Transfinite Automata Recursions and Weak Second Order Theory of Ordinals
We identify the ordinal α with the set of all ordinals x < α. The weak second order theory of [α, <] is the interpreted formalism WST [α, <] which makes use of: (a) the propositional connectives with usual interpretation; (b) a binary relation letter < interpreted as ordering relation on α; (c) individual variables t,x,y,z,..., ranging over α; (d) monadic predicate variables i,j,s,r,..., ranging over finite subsets of α; (e) quantifiers ∀, ∃ for both types of variables. The purpose of this paper is to provide, for any ordinal α, a clear understanding of which relations R on finite subsets of α can be defined by formulas Σ(i 1 ,...,i n ) of WST[α, <]. In addition we obtain a decision method for truth of sentences Σ in WST[α, <].
KeywordsFinite Subset Order Theory Finite Automaton Recursive Operator Predicate Variable
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