Tarski's indefinability theorem shows us that truth is not definable in arithmetic. The requirement to define truth for a language in a stronger language (if contradiction is to be avoided) lapses for particularly weak languages. A weaker language, however, is not necessary for that lapse. It also lapses for an adequately weak theory. It turns out that the set of G{\"o}del numbers of sentences true in arithmetic modulo $n$  is definable in arithmetic modulo $n$

Shackel, Nicholas

English

Online Research @ Cardiff

South American Journal of LogicVol. 4, n. 1, pp. 121–131, 2018ISSN: 2446-6719A Simple Theory Containing its Own TruthPredicateNicholas ShackelAbstractTarski’s indefinability theorem shows us that truth is not definable inarithmetic. The requirement to define truth for a language in a strongerlanguage (if contradiction is to be avoided) lapses for particularly weaklanguages. A weaker language, however, is not necessary for that lapse.It also lapses for an adequately weak theory. It turns out that the set ofGödel numbers of sentences true in arithmetic modulo n is definable inarithmetic modulo n.Keywords: Truth, Predicate, Definable, Arithmetic modulo nIntroductionTarski’s indefinability theorem gives us that the set of Gödel numbers of sen-tences true in [arithmetic] is not definable in arithmetic (Boolos and Jeffrey1989: 176). The requirement to define truth for a language in a stronger lan-guage (if contradiction is to be avoided) lapses for particularly weak languages,for example, Myhills system which lacks negation (Quine 1980:137 fn 10). Buta weaker language is not a prerequisite for that lapse. It also lapses for an ad-equately weak theory. It turns out that the set of Gödel numbers of sentencestrue in arithmetic modulo n is definable in arithmetic modulo n.Notation• L is the first order language of arithmetic.• Sent(L) is the set of closed sentences of L.• 0M , +M , ×M etc. is what is denoted in a model M by the respectivemembers of L.122 N. Shackel• φ and ψ are always closed sentences of L .• n is the symbol ss . . . s(0): n repetitions of s.• [m] is the equivalence class of natural numbers defined by the relation‘having the same remainder on division by n’: alternatively, and equiva-lently, the element m+nZ in the quotient ring Z/nZ, (i.e. in Zn).• # is a Gödel coding• pφq is the Gödel numeral for #φ• M is a model. In speaking of models M , I use ‘M ’ to refer both tothe model and its domain, unless the context requires them to be distin-guished.1 Qn: Modulo n arithmeticWe meet modular arithmetic in number theory, where congruence modulo naids the study of divisibility and in algebra as the quotient ring Z/nZ, whoseelements defined in terms of cosets of the ideal nZ turn out to be the equivalenceclasses induced by the congruence relation. Modular arithmetic is the structureof the ring of integers Zn.For the purposes of this paper, Qn is modulo n arithmetic, for n>1, withthe following 7 axioms and 1 axiom schema:1. n = 02. ∀x : (x = 0 ∨ x = 1 ∨ . . . ∨ x = n− 1)3. s(0) 6= 04. ∀x : x+ 0 = x5. ∀x∀y : x+ s(y) = s(x+ y)6. ∀x : x× 0 = 07. ∀x∀y : x× s(y) = x× y + x8. Axiom schema. For any m that is a factor of n, m6= 0.11Factors of n, or multiples of such factors, are divisors of zero modulo n, which fact preventsus proving the inequation m 6=0 from the axioms already given. We need the inequations toguarantee the isomorphism of all models of QnA Simple Theory Containing its Own Truth Predicate 123Properties of Qn(i) Qn ` ∀x∀y(s(x) = s(y)→ x = y)).(ii) Qn ` ∀x∃y(x = s(y)).(iii) m 6≡ k (mod n) iff Qn ` m 6= k.(iv) If j ≡ m+ k (mod n) then Qn ` j = m + k.