“I know what you’re thinking about,” said Tweedledum: “but it isn’t so, nohow.”
with the set of formulas in Lthat do not contain fand are deducible from E inL.
Proof.
Translation of formulas. Supposen1. (The casen= 0 is analogous, and is simpler, so we shall omit it.) The first effect of adding f is to increase the set of terms: L includes terms of the formf(t1, . . . , tn), wheref can occur in t1, . . . , tn, and so on. In order to decrease the number of references tof, we must say “f(t1, . . . , tn)” in a roundabout way: “that x for which P(x, t1, . . . , tn).”
This is the basic idea behind the translation of formulas. We now give a precise inductive definition.
(a) A termf(t1, . . . , tn) is called a simplef-term iffdoes not occur int1, . . . , tn. (b) LetQbe an atomic formula inL. Iff does not occur inQ, we letQbe its own translation. Iff occurs inQ, then there exists a simplef-termf(t1, . . . , tn) that occurs in Q. We take the very first occurrence of a simple f-term in Q, then take a variable symbol xthat does not occur in Q, substitute it in place of this occurrence, thereby obtaining a formula Q∗, and finally construct the formula
Q(1):∃x(P(x, t1, . . . , tn)∧Q∗(x)).
We apply this procedure toQ(1)to obtainQ(2), and so on. After a finite number of steps we obtain a formula Q(i) =Q in whichf does not occur. ThisQ is the translation ofQ.
(c) If Q is not an atomic formula, it has the form ¬Q1 or Q1∗Q2 (where ∗ is a connective), or else ∀y Q1 or ∃y Q1. In all casesQis translated automati- cally using the translations of Q, Q1, Q2, i.e., by “from Qproduce Q” to the component parts.
Translation of deductions. The problem is the following: LetQ1, . . . , Qn=Q be a deduction of Qfrom E, and letQ be the translation ofQ. We must con- struct a deduction ofQfromE. The most obvious idea is to write the sequence of translationsQ1, . . . , Qn. Why isn’t this a deduction ofQ fromE, since MP and Gen are translated in a trivial way, and tautologies are translated as tau- tologies? Because, for example, the logical axiom∀x R(x)⇒R(f) might appear in this sequence, and this formula stops being an axiom after it is translated, if f occurs in R. Hence, we must fill in the sequence Q1, . . . , Qn by adding deductions from E of certain of its terms. This is a rather cumbersome com- binatoric procedure, which one can read in §74 of Kleene’s book Introduction to Metamathematics (Van Nostrand, New York–Toronto, 1952). (The moral of the story is that new notation really does economize on time and space.)
Instead of using this procedure, we shall give an ineffective proof thatE Q using the deducibility criterion in 6.3. We state this criterion once more:
(a)If Q is true in any model ofE, thenE Q. SinceE contains the axioms of equality, we can slightly strengthen this as follows:
(b)If Q is true in any normal model ofE thenQ is true in any model ofE. Recall that = is interpreted as equality in a normal model. On the other hand, in §4 we showed that in any model = is interpreted as an equivalence relation that is compatible with the interpretation of all the constants, functions, and relations. Factoring out by this equivalence relation leads to a normal model, in which the truth values of all the formulas remain as before.
(c) The normal models of E (in the language L) coincide with the normal models of E (in the language L).
More precisely, we can give the following natural one-to-one correspondence between them that preserves the truth function. We shall limit ourselves to the casen1. Letφbe a normal interpretation ofL inM for which|Q|φ= 1 for allQ ∈ E. In particular, sinceE ∃!x P, we have
|∃!x P(x, y1, . . . , yn)|φ= 1.
Computing the truth value on the left at a pointξ∈M and using the normal- ity of the model, we then find that to every n-tuple yξ1, . . . , ynξ ∈ Mn there corresponds a unique xξ ∈M such that|P(xξ, yξ1, . . . , ynξ)|φ = 1 (this is not the standard notation, but the meaning is clear). We now interpret the symbol f (which is the new symbol in the languageL) as the functionMn→M that takes yξ1, . . . , ynξto xξ. We obviously obtain a normal model forE inL.
Conversely, any normal model for E can be restricted to L to obtain a normal model forE.
(d) IfQis deducible fromE inL, thenQ is true in any normal model forE. Proof.Qis true in any modelφforE. To prove thatQ is true, we begin with atomic formulasQthat containf. In the notation in the first part of the proof (translation of formulas), we constructQ∗and thenQ(1)=∃x(P(x, t1, . . . , tn)∧ Q∗(x)). To verify that|Q(1)|φ= 1, for each pointξ∈M we must find a variation ξ ofξalongxfor which
|P|φ(ξ) = 1 and |Q∗(x)|φ(ξ) = 1.
We determinexξ from the condition|P(xξ, tξ1, . . . , tξn)|φ = 1. The description in (c) of the interpretation off shows that we now have|Q∗|φ(ξ) =|Q|φ(ξ) = 1.
