Zermelo-Fraenkel Set Theory

CHAPTER 2. AXIOMS 9
2. (x, y) = (a, b) ⇒ x = a ∧ y = b.
9 ♣ Show that if {{x}, {∅, y}} = {{a}, {∅, b}}, then x = a and y = b. Thus, defining
(x, y) = def {{x}, {∅, y}} would also be a good ordered pair.
10 ♣ Show: if a ≠ ∅, then ⋂ a = def {x | ∀y ∈a(x ∈ y)} exists as a set. This even holds
when a is a proper class.
What about ⋂ ∅?
11 ♣ Show, using the Foundation Axiom:
1. No set a is an element of itself. (Hint: consider {a}.)
2. There are no sets a 1 , a 2 , a 3 , a 4 such that a 1 ∈ a 2 ∈ a 3 ∈ a 4 ∈ a 1 .
12 ♣ Suppose that ∀x ∈ a ∃!y Φ holds. Show that {(x, y) | x ∈ a ∧ Φ} is a set (and,
hence, a function).
(Hence: if H is an operation on the set a, a function f on a is defined by putting, for
x ∈ a: f(x) = def H(x).)
13 ♣ Why do you think the variable b is not allowed to occur free in Φ in Separationand
Substitution Axiom?
14 ♣ Suppose that the operation F maps the class K injectively into the class L. Assume
that K is a proper class. Show that L also is proper.
In particular, if K ⊂ L and K is proper, then so is L. Example: R = {x | x ∉ x} ⊂
{x | x = x} = V.
15 ♣ Show that the following classes are proper:
• R = def {x | x ∉ x}, the Russell class of sets that do not belong to themselves,
• V = {x | x = x}, the class of all sets,
• {{x, y} | x ≠ y}, the class of all two-element sets,
• Q n = def {x | ¬∃x 1 , . . . , x n (x 1 ∈ x ∧ x 2 ∈ x 1 ∧ · · · ∧ x n ∈ x n−1 ∧ x ∈ x n )}, Quine’s
classes,
• G = def {x | ∀a [x ∈ a ⇒ ∃y ∈ a(y ∩ a = ∅)]}, the class of grounded sets.
Note that G ⊂ Q n ⊂ R ⊂ V.
N.B.: The Axiom of Foundation says precisely that V = G. Indeed,
means that
which amounts to
which is the Foundation Axiom.
V = G, or: ∀b(b ∈ G)
∀b∀a (b ∈ a ⇒ ∃x ∈ a(x ∩ a = ∅)) ,
∀a (∃b(b ∈ a) ⇒ ∃x ∈ a(x ∩ a = ∅)) ,