(v) If j ≡ m× k (mod n) then Qn ` j = m× k.The intended model M of Qn is (D, [0], σ,⊕,⊗), defined by:1. dom(M) = {[0], [1], . . . , [n− 1]}2. for all k < n− 1, (σ[k]) = [k + 1]3. σ([n− 1]) = [0]4. [k]⊕ [l] = [k + l]5. [k]⊗ [l] = [k × l]Clearly, M|= Qn. M is just the ring Zn (or isomorphic to Zn if you prefer)since for 0 ≤ m < n, [m] is the equivalence class of m (mod n), and so it is alsothe relevant coset of the ideal nZ, and hence M = the quotient ring Z/nZ. Itis obvious from arithmetic that M satisfies R1 - 8. Straightforwardly we haveidentities, inverses, associativity of and closure under ⊕ and ⊗, commutativityof ⊕ and left and right distributivity of ⊗ over ⊕.Interpretations of Qn will have to use 0 to name [0]. However, for somen the numerals naming the other members of M will be permutable whilstpreserving truth. We shall make use of interpretations of Qn that use thenumeral m to name [m].Proposition 1.1 If M |= Qn, then the domain of M has n members.Proof. By the domain axiom, M has the form {0M, . . . , (n − 1)M} so thedomain has ≤ n members. There are n distinct elements by the inequations(property (iii)) so the domain has ≥ n members. Theorem 1.2 All models of Qn are isomorphic to M.124 N. ShackelProof.Let M∗ |= Qn. Let ρ be the function ρ : M 7→M∗ such that: ρ : mM 7→ mM∗.ρ is a surjection. Suppose x ∈ M∗. Then x = mM∗ for some m in L. HencemM ∈M and ρ(mM) = mM∗ρ is an injection. Suppose for a contradiction that mM 6= kM but ρ(mM) =ρ(kM). Then Qn ` m 6= k (property (iii)) and hence, since Qn is sound,mM∗ 6= kM∗ . But if ρ(mM) = ρ(kM), then mM∗ = kM∗ , which is thecontradiction we sought.Preservation of successor. We need to show that ρ(σ(mM)) = σ∗(ρ(mM)).LHS of equation.σ(mM) = σ([m]) = [m+ 1] = (m+ 1)Msoρ(σ(mM)) = ρ((m+ 1)M) = (m+ 1)M∗RHS of equation. We haveσ∗(ρ(mM)) = σ∗(mM∗)and sinceQn ` s(m) = s(m+ 0) = m+ 1then soundness givesσ∗(mM∗) = σ∗((m+ 0)M∗) = (m+ 1)M∗whence LHS = RHS.Preservation of addition.Suppose [m] ⊕ [k] = [j]. Then Qn ` j = m + k (property (iv)), when bysoundnessmM∗+M∗kM∗= jM∗i.e. ρ(mM) +M∗ρ(kM) = ρ(jM)i.e. ρ([m]) +M∗ρ([k]) = ρ([m]⊕ [k])Preservation of multiplication. Similar proof to addition with ⊗ for ⊕, × for+ and the use of property (v) instead of property (iv). Theorem 1.3 Qn is not weaker than the theory of rings.Proof. Since M is a ring and all models of Qn are isomorphic, there cannotbe non-standard models (in which, e.g. ∀x∀y : x+ y = y+x is not true) whichcan be used to show Qn is weaker. A Simple Theory Containing its Own Truth Predicate 125Theorem 1.4 Qn is complete.Proof. Suppose Qn is not complete. Then there is a sentence φ such that φis true in at least one model and false in at least one model of Qn. But this isimpossible, since all models are isomorphic. Corollary 1.5 For any model, M,Qn ` φ iff M |= φ.Definition 1.6 The truth set Tn = {φ ∈ Sent(L) : M |= φ}Corollary 1.7 All models of Qn have the same truth set (which is just the setof theorems of Qn)2 Finite ModelsTheorem 2.1 For any finite model M the truth set is decidable.Proof. First of all, the truth value of any atomic formulae can be determinedby a finite calculation, as can any finite Boolean combinations of such formulae.Let φ be a formula of the form ∀x : χ(x), where χ has only x free. Then,M |= φ iff M |= χ(0)∧χ(1)∧· · ·∧χ(n− 1). Likewise, an existentially quantifiedformula ∃x : χ(x) can be expressed as a finite disjunction and M satisfies it iffM satisfies the finite disjunction. Hence, any singly quantified formula can bere-expressed as a finitely long Boolean combination.An example will illustrate the general proof. Let φ be a closed prenexformula ∀x1∃x2 : t(x1, x2) = u(x1, x2). We now inflate by use of the facts justmention to get a formula satisfied by M iff M |= φ. From the outside movingin we get∃x2 : t(0, x2) = u(0, x2) ∧ ∃x2 : t(1, x2) = u(1, x2) ∧ . . .· · · ∧ ∃x2 : t(n− 1, x2) = u(n− 1, x2)and then(t(0, 0) = u(0, 0) ∧ t(1, 0) = u(1, 0) ∧ · · · ∧ t(n− 1, 0) = u(n− 1, 0))∨ . . . . . .∨ . . . . . .∨ (t(0, n− 1) = u(0, n− 1) ∧ t(1, n− 1) = u(1, n− 1) ∧ . . .