This textbook gives an introduction to axiomatic set theory and examines the prominent questions that are relevant in current research in a manner that is accessible to students. Its main theme is the interplay of large cardinals, inner models, forcing and descriptive set theory.

∼ {(dom(π ), x)}. Then σ ≤ F (with dom(σ ) = dom(π ) + 1), π σ. Contradiction! If f : α → A, where α is an ordinal (or α = OR), then f is also called a sequence and we sometimes write ( f (ξ ) : ξ < α) instead of f . 3.3 Problems 3.1 Use the recursion theorem 3.13 to show that there is a sequence (Vα : α ≤ OR) which satisfies (3.1). Show that every Vα is transitive and that Vβ ∧ Vα for β ⇒ α. 30 3 Ordinals Show that V0 = →, Vα+1 = P(Vα ) for every α, and Vλ = limit ordinal λ. α<λ Vα for

([ f ]) = ρU ( f )(χ ) for all f : β ≤ V. (4.6) This is because ρU ( f )(χ ) may be written in a cumbersome way as σ −1 ([c f ]) σ −1 ([id]) , which with the help of Ło´s’ Theorem is easily seen to be equal to σ −1 ([ f ]). In particular, N = ult(V ; U ) = {ρU ( f )(χ ) : f : β ≤ V } . (4.7) χ is often called the generator of U . It is also easy to see that for X ⊂ β, X ∼ U ≡⇒ χ ∼ ρ(X ). (4.8) Now let ρ : V ≤ M be any elementary embedding with critical point β, where M is an inner model.

Let U be a < β-closed normal ultrafilter on β, and let (X s : s ∼ [β]<α ) be a family such that X s ∼ U for every s ∼ [β]<α . Let us define Δs∼[β]<α X s = {τ ∼ β : τ ∼ X s }. s∼[τ ]<α 4.4 Problems 65 Show that Δs∼[β]<α X s ∼ U . 4.27. Let β be a measurable cardinal. (a) Let U be a measure on β. Show that β +ult(V ;U ) = β +V and that 2β < ρUV (β) < ((2β )+ )V . [Hint: If Φ < ρUV (β), then Φ is represented by some f : β ≤ β, and there are 2β functions from β to β.] Conclude that U∼ / ult(V ;

cardinal ν with cf(ν) > Φ (cf. Problem 6.9). We now consider variants of Cohen forcing for cardinals above Φ. Definition 6.34 Let ν ≡ Φ be a cardinal. Let Cν = <ν ν, i.e., the set of all f such that there is some Δ < ν with f : Δ ⇒ ν. For p, q ∈ Cν , let p → q iff p ∅ q (iff ⊂Δ p Δ = q). The partial order (Cν , →) is called Cohen forcing at ν. Of course, CΦ = C. If ν <ν = ν (which is true for ν = Φ and only possible for regular ν), then Card(Cν ) = ν, so that in this case forcing with Cν

Corollary 7.21.) 1 -formula is equivalent to a δ HC -formula in 7.11. Let n < Φ. Show that every δn+1 n 1 (z), where z ∼ Φ Φ. There is then the following sense. Let A ∈ Φ Φ be δn+1 a δn -formula ζ(v, w) such that for all x ∼ Φ Φ, x ∼ A ∩∅ HC |= ζ(x, z). Conclude that if z ∼ Φ Φ, ζ(v) is δ1 , and V |= ζ(z), then L[z] |= ζ(z). [Hint. Corollary 7.21.] Show also that it is consistent to have some a ∼ HC and δ1 -formula ζ(v) such that V |= ζ(a), but L[a] |= ¬ζ(a). [Hint. a = Φ1L , ζ(v) ˙ “v is