Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence

Notes on Cardinals That Are Characterizable by a Complete (Scott) Sentence

Building on [4], [8] and [9] we study which cardinals are characterizable by a Scott sentence, in the sense that φM characterizes κ, if it has a model of size κ, but not of κ. We show that if אα is characterizable by a Scott sentence and β < ω1, then אα+β is characterizable by a Scott sentence. If 0 < γ < ω1, then the same is true for 2אα+γ . Also, אא0 α is characterizable by a Scott sentence. ...

Second-order Characterizable Cardinals and Ordinals Second-order Characterizable Cardinals and Ordinals

The notions of finite and infinite second-order characterizability of cardinal and ordinal numbers are developed. Several known results for the case of finite characterizability are extended to infinite characterizability, and investigations of the secondorder theory of ordinals lead to some observations about the Fraenkel-Carnap question for well-orders and about the relationship between ordin...

Notes on ordinals and cardinals

Notes on the theory of cardinals

Lemma 3 On a well-ordered set, a strictly monotone function is extensive. Proof. Assume that S = {x ∈ X | x > f(x)} 6= ∅. Then, let a be the least element of S (X is well ordered). Hence, a > f(a). By strict monotony, f(a) > f(f(a)). Therefore, f(a) ∈ S and a ≤ f(a). Contradiction. Lemma 4 If f is a monotone injection from α to β, then α ≤ β. Proof. Since α is well ordered, f is extensive. Henc...

Scott Wallsten talk notes