· · · ∧ t(n− 1, n− 1) = u(n− 1, n− 1))This is a Boolean combination of a finite number of atomic formulae so itstruth value can be worked out by a finite calculation.126 N. ShackelEvidently, any closed prenex formula with m quantifiers at the front can beinflated in stages in this way to a Boolean combination of nm atomic formulaeeach of the form t(x1, . . . , xm) = u(x1, . . . , xm) Once again, its truth value canbe worked out by a finite calculation.Therefore a Turing machine can determine whether a closed formula be-longs to Tn, and hence the characteristic function of Tn is recursive. Theorem 2.2 The truth set for models of Qn is decidable.Proof. Apply proposition 1.1, theorem 2.1 and corollary 1.7. Models of Qnare finite so have decidable truth sets, which are all identical. Theorem 2.3 Qn is decidable.Proof. For any model M of Qn, the truth set Tn of M is decidable. Qn ` φiff M |= φ so any decision procedure for Tn will yield a decision procedure forQn 3 The Truth Predicate for QnA truth predicate for Qn, satisfying Tarski’s material adequacy condition wouldbe a predicate Tn(x) such that for all φ,Qn ` Tn(pφq)↔ φLemma 3.1 pφqM = [#φ]Proof. pφq is the Gödel numeral for #φ so the object in M named by it is[#φ] Definition 3.2 Let Θ be any theory in the language L, M be any model ofΘ. An acceptable M coding for a theory Θ is a Gödel coding # suchthat for any closed formulas φ, ψ, if pφqM = pψqM then for any model M∗ ofΘ,M∗ |= φ iff M∗ |= ψWe are interested only in acceptable M codings for Qn, which from hereonwe call acceptable codings. Not all codings are acceptable for Qn. The simplestunacceptable coding can be got by an injection from expressions into the setof multiples of n (see proposition 4.1 below).A Simple Theory Containing its Own Truth Predicate 127Lemma 3.3 For acceptable codings #, for any closed formulas φ, ψ, any ofthe following equivalent propositions:1. (a) pφqM = pψqM(b) #φ ≡ #ψ (mod n)(c) [#φ] = [#ψ]implies any of the following equivalent propositions:2. (a) M |= φ iff M |= ψ(b) φ ∈ Tn iff ψ ∈ Tn(c) M |= φ↔ ψ(d) Qn ` φ↔ ψProof. The definition gives us that if 1(a) then 2(a) and so equivalents to1(a) imply equivalents to 2(a). Lemma 3.4 Let # be an acceptable coding and φ, ψ closed formulae. If #φ ≡#ψ (mod n) then φ ∈ Tn iff ψ ∈ Tn.Proof. Using the aforementioned equivalences Remarks 3.5 Essentially, this lemma is telling us that an acceptable codingis a Gödel coding such that if the Gödel numbers of two closed sentences inL are in the same equivalence class modulo n, then they have the same truthvalue.Lemma 3.6 For any Qn an acceptable Gödel coding exists.Proof. Let P be the set of primes and R the set of primes in the primefactorisation of n. Use P\R as the basis for a powers of primes Gödel codingg. Then define the following function # : L 7→ N#φ ={n× g(φ) if φ ∈ Tng(φ) if φ /∈ TnBy the unique prime factorisation theorem # is an injection and#φ ∈ Tn iff [#φ] = [0] (since for all φ, n is a factor of n× g(φ))#φ /∈ Tn iff [#φ] 6= [0] (since for all φ, n is a not factor of n× g(φ))If the characteristic functions of the conditions for a function defined by casesfrom recursive functions are themselves recursive then the function defined by128 N. Shackelcases is also recursive. Since Tn is decidable the characteristic function formembership of Tn is recursive. Both g and multiplication by n are recursivefunctions. Hence # is recursive. Therefore # is a Gödel coding. For any φ, ψ, φand ψ will have Gödel numbers congruent modulo n only if they have the sametruth value. Therefore # is an acceptable coding. Definition 3.7 We make use of an acceptable coding, #.Let Kn = {k1, . . . , ki} be the set such thatk ∈ Kn iff 0 ≤ k < n and there exists a φ such that φ ∈ Tn andk ≡ #φ (mod n)So Kn is the set of numbers between 0 and n − 1 which are congruent to theGödel numbers of the true sentences of Qn.Lemma 3.8 If # is acceptable then for any φ, φ ∈ Tn iff for some k ∈Kn, [#φ] = [k].Proof. Suppose that φ ∈ Tn. By the division algorithm for some r suchthat 0 ≤ r < n, #φ ≡ r (mod n) and hence r ∈ Kn. Suppose that forsome k ∈ Kn, k ≡ #φ (mod n). Then for some ψ ∈ Tn, k ≡ #ψ (mod n),so #φ ≡ #ψ (mod n), hence φ ∈ Tn iff ψ ∈ Tn (because # is acceptable),whence φ ∈ Tn Theorem 3.9 Let the open L-formula Tn(x) be the open formula (x = k1∨x =k2 ∨ · · · ∨ x = ki). Tn(x) defines truth in Qn (and hence in M).Proof. We make use of an acceptable coding, #.For any φ ∈ Sent(L), Qn ` Tn(pφq)iff M|= Tn(pφq) (soundness and completeness)iff M|= [(pφq = k1) ∨ (pφq = k2) ∨ · · · ∨ (pφq = ki)iff either [#φ] = [k1] or[#φ] = [k2] or ... or [#φ] = [ki]iff, for some k, [#φ] = [k] and k ∈ Kniff φ ∈ Tn (by lemma 3.8)Theorem 