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

k 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, denotedPx.. As an exercise, using the

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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, denotedP(x)

A natural operation on sets, definable inLZF, is the successor set operation which,

applied to any setx, 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 thatx  = xfor any setx It then follows that any setxsatisfying 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 setxhas a disjoint element (i.e., an element having no common elements withx), 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

LZF, where ¯x is a tuple of free variables x1, , x n (to be treated as parameters),

it states:

SEP: ∀ y ∀ x1 ∀ x n∃ z ∀ u(u ∈ z ↔ u ∈ y ∧ φ(¯x, u))

Intuitively, this axiom states that, given any sety, for any fixed values of the parameters

¯

xthere exists a set consisting of exactly those elements ofuthat satisfy the property of sets defined by the formulaφ(¯x, u)

The last axiom in ZF is again a scheme, called the Axiom Scheme of Replacement:

for any formulaφ(¯x, y, z) fromLZF, where ¯xis a tuple of free variablesx1, , x n(to

be treated as parameters), it states:

REP:

∀ x1 ∀ x n ∀ u(∀ y(y ∈ u → ∃! zφ(¯x, y, z))→

∃ v ∀ z(z ∈ v ↔ ∃ y ∈ uφ(¯x, y, z))).

Intuitively, this axiom states that if, for any fixed values of the parameters ¯x, the formula

φ(¯x, y, z) defines a functional relationf φ, x¯(y, z), then the image of any setuunder that relation is again a set (v)

Another important axiom was added to ZF later, namely the axiom of choice, stating

that for every setxof non-empty sets there exists a “set of representatives” of these sets, which has exactly one element in common with every element ofx:

AC: ∀ x(∀ y(y ∈ x → y = ∅) → ∃ z ∀ y(y ∈ x → ∃! u(u ∈ y ∧ y ∈ z)))