of the expressionsxk=x··· in any term inT m0are obviously independent of one another. All such occurrences are considered free.
(b0)F l1 is the least set of expressions that contains all expressions of the formt1=t2 (whereti∈T m0)and is closed with respect to forming the expres- sions (P1)↓(P2), wherePi ∈F l1. In other words,F l1 is the logical closure of the set of atomic formulas{t1=t2|ti∈T m0}.
Choosing a point ξ determines a truth value for any formula P ∈ F l1 by induction on the number of times ↓occurs:
|t1=t2|(ξ) =
1, iftξ1=tξ2; 0, otherwise;
|(P1)↓(P2)|(ξ) =
1, if|P|1(ξ) =|P2|(ξ) = 0, 0, otherwise.
All occurrences of variables in elements ofF l1 are independent of one another, and are considered free.
Now let i 1, and suppose that the sets T m2k−2, F l2k−1 are already defined for ki along with the interpretations and the division into free and bound occurrences of variables. We define the next sets T m2i and F l2i+1 as follows.
(ai)T m2i consists of the class terms of rank2i:
T m2i−2∪ {xk(P)|k1, P ∈F l2i−1}
(T m0 need not be included when i = 1). These elements have the following interpretation:
(xk(P))ξ =
xξ
k|ξruns through the variations ofξalongxk
for which|P|(ξ) = 1
.
All occurrences of the variable xk in xk(P) are considered bound, and the occurrences of other variables remain the same (free or bound) as inP.
(bi)F l2i+1 is the logical closure of the set of expressions
F l2i−1∪ {xk(P) =xk(Q)|k1;P, Q∈F l2i−1} ∪ {Tk¯|k1, T ∈T m2i} Thetruth functionis defined as follows: if we setxk(P) =T1 andxk(Q) =T2, then
|xk(P) =xk(Q)|(ξ) =
1, ifT1ξ =T2ξ as subsets ofN, 0, otherwise;
|T|¯k(ξ) =
1, ifk∈Tξ, 0, otherwise.
The function || is extended to the logical closure in the same way as in (b0).
All occurrences of variables in xk(P) =xkQand in Tk¯ are the same (free or
bound) as in the corresponding class term. Composition using the connective
↓ does not change the nature of the occurrence. As in Section 2.10, one can prove that |P|(ξ) depends only on the ξ-values of the variables that have free occurrences in the formula P∈ ∪∞i=0F l2i+1.
This finishes the description of the syntax and semantics of SAr.
In conclusion, we show that the classes of sets in∪r1Nrthat are definable by formulas in L1Ar and in SAr coincide. This result is not used in the proof of Tarski’s theorem in the next section. However, the result itself and the method of proof are instructive, and we shall return to these ideas in Part III of the book.
Let L1Ar have a countable set of variables. If we denote them byx1, x2, . . . , xn, . . .and identifyxiwithx...(i−1 primes), we can also identify the interpre- tation classes for L1Ar and SAr in the obvious way. Our claim that the classes of definable sets coincide is then an immediate consequence of the following stronger fact:
10.4.Proposition.Two translation mappings
{formulas of L1Ar}{formulas of SAr} can be explicitly defined with the following properties:
(a) At every pointξthe truth values of any formula and its translation coincide.
(b) The sets of free variables of any formula and its translation coincide.
We note that the mappings we define will not be inverse to each other!
Proof.
(a)The translation from L1Ar to SAr. The translation of a formulaP will be denoted by “P”. We first translate atomic formulas, and then use induction on the length. The alphabet of SAr does not have addition, but it has both multiplication and raising to a power, so that in place ofz=x+ywe can write 2z= 2x·2y.
(a1) Atomic formulas. They have the form t1 = t2. By “carrying out the operations,” we replace every nonzero term in L1Ar by a “normalized term,”
i.e., a polynomial of the form Σxii1· · ·xinn, where the monomials are written in the form (· · ·(x1·x1)· · ·x1)·x2). . .), then arranged in lexicographic order, and finally separated by parentheses: (· · ·((m1+m2) +m3) +· · ·). It is clear how to correlate such a term t to the term “¯2↑ t” in SAr. For example, “¯2 ↑ ((x1)·(x1) +x2)” is (¯2↑(x1)·(x1))·(¯2↑(x2)). By definition, the translation
“¯2 ↑ ¯0” is ¯1. Then we define the translation of the formula t1 = t2 to be
“2↑t1”= “2↑t2”. It is clear that such a formula and its translation have the same variables and are true at the same points ξ.
(a2) If “Q”, “Q1”, and “Q2”, have already been defined, then “¬Q” is defined as “Q”↓“Q”. We similarly construct “Q1∗Q2” for the other connectives (see “Digression: Syntax” in Chapter I).
(a3) If “Q” has already been defined, then “∀xkQ” is defined as xk(“Q”) =xk(xk =xk).
Both the formula and its translation are true at a pointξif and only ifQ(and
“Q”) are true at all variationsξ ofξ alongxk. They also have the same free variables, since by induction, we may assume that this is the case for Q and
“Q”.
(a4) By definition, “∃xkQ” coincides with “¬∀xk¬Q”.
(b) The translation from SAr to L1Ar. As before, we let “P” denote the translation of a formulaP, although this timeP will be a formula in SAr and
“P” will be a formula in L1Ar.
There is a subtle point here, namely, how to translatex1=x2↑x3. It will be shown in Part II of the book that such a translation exists, and can even be taken in the form ∃x4· · · ∃xnp(x1, x2, x3, x4, . . . , xn), where pis an atomic formula in L1Ar. Here we shall take this fact on faith, and choose a translation
“x1=x2↑x3” once and for all.
(b1)Translation of formulas in F l0. The following rules give an inductive definition:
“t1 = t2” has exactly the same form if t1, t2 ∈ {variables} ∪ {¯1,¯1¯1, . . .} (of course, in the sense that x... is replaced by xk and ¯1· · ·¯1 is replaced by (· · ·(¯1 + ¯1) + ¯1) +· · ·)).“xk =t1·t2” has the form ∃xi∃xj(“xi =t1”∧“xj = t2”∧xk=xi·xj) and “xk =t1↑t2” has the form∃xi∃xj(“xi=t1”∧“xj=t2”
∧xk=xi↑xj), wherexiandxj are the first two variables not occurring int1or t2. We similarly translate formulas with the left- and right-hand sides permuted, and also with ¯1· · ·¯1 instead ofxk. We further stipulate that “t1=t2” has the form ∃xi(“xi =t1”∧“xi =t2”), wherexi is the first variable not occurring in t1ort2, and where we assume only that neithert1 nort2is a variable or ¯1· · ·¯1.
It is clear that the truth function and the set of free variables are preserved under these translations.
(b2) Suppose that the formulas inF l2i−1have already been translated. Let
“xk(P1) =xk(P2)” be ∀xk(“P1”⇔“P2”), and
“xk(P)¯n” be “P”(¯n),
where on the right ¯n= (· · ·(¯1 + ¯1) + ¯1) +· · ·) is substituted in place of all free occurrences of xk in “P”. This completes the proof.