Relative strength of Malitz quantifiers.
Steven Garavaglia · Notre Dame Journal of Formal Logic · 1978
In this paper I will solve a problem concerning Malitz quantifiers which was posed in [l].Before stating this problem I will introduce some notation which will be used in the proof.If X is a set then c(X) is the cardinality of X and [X] n is the set of w-element subsets of X. S n is the set of permutations of {l, 2, . .., n}.If 3ί is a structure 1311 denotes the domain of 31.If ^ is a first-order language, -C(|3ί|) is the result of adjoining to ω by exhibiting two structures 31 and 53 of the same similarity type such that 31 and 53 satisfy the same sentences in jC a but do not satisfy the same sentences in Jtf* 1 .1Let n be any fixed positive integer and let a be any fixed uncountable cardinal.£ will be a first-order language with equality whose only nonlogical symbol is an (n + l)-ary predicate symbol R.Definition 1: If 31 is an ^-structure, γ is a finite subset of 1311, σe S w+1 , and t l9 . .., t n+1 eγ u {#i, . .., Xk, 3>i, .., yjthen σ(t l9 . .., t n+ι ) is the (n + 1)tuple U σ(l) , . .., t σ(n+ι) )and σR(t l9 . .., 4+i) is the *C( 1311)-formula R(t σ{ι) , ., ίτ(«+i))1.For α = coi, this result was obtained independently by Andreas Baudisch under the assumption Oωi