Thus, truth is preserved in going fromQtoQ(1). Repeating this procedure, we find that Q is true for atomic formulas Q. Finally, the truth of Q in the general case is proved by induction on the number of connectives and quan- tifiers. Combining the results (a)–(d), we then obtain E → Q, which which
completes the proof of Proposition 8.3.
8.4. Examples
(a) In L1Set the following formula is deducible from the axioms of extensionality and pairing (and also the axioms of equality and the logical axioms):
∃!x∀z(z∈x⇔z=u∨z=v).
Using Proposition 8.3, we see that we may add to L1Set a new degree 2 function symbol{}, “unordered pair,” without changing the set of formulas in L1Set that are deducible from the Zermelo–Fraenkel axioms. Therefore, without hesitation we may use not only the abbreviated notation “x={u, w}” as before, but also terms that are put together using the symbol {}. In particular (here the use of{}is not normalized, but is in agreement with tradition): (b) We can introduce notation for the finite ordinals
∅,{∅},{∅,{∅}}, . . .
as terms in their own right in our language extension, and then embed formal arithmetic in formal set theory.
(c) After deducing the formula
∃!x(“xis an ordinal”∧“xis not finite”∧“∀ordinaly < x, y is finite”) from the Zermelo–Fraenkel axioms, we can introduce a new constant ω0, and then continue to introduce names of more and more ordinals that are demon- strably uniquely characterized by formulas in L1Set (or in language extensions that are formed in the same way).
We shall make use of this new freedom of action in Chapter III.
9 Undefinability of Truth: The Language SELF
9.1. When modeled in formal languages, arguments of the “liar paradox” type lead to important theorems on the limitations of the modes of expression and proof in these languages. The best known of these theorems are Tarski’s theorem on the undefinability of the set of true formulas and G¨odel’s theorem on the impossibility of effectively axiomatizing arithmetic.
The next three sections are devoted to Tarski’s theorem. Our presentation is based on an excellent article by Smullyan (Languages in which self-reference is possible, J. Symb. Logic. vol. 22, no. 1 (1957), 55–67).
In this section we describe the extremely elementary language SELF (which does not belong toL1), which was designed to illustrate self-reference and which graphically demonstrates the idea of such a construction. In §10 we introduce the language SAr, which is just as expressive as L1Ar, but does not belong to L1. Its syntax is close to that of SELF, which greatly simplifies proofs. Finally, in §11 we use a method of Smullyan to prove Tarski’s theorem for SAr.
9.2. The language SELF(Smullyan’s Easy Language For self-reference) The alphabetof SELFE,∗ (symmetric quotes),r (relation of degree 1 ),¬ (negation).
Thesyntaxof SELF. The distinguished expressions are labels, displays, for- mulas, and names. The label of any expressionP is ∗P∗ (“P in quotes”). The display of any expressionP isP∗P∗(“something with a label”).Formulasare expressions of the formrE . . . E∗P∗or¬rE . . . E∗P∗, whereEappearsk0
times after r. We use the abbreviated notation rEk ∗P∗ and ¬rEk∗P∗ for formulas. Finally, we introduce the binary relation “is the name of” on the set of all distinguished expressions. This relation is defined recursively:
(a) The label ofP is a name ofP.
(b) IfP is a name ofQ, thenEP is a name of the display ofQ, i.e., a name of the expressionQ∗Q∗.
9.3. Remarks
(a) IfP is a name ofQ, then the display ofQhas at least two different names:
EP and ∗Q∗Q∗∗. Thus, an expression can have several names. But con- versely, an expression is uniquely determined if we know its name; names all have the formEk∗P∗, k0. We shall writeN(Q) in place of “one of the names ofQ.”
(b) Every formula has the form rN(Q) or ¬rN Q. In 9.4 we interpret such a formula as the statement, “The expressionQ has (or does not have) the propertyR,” and it is natural that the formula, in saying something about Q, “callsQby name.”
(c) The expressionE∗E∗is one of two possible names for itself. In exactly the same way, the formula rE∗rE∗ “says something about itself” (see 9.5).
The language SELF was constructed precisely in order to produce these effects of self-reference with the fewest possible modes of expression.
9.4.The standard interpretations. In order to give one of the standard interpre- tations of the language SELF, we choose any set (property)Rof expressions of the language and introduce the truth function| |Ron the formulas by stipulating
1− |¬rN(Q)|R=|rN(Q)|R=
1, ifQ∈R, 0, otherwise.
We say that a formula is R-true (R-false) if the value of | |R in the formula equals 1 (resp. 0).
9.5. Undefinability Theorem.For any property R, R∩ {f ormulas} =
R-truef ormulas, R-f alsef ormulas.
Proof.
(a) The formulaQ=¬rE∗ ¬rE∗ isR-true⇔rE∗ ¬rE∗ isR-false⇔Q∈R, since E∗ ¬rE∗ is a name of the display of ¬rE, i.e., a name of Q. Thus, Q cannot both lie inRand be true, which proves the first part of the theorem. The connection with the liar paradox becomes clear if we note that Q says about itself, “I do not have the propertyR.”
(b) Analogously, the formula rE∗rE∗says about itself, “I have the property R,” and so cannot both lie inRand beR-false.