Σ₁ compactness for next admissible sets

Extract

Let σ be any sequence B₀, B₁ …, B_n, … of transitive sets closed under pairs with for each n. In this paper we show that the smallest admissible set A_σ with σ ∈ A_σ is Σ₁ compact. Thus we have an entirely new class of explicitly describable uncountable Σ₁ compact sets.

The search for uncountable Σ₁ compact languages goes back to Hanf's negative results on compact cardinals [7]. Barwise first showed that all countable admissible sets were Σ₁ compact [1] and then went on to give a characterization of the Σ₁ compact sets in terms of strict reflection [2]. While his characterization has been of interest in understanding the Σ₁ compactness phenomenon it has led to the identification of only one class of uncountable Σ₁ compact sets. In particular, Barwise showed [2], using the above notation, that if ⋃_nB_n is power set admissible it satisfies the strict reflection principle and hence is Σ₁ compact. (This result was obtained independently by Karp using algebraic methods [9].)

In proving our compactness theorem we follow Makkai's approach to the Barwise Compactness Theorem [12] and use a modified version of Smullyan's abstract consistency property [14]. A direct generalization of Makkai's method to the cofinality ω case yields a proof of the Barwise-Karp result mentioned above [6]. In order to obtain our new result we depart from the usual definition of language and use instead the indexed languages of Karp [9] in which a conjunction is considered to operate on a function whose range is a set of formulas rather than on a set of formulas itself.