Uncountable cardinals have the same monadic ∀11positive theory over large sets

Athanassios Tzouvaras
2004 Fundamenta Mathematicae  
We show that uncountable cardinals are indistinguishable by sentences of the monadic second-order language of order of the form (∀X)φ(X) and (∃X)φ(X), for φ positive in X and containing no set-quantifiers, when the set variables range over large (= cofinal) subsets of the cardinals. This strengthens the result of Doner-Mostowski-Tarski [3] that (κ, ∈), (λ, ∈) are elementarily equivalent when κ, λ are uncountable. It follows that we can consistently postulate that the structures (2 κ , [2 κ ] >κ
more » ... es (2 κ , [2 κ ] >κ , <), (2 λ , [2 λ ] >λ , <) are indistinguishable with respect to ∀ 1 1 positive sentences. A consequence of this postulate is that 2 κ = κ + iff 2 λ = λ + for all infinite κ, λ. Moreover, if measurable cardinals do not exist, GCH is true. Preliminaries. Let L = {≺} be the first-order language of order with equality. The following is a special case of a much more general result proved in [3]: Theorem 1.1. For any uncountable cardinals κ, λ (κ, <) ≡ L (λ, <), where < is the natural ordering of ordinals (i.e., < = ∈). Proof. The reader is warned that there is a mistake in the definition of congruence modulo ω ω given in [3, p. 51], as Professor Doner kindly informed me. The correct definition, which can be found e.g. in [2] , is as follows: The ordinals α, β are congruent modulo ω ω if there are ξ, η and δ < ω ω such that α = ω ω · ξ + δ, β = ω ω · η + δ and either ξ = η = 0 or both ξ = 0 and η = 0. Since for all cardinals κ, λ > ω, κ = ω ω · κ and λ = ω ω · λ, it follows that κ, λ are congruent modulo ω ω . So the claim follows from Corollary 44 of [3]. Henceforth we write (κ, <) ≡ (λ, <) instead of (κ, <) ≡ L (λ, <). Let x, y, z, . . . range over the individual variables of L. The monadic (secondorder ) extension of L, denoted by L mon , is L augmented with ∈ and set variables X, Y, Z, . . . . Hence L mon has x ∈ X as additional atoms. The stan-2000 Mathematics Subject Classification: 03E10, 03E70. Key words and phrases: monadic second-order language of order, positive formula.
doi:10.4064/fm181-2-3 fatcat:2zsb5csljzczpen7ttizdgsoji