formulas into formulas (induction on the length). Iftis free for each free occur- rence ofxin Pwe simply say thattis free forxin P.
1.7. We shall start working with Definitions 1.5 and 1.6 in the next section.
Here we shall only give some intuitive explanations.
Definition 1.5 allows us to introduce the important class ofclosedformulas.
By definition, this consists of formulas without free variables. (They are also called sentences.) The intuitive meaning of the concept of a closed formula is as follows. A closed formula corresponds to an assertion that is completely deter- mined (in particular, regarding truth or falsity); indeterminate objects of the theory are mentioned only in the context “all objectsxsatisfy the condition. . .”
or “there exists an objecty with the property. . . .” Conversely, a formula that is not closed, such as x∈ y or ∃x(x∈ y), may be true or false depending on what sets are being designated by the names xand y (for the first) or by the namey(for the second). Here truth or falsity is understood to mean for a fixed interpretation of the language, as will be explained in§2.
In particular, Definition 1.6 gives the rules of hygiene for changing notation.
If we want to call an indeterminate object x by another name y in a given formula, we must be sure that x does not appear in the parts of the formula where this name y is already being used to denote anarbitrary indeterminate object (after a quantifier). In other words,y must be free forx. Moreover, if we want to say that xis obtained from certain operations on other indeterminate objects (x= a term containingy1, . . . , yn), then the variablesy1, . . . , yn must not be bound.
There is a close parallel to these rules in the language of analysis: in- stead ofx
1 f(y)dy we may confidently writex
1 f(z)dzbut we must not write x
1 f(x)dx; the variabley is bound, in the scope of
f(y)dy.
2.2. Primary mappings
(a) An interpretation of the constants is a map from the set of symbols for constants (in the alphabet ofL) toM that takes a symbolctoφ(c)∈M.
(b) An interpretation of the operations is a map from the set of symbols for operations (in the alphabet of L) that takes a symbol f of degree r to a functionφ(f) onM × · · · ×M =Mrwith values inM.
(c) An interpretation of the relations is a map from the set of symbols for relations (in the alphabet of L) that takes a symbol p of degree r to a subsetφ(p)⊂Mr.
Secondary mappings. Intuitively, we would like to interpret variables as names for the “generic element” of the set M, which can be given specific values inM. We would like to interpret the termf(x1, . . . , xr) as a functionφ(f) ofr arguments that run through values inM, and so on.
In order to give a precise definition, we introduce the interpretation class M:
M = the set of all maps toMfrom the set of symbols for variables in the alphabet ofL.
Thus, every point ξ ∈ M correlates to any variable x a value φ(x)(ξ) ∈ M, which we shall usually denote simply byxξ. This allows us to consider variables as functions onM with values in M. More generally:
2.3. The interpretation of terms correlates to each termta functionφ(t) onM with values in M. This correspondence is defined inductively by the following compatibilities:
(a) Ifc is a constant, thenφ(c) is the constant function whose value is defined by the primary mapping.
(b) Ifxis a variable, thenφ(x) isφ(x)(ξ) as a function ofξ.
(c) Ift=f(t1, . . . , tr), then for allξ∈M,
φ(t)(ξ) =φ(f)(φ(t1)(ξ), . . . , φ(tr)(ξ)),
where the φ(ti)(ξ) are defined by the induction assumption, and φ(f) : Mr → M is given by the primary mapping. Instead of φ(t)(ξ) we shall sometimes write simplytξ.
2.4. Interpretation of atomic formulas. An interpretation φ assigns to every formula P in L a truth function|P|φ. This is a function on the interpretation class M that takes only the values 0 (“false”) and 1 (“true”). It is defined for atomic formulas as follows:
|p(t1, . . . , tr)|φ(ξ) =
1, if tξ1, . . . , tξr ∈φ(p), 0, otherwise.
Intuitively, a statementpabout the namest1, . . . , trfor objects inM becomes true if the objects named byt1, . . . , tr satisfy the relation named byp.
