Notes on Forcing

1. Extensionality (definition of =): ∀x∀y (x = y iff ∀z z ∈ x iff z ∈ y). 2. Pairing (pairs exist): ∀x∀y∃z∀w (w ∈ z iff (w = x or w = y)). 3. Union (unions exist): ∀x∃z∀w (w ∈ z iff ∃y (w ∈ y and y ∈ x)). 4. Power set (power sets exist): ∀x∃z∀w(w ∈ z iff w ⊂ x). 5. Regularity (∀x (x,∈) has a minimal element): ∀x∃y (y ∈ x and y ∩ x = ∅). 6. Infinity (there is an infinite set): ∃x(x 6= ∅ and ∀y ( if y ∈ x then y ∪ {y} ∈ x)). 7. Choice (choice functions exist): ∀x 6= ∅ (if ∅ / ∈ x then ∃f (f a function, dom f = x and ∀y ∈ x f(y) ∈ y). 8. Comprehension Schema (restricted formulas define sets): Let φ be a formula. ∀x∃z (y ∈ z iff (y ∈ x and φ(y)). 9. Replacement Schema (ranges of functions exist): Let φ be a formula. ∀x(if ∀y ∈ x (φ(y, z) and φ(y, w)⇒ z = w) then ∃u ∀v (v ∈ u iff ∃y ∈ x φ(y, v))).1