Logic as a tool a guide to formal logical reasoning ( PDFDrive ) 257

Applications: Mathematical Proofs and Automated Reasoning

233

As an exercise, using EXT prove that for any sets x there is only one powerset of x,
denoted P(x).
A natural operation on sets, definable in LZF , is the successor set operation which,
applied to any set x, produces the set

x := x ∪ {x}.
The infinity axiom states the existence of a set containing the empty set and closed under
the successor set operation:
INF: ∃x(∅ ∈ x ∧ ∀y (y ∈ x → y ∈ x)).
As an exercise, using the other axioms (some are yet to come), prove that x = x for any
set x. It then follows that any set x satisfying the formula above must indeed be infinite.
“Infinite” here means that it is bijective (see Section 5.2.3) with a proper subset of itself,
in this case the subset obtained by removing ∅.
We next have the regularity axiom or the foundation axiom:
REG: ∀x(x = ∅ → ∃y ∀z (z ∈ y → z ∈
/ x))
which states that every non-empty set x has a disjoint element (i.e., an element having no
common elements with x), therefore preventing the existence of an infinite descending
chain of set memberships. In particular, this axiom forbids any set to be an element of
itself (show this as an exercise).
The next axiom of ZF is actually a scheme, called the Axiom Scheme of Separation
or Axiom Scheme of Restricted Comprehension: for any formula φ(¯
x, u) from
¯ is a tuple of free variables x1 , . . . , xn (to be treated as parameters),
LZF , where x
it states:

x, u)).
SEP: ∀y ∀x1 . . . ∀xn ∃z ∀u(u ∈ z ↔ u ∈ y ∧ φ(¯
Intuitively, this axiom states that, given any set y , for any fixed values of the parameters
x¯ there exists a set consisting of exactly those elements of u that satisfy the property of
x, u).
sets defined by the formula φ(¯
The last axiom in ZF is again a scheme, called the Axiom Scheme of Replacement:
¯ is a tuple of free variables x1 , . . . , xn (to
for any formula φ(¯
x, y, z ) from LZF , where x
be treated as parameters), it states:
REP:

x, y, z )) →
∀x1 . . . ∀xn ∀u(∀y (y ∈ u → ∃!zφ(¯
∃v ∀z (z ∈ v ↔ ∃y ∈ uφ(¯
x, y, z ))).
¯, the formula
Intuitively, this axiom states that if, for any fixed values of the parameters x
φ(¯
x, y, z ) defines a functional relation fφ,x¯ (y, z ), then the image of any set u under that
relation is again a set (v).
Another important axiom was added to ZF later, namely the axiom of choice, stating
that for every set x of non-empty sets there exists a “set of representatives” of these sets,
which has exactly one element in common with every element of x:
AC: ∀x(∀y (y ∈ x → y = ∅) → ∃z ∀y (y ∈ x → ∃!u(u ∈ y ∧ y ∈ z ))).