2.5. Interpretation of formulas. The truth function for nonatomic formulas is defined inductively by means of the following relations (for brevity, we have omitted parentheses and explicit mention ofφandξ):
|P ⇔Q|=|PQ|+ (1− |P|)(1− |Q|) :
P ⇔Qis true when eitherP andQare both true orP andQare both false.
|P ⇒Q|= 1− |P|+|PQ|: P ⇔Qis false only whenP is true andQis false.
|P∨Q|= max(|P|,|Q|) : P∨Qis false only whenP andQare both false.
|P∧Q|= min(|P|,|Q|) : P∧Qis true only when P andQare both true.
|¬P|= 1− |P|:
¬P is false only whenP is true.
Finally, we must describe what happens when quantifiers are introduced.
Suppose that ξ∈M andx is a variable. By avariation of ξ along xwe mean any pointξ ∈M for whichyξ =yξ wheneveryis a variable different fromx.
Then
|∀xP|(ξ) = min
ξ |P|(ξ),
|∃xP|(ξ) = max
ξ |P|(ξ), where ξ runs through all variations ofξalongx.
A formulaP is calledφ-true if|P|φ(ξ) = 1 for allξ∈M. The interpretation φ (or M) is called a model for a set of formulas E if all the elements ofE are φ-true.
2.6.Example: Standard Interpretation ofL1Ar. This is the interpretation in the set N of nonnegative integers, in which ¯0,¯1 are interpreted as 0, 1, respectively, and +,·, = are interpreted as addition, multiplication, and equal- ity, respectively.
2.7.Example: Standard Interpretation of L1Set. This is the interpreta- tion in the von Neumann universe V, in which∅ is interpreted as the empty set, ∈is interpreted as the relation “is an element in,” and = is interpreted as equality.
All of the examples of translations in Chapter I were based on these stan- dard interpretations. The relationship between those examples and the above definitions is as follows. Let Π(x, y, z) be a statement in argot about the
indeterminate sets x, y, z in V; and let P(x, y, z) be a translation of Π into the language L1Set. Then for any point ξ interpreting x, y, z as the names of sets xξ, yξ, zξ in the von Neumann universe, we have:
Π(xξ, yξ, zξ) is true⇔ |P(x, y, z)|(ξ) = 1.
Thus, every formula expresses, or defines, a property of objects in the interpre- tation set:
2.8. Definition. A setS ⊂Mr, r1, is calledφ-definable (by the formulaP in Lwith the interpretationφ) if there exist variablesx1, . . . , xrsuch that
|P|φ(ξ) = 1⇔
xξ1, . . . , xξr
∈S for allξin M.
One of the most important problems concerning formal languages is to understand the structure of the sets of
φ-true formulas inL;
φ-definable sets in
r1
Mr.
2.9. Example. The sets definable by means of L1Ar with the standard inter- pretation constitute the smallest class of sets in
r1Nr that (a) contains all sets of the form
{ k1, . . . , kr|F(k1, . . . , kr) = 0} ⊂Nr, whereF runs through all polynomials with integral coefficients;
(b) is closed relative to finite intersections, unions, and complements (in the appropriateNr);
(c) is closed relative to the projections pri:Nr→Nr−1: pri k1, . . . , kr= k1, . . . , ki−1, ki+1, . . . , kr.
In fact, sets of type (a) are defined by atomic formulas of the formtF1 =tF2, wheretF1 is a term corresponding to the sum of the monomials inF with posi- tive coefficients, andtF2 corresponds to the sum of the monomials with negative coefficients. Further, if S1, S2⊂Nr are definable by formulasP1, P2 (with the same variables), then S1∩S2 is definable byP1∧P2, S1∪S2 is definable by P1∨P2, and Nr \S1 is definable by ¬P1. Finally, the set pri(S1) is defin- able by the formula ∃xi(P1). The connectives⇒ and ⇔ and the quantifier ∀ give nothing new, since without changing the set being defined, we may replace them by combinations of the logical operations already discussed: ∀x may be replaced by¬∃x¬, and so on.
