of arity ~ p and does not belong to s1 OJ (suggested by C. Benzaken). 9. 3). 3). 3). Together these four sets, for each p and for w, we have the four duals, which are also closed under formulation.

This is a consequence of the fact that s1 Pis closed under formulation, and of the preceding assertion. s#OJ. 91 2 , since the two systems + - + and - + + each give it the value - while there is no rank for which both have -. If rx is positive, then rx belongs to s1 P if and only if every reduction of arity ~p belongs to s1 w· t> By 2. 8, it is sufficient to assume that rx does not belong to s1 P and to find a reduction of arity ~ p which does not belong to s1OJ" Take h ~ p systems of values which give rx the value -, such that for each rank the value + appears at least once.

The connection IX is said to be positive if it is i-positive for every i ~the arity. In other words, the value taken by IX remains the same or varies in the same sense as do the values of the system. The connections +, -, CONNECTIONS AND CONNECTIVE FORMULAS 35 1\ and V are positive; negation I is not; implication = is 2-positive, but not 1-positive. A connection is positive if and only if it is one of the connections +, - or is representable by a formula in /\, V. Indeed, let IX be positive, and consider every system of h values {where h = arity of IX) which gives IX the value +.