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