This first description ofarithmetical sets, i.e., L1Ar-definable sets, will be greatly amplified in the second and third parts of the book. At this point it is not immediately clear how to develop the subtler properties of definability,
such as the definability of the set of prime numbers in N (see Example 3.14 in Chapter I), the definability of the set of partial fractions in the continued fraction expansion of √3
2, or the definability of the set of pairs { i, ith digit in the decimal expansion ofπ} ⊂N2.
However, as we shall see in§11 and in Chapter VII, the “G¨odel numbers of the true formulas of arithmetic” form a much more complicated set, and this set is not definable.
We now give several simple technical results.
2.10. Proposition. Let P be a formula in L, φ an interpretation in M, and ξ, ξ ∈M. Suppose thatxξ coincides withxξ for all variables xoccurring freely in P. Then |P|φ(ξ) =|P|φ(ξ).
2.11.Corollary.In any interpretation the closed formulas P have well-defined truth values: |P|φ(ξ)does not depend on(ξ).
Proof.
(a) Let tbe a term, and suppose that for any variablexin twe havexξ =xξ. Then Lemma 1.4 and induction on the length oft givetξ =tξ.
(b) Assertion 2.10 holds for atomic formulas P of the form p(t1, . . . , tr).
In fact,
|P|(ξ) =
1, iftξ1, . . . , tξr ∈φ(P), 0, otherwise,
and similarly for|P|(ξ). But if ξand ξ coincide on all the variables in P (all of which occur freely), then a fortiori they coincide on all the variables in ti, and by part (a), we havetξi =tξi, i= 1, . . . , r. Therefore |P|(ξ) =|P|(ξ).
(c) We now use induction on the total number of connectives and quantifiers inP. IfP has the form¬QorQ1∗Q2, then 2.10 forPfollows trivially from 2.10 forQ, Q1, Q2. Now suppose thatP has the form∀x(Q), and that 2.10 holds for Q. (The case ∃x(Q) can be treated analogously or can be reduced to the case
∀xby replacing∃xby¬∀x¬.) By definition, we have
|∀xQ|(ξ) =
1, if|Q|(η) = 1 for variationsηofξ alongx, 0, otherwise;
|∀xQ|(ξ) =
1, if|Q|(η) = 1 for variations η ofξalongx, 0, otherwise.
On the right we may letη and η vary in addition on all variables that do not occur freely in Q. The assertions after the word “if” remain true or false in this wider range of values if they were true or false before, by the induction hypothesis on Q. But then η and η run through the same values, because ξ
and ξ differ only on variables that do not occur freely in Q, and on x. The
proposition is proved.
The following almost obvious fact is the basis for many phenomena that attest to the inadequacy of formal languages for completely describing intuitive concepts (see “Skolem’s paradox” below):
2.12. Proposition. The cardinality of the class of φ-definable sets does not exceed
card(alphabet ofL) +ℵ0.
Here and below, by “card(alphabet of L)” we mean the cardinality of the al- phabet of L without the set of variables.
Proof. If the language hasℵ0variables, then there are at most card(alphabet ofL) +ℵ0 formulas.
If, on the other hand, it has an uncountable set of variables, then we note that every definable set can be defined by a formula whose variables belong to a fixed countable subset of the variables that is chosen once and for all.
2.13. Corollary. If M is infinite and card(alphabet of L) < 2cardM, then
“almost all” sets are undefinable.
Thus, the only way to define all subsets of M is to include a tremendous number of names in the language. For languages that are to describe actual mathematical reasoning this is an unrealistic program. Essentially, any finitely describable collection of modes of expression allows us to define only a countable number of sets. However, it is often technically useful to include in the alphabet, for example, names for all the elements ofM.
In the following sections we proceed to study systematically sets of true formulas.