Computability and Logic

282 SECOND-ORDER LOGIC
is true. But N0 is true, since the individual a that is the denotation of 0 is in the set
A that is the denotation of N, and ∀x(Nx → Nx ′ ) is true, since if any individual is in
A, so is the value obtained when the function f that is the denotation of ′ is applied
to that individual as argument. Hence ∀xNx must be true, and this means that every
individual in the domain is in A, so the domain, being just A, is enumerable.
Conversely, suppose that the domain of an interpretation M is enumerable. Fix
an enumeration of its elements: m 0 , m 1 , m 2 , and so on. Let a be m 0 , and let f be the
function that given m i as argument yields m i+1 as value, and add a constant 0 and a
one-place function symbol ′ to denote a and f . Given any subset A of the domain,
suppose we add a one-place predicate N to denote A. Then if N0 is true, a = m 0
must belong to A, and if ∀x(Nx → Nx ′ ) is true, then whenever m i belongs to A,
f (m i ) = m i+1 must belong to A. So if both are true, every element m 0 , m 1 , m 2 ,...
of the domain must belong to A, and therefore ∀xNx is true. Thus
(N0 & ∀x(Nx → Nx ′ )) →∀xNx
is true if N is taken to denote A, and therefore A satisfies
and since this is true for any A,
is true, and therefore
or Enum is true in M.
(X0 & ∀x(Xx → Xx ′ )) →∀xXx
∀X((X0 & ∀x(Xx → Xx ′ )) →∀xXx)
∃z∃u∀X((Xz& ∀x(Xx → Xu(x))) →∀xXx)
22.4 Example (The ‘axiom’ of infinity). Let Inf be the sentence
∃z∃u(∀xz≠ u(x)&∀x∀y(u(x) = u(y) → x = y)).
Then Inf is true in an interpretation if and only if its domain is infinite. The proof is left to
the reader.
22.5 Proposition. The downward and upward Löwenheim–Skolem theorems both fail
for second-order logic.
Proof: Inf & ∼Enum and Inf & Enum are both second-order sentences having
infinite but no finite models. The former has only nonenumerable models, contrary
to the downward Löwenheim–Skolem theorem; the latter only denumerable models,
contrary to the upward Löwenheim–Skolem theorem.
It is an immediate consequence of the downward and upward Löwenheim–Skolem
theorems that if a first-order sentence or set of such sentences has an infinite model,
then it has nonisomorphic infinite models. Even this corollary of the Löwenheim–
Skolem theorems fails for second-order logic, as the next example shows.