3.10 Tn(x) is a truth predicate satisfying Tarskis material adequacycondition, i.e. for all φ ∈ Sent(L), Qn ` Tn(pφq)↔ φA Simple Theory Containing its Own Truth Predicate 129Proof. Given any φ, M |= Tn(pφq) iff φ ∈ Tn iff M|= φ, hence M|=Tn(pφq)↔ φ, whence by completeness we have Qn ` Tn(pφq)↔ φ We have now proved our main result. The set of Gödel numbers of sentencestrue in arithmetic modulo n is definable in arithmetic modulo n, Qn, by thepredicate Tn(x). Hence for all n, arithmetic modulo n is a theory containingits own truth predicate.4 Coding, representability and acceptability.Clearly, the set of Gödel numbers for a set of expressions, and representabilityin general, is relative to coding: but in Q (Robinson arithmetic) and extensionsof Q all codings can be shown to be N acceptable, simply because Gödel codingsmust be injections into N. For the same reason, for all the non-standard modelsNS of Q of which I am aware, in which the domain is got by adding extraelements to N, all codings are NS acceptable. So in general, relativity tocoding is irrelevant and not discussed for that reason.However, when using L for a theory of a finite model then the syntax mayno longer be properly represented in the theory. Although any coding willstill be injective into N, and although there will still be an injection from Nto the numerals of L , any theory which induces a partition on the numeralsis in danger of proving that for distinct φ, ψ, pφq = pψq. Of course, it is onlyin danger of doing this if at least one equivalence class has more than onemember. In Qn the relation inducing the partition can be defined by x ∼ yiff Qn ` x = y, and since there are infinite numerals and only n equivalenceclasses of numerals, Qn is in that danger. Indeed, there is a coding for Qnwhich ensures that for all φ, ψ,Qn ` pφq = pψq.Proposition 4.1 For any n > 1 there exists a Gödel coding of a languageL which makes the code of every formula and its negation congruent to zeromodulo n.Proof. Take any Gödel coding #. # maps L into N (is an injection). Let gbe a function g : N 7→ N, g : x 7→ nx. Then let f : L 7→ N, f : x 7→ g(#x).Both g and # are injections and hence f is an injection from L into N andas such is a Gödel coding. Under f all formulas of L are congruent to 0 modulon, a fortiori , so is each formula and its negation. Hence the need for an acceptable coding if anything of significance is to begot out of a coding. But even with an acceptable coding, coding of the syntaxof L into any model of Qn will be many-one and consequently the capacity for130 N. Shackelrepresentation of relations of the models must be fairly crude. The proof aboveshows that being the code of a true sentence is definable. Adapting the Qnacceptable coding given above in a fairly obvious manner would create whatKetland calls a Frege coding.Definition 4.2 A Frege Coding Fc. Given a model M , pick any two ele-ments of the domain, say a and b. Then define the coding as follows, for anyφFc(φ) ={a if M |= φb if M |= ¬φIntuitively, a and b are just the two truth values, ‘The True’ and ‘The False’.Definition 4.3 A Frege Coding for Qn. In our case we can define Fc ,using the acceptable coding of lemma 3.6.Fc(φ) ={[1] if #φ ≡ 0 (mod n)[0] if #φ 6≡ 0 (mod n)It might be that the capacity for representation in Qn is so crude that verylittle else is representable or definable. I dont know. I havent investigatedrepresentability or definability in general for Qn. The following results may berelevant to the problems of definability and representability in Qn:Corollary 4.4 Given an acceptable coding, the diagonal function is not rep-resentable in Qn relative to that coding.Proof. If the diagonal function were representable, we could derive a contra-diction in the standard way. But Qn is consistent (it has a model). Corollary 4.5 Introduce a new predicate T ∗(x) and let DT ∗ be the set of allaxioms T ∗(pφq ↔ φ), with φ ∈ Sent(L). Then Qn ∪ DT ∗is a conservativeextension of Qn .Proof. Take any model M . Define an expansion (M,E) by interpretingthe new predicate T ∗(x) as the set E, where E is got from the set Kn definedabove as followsmM ∈ E iff for some k ∈ Kn,m ≡ k (mod n).Then (M,E) |= Qn ∪DT ∗. (Plainly E is definable in Qn). A Simple Theory Containing its Own Truth Predicate 1315 Final remarksFirst we note some further questions worth asking. 