The Collected Works of J. Richard Büchi pp 437-457 | Cite as

# Transfinite Automata Recursions and Weak Second Order Theory of Ordinals

## Abstract

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*, <].

## Keywords

Finite Subset Order Theory Finite Automaton Recursive Operator Predicate Variable## Preview

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## References

- [1]J. R. Büchi, Weak second order arithmetic and finite automata.
*Zeitschrift für Math. Log. und Grundl. der Math.***6**(1960), pp. 66–92.MATHCrossRefGoogle Scholar - [2]J. R. Büchi, On a decision method in restricted second order arithmetic.
*Logic, Method. and Phil, of Sc, Proc.*1960*Int. Congress*, Stanford Univ. Press, 1962.Google Scholar - [3]S. Feferman, Some recent work of Ehrenfeucht and Fraïssé. Summer Institute for Symbolic Logic, Cornell Univ. 1957, Commun. Research Div., Institute for Defense Analysis, 1960, pp. 201–209.Google Scholar
- [4]S. Feferman and R. L. Vaught, The first order properties of products of algebraic systems.
*Fund. Math.***47**(1959), pp. 57–103.MathSciNetMATHGoogle Scholar - [5]
- [6]A. Ehrenfeucht, Application of games to some problems of mathematical logic.
*Bull, de l’Acad. Pol. Sci.***5**(1957), pp. 35–37.MathSciNetMATHGoogle Scholar