Computability and Logic

252 NORMAL FORMS
as ‘set (of natural numbers)’ is true just of the sets (of natural numbers) in the
domains.
19.10 Example (The ‘Skolem paradox’). There is no denying that the state of affairs
thought to be paradoxical does obtain. In order to see how it arises, we first need an
alternative account of what it is for a set E of sets of natural numbers to be enumerable,
and for this we need to use the coding of an ordered pair (m, n) of natural numbers by a
single number J(m, n), as described in section 1.2. We call a set w of natural numbers an
enumerator of a set E of sets of natural numbers if
∀z(z is a set of natural numbers & z is in E →
∃x(x is a natural number &
∀z(∀y(y is a natural number → (y is in z ↔ J(x, y) isinw)))).
The fact about enumerators and enumerability we need is that a set E of sets of natural
numbers is enumerable if and only if E has an enumerator.
[The reason: suppose E is enumerable. Let e 0 , e 1 , e 2 , ...be an enumeration of sets of
natural numbers that contains all the members of E, and perhaps other sets of natural
numbers also. Then the set of numbers J(x, y) such that y is in e x is an enumerator of E.
Conversely, if w is an enumerator of E, then letting e x be the set of those numbers y such
that J(x, y) isinw, we get an enumeration e 0 , e 1 , e 2 , ...that contains all members of E,
and E is enumerable.]
We want now to look at a language and some of its interpretations. The language contains
just the following: constants 0, 1, 2, ..., two one-place predicates N and S, a two-place
predicate ∈, and a two-place function symbol J. An interpretation I of the kind we are
interested in will have as the elements of its domain all the natural numbers, some or all of
the sets of natural numbers, and nothing else. The denotations of 0, 1, 2, and so on will be
the numbers 0, 1, 2, and so on. The denotation of N will be the set of all natural numbers,
and of S will be the set of all sets of natural numbers in the domain; while the denotation
of ∈ will be the relation of membership between numbers and sets of numbers. Finally, the
denotation of J will be the function J, extended to give some arbitrary value—say 17—for
arguments that are not both numbers (that is, one or both of which are sets). Among such
interpretations, the standard interpretation J will be the one in which the domain contains
all sets of natural numbers.
Consider the sentence ∼∃wF(w) where F(w) is the formula
Sw & ∀z(Sz →∃x(Nx & ∀y(N y → (y ∈ z ↔ J(x, y) ∈ w)))).
In each of the interpretations I that concern us, ∼∃wF(w) will have a truth value. It will be
true in I if and only if there is set in the domain of I that is an enumerator of the set of all
sets of numbers that are in the domain of I , as can be seen by compairing the formula F(w)
with the definition of enumerator above. We cannot say, more simply, that the sentence
is true in the interpretation if and only if there is no enumerator of the set of all sets of
numbers in its domain, because the quantifier ∃w only ‘ranges over’ or ‘refers to’ sets in
the domain.
There is, as we know, no enumerator of the set of all sets of numbers, so the sentence
∼∃wF(w) is true in the standard interpetation J , and can be said to mean ‘nonenumerably
many sets of numbers exist’ when interpreted ‘over’ J , since it then denies that there is an