1) Are there codings bywhich we can show M to be an acceptable structure? 2) Could it be thatacceptability is not a problem in free logic?A reservation that might be raised is whether Tn(x) is properly called atruth predicate? Certainly we have seen that it satisfies Tarksi’s materialadequacy condition. Nevertheless, you might reject it because you deny thecoding we used being a Gödel coding. However, since such coding can never beone-one into a finite model, it is the best coding we can get, and it is sufficientfor truth. So this reservation appears pedantic and we set it aside.In theorem 3.10 we proved our main result. Since for all n, the set ofGödel numbers of sentences true in arithmetic modulo n is definable by thepredicate Tn(x), arithmetic modulo n is a simple theory containing its owntruth predicate.2References[1] Boolos, G. and Jeffrey, R. C. Computability and Logic. 3rd ed. CambridgeUniversity Press, 1989.[2] Quine, W. V. From a Logical Point of View : 9 Logico-PhilosophicalEssays. 2nd ed. Harvard University Press, 1980.Nicholas ShackelPhilosophyCardiff UniversityGreat BritainOxford Uehiro Centre for Practical EthicsUniversity of OxfordGreat BritainE-mail: shackeln@cardiff.ac.uk2This paper has benefited from extensive discussion with Jeffrey Ketland.

A simple theory containing its own truth predicate

PhilPapers

South American Journal of LogicVol. 4, n. 1, pp. 121–131, 2018ISSN: 2446-6719A Simple Theory Containing its Own TruthPredicateNicholas ShackelAbstractTarski’s indefinability theorem shows us that truth is not definable inarithmetic. The requirement to define truth for a language in a strongerlanguage (if contradiction is to be avoided) lapses for particularly weaklanguages. A weaker language, however, is not necessary for that lapse.It also lapses for an adequately weak theory. It turns out that the set ofGo¨del numbers of sentences true in arithmetic modulo n is definable inarithmetic modulo n.Keywords: Truth, Predicate, Definable, Arithmetic modulo nIntroductionTarski’s indefinability theorem gives us that the set of Go¨del numbers of sen-tences true in [arithmetic] is not definable in arithmetic (Boolos and Jeffrey1989: 176). The requirement to define truth for a language in a stronger lan-guage (if contradiction is to be avoided) lapses for particularly weak languages,for example, Myhills system which lacks negation (Quine 1980:137 fn 10). Buta weaker language is not a prerequisite for that lapse. It also lapses for an ad-equately weak theory. It turns out that the set of Go¨del numbers of sentencestrue in arithmetic modulo n is definable in arithmetic modulo n.Notation• L is the first order language of arithmetic.• Sent(L) is the set of closed sentences of L.• 0M , +M , ×M etc. is what is denoted in a model M by the respectivemembers of L.122 N. Shackel• φ and ψ are always closed sentences of L .• n is the symbol ss . . . s(0): n repetitions of s.• [m] is the equivalence class of natural numbers defined by the relation‘having the same remainder on division by n’: alternatively, and equiva-lently, the element m+nZ in the quotient ring Z/nZ, (i.e. in Zn).• # is a Go¨del coding• pφq is the Go¨del numeral for #φ• M is a model. In speaking of models M , I use ‘M ’ to refer both tothe model and its domain, unless the context requires them to be distin-guished.1 Qn: Modulo n arithmeticWe meet modular arithmetic in number theory, where congruence modulo naids the study of divisibility and in algebra as the quotient ring Z/nZ, whoseelements defined in terms of cosets of the ideal nZ turn out to be the equivalenceclasses induced by the congruence relation. Modular arithmetic is the structureof the ring of integers Zn.For the purposes of this paper, Qn is modulo n arithmetic, for n>1, withthe following 7 axioms and 1 axiom schema:1. n = 02. ∀x : (x = 0 ∨ x = 1 ∨ . . . ∨ x = n− 1)3. s(0) 6= 04. ∀x : x+ 0 = x5. