Model-completeness in a first order language with a generalized quantifier
Shlomo Vinner · Pacific Journal of Mathematics · 1975
The concept of Model-Completeness is defined in a first order language with a generalized quantifier.A necessary and sufficient condition is given for that Model-Completeness and its relation to categoricity is discussed.Some results of this paper were obtained in the author's thesis [12] and were announced in [11], They, together with other results of [12] were improved independently by the author and by S. Shelah.A suggestion of S. Shelah made some proofs simpler and due to it, better results were obtained in Theorem 1.5.The author wishes to thank S. Shelah for his remarks.Let L be a first order language with equality and let L(Q) be the language obtained from L by adding a new quantifier Q.Let α, β denote infinite cardinals.We define α-satisfaction for L(Q) by interpreting Q as "there exist at least a elements".If a sentence φ of L(Q) is a-satisfied in a model 91 for L we write 9ll= α φ and we say that 91 is an a-model for φ.Let 31,93 be two models for L, |9l|^α and 91 C 93. Write 21 [α,, ,αj iff 93h α φ[α,, ,α n ].Let T be an ordinary first order theory (namely a theory in L) that has infinite models.Define T(Q) = TU{Qx[x =x]}.Call T a-model-complete if for every 21,93 which are α-models forT(Q) and 91 C 93 also 9l H o is given in section 1.Let T be as before.Define T(a) = {φ: φ is a sentence in L(Q) and for every 91, if 9lh α T(Q) then 9ίh α φ}.Call T α-complete if for every sentence φ in L(Q) either φ G T(a) ory φ G Γ(α).In §2, it is shown that if T is categorical in one uncountable power, it is incomplete and for every a^H*: T(a)=T(Ho).If T is also modelcomplete (in the usual sense) then it is α-model-complete for every α^No and T(a) is decidable provided T is axiomatic.1. ci-Model-Completeness.DEFINITION 1.1.Let φ(x,x u,jc m ) be a formula in L such that x, jt,, ,jc m are exactly all its free variables.Let 91 be a model for L and let α,, ,α m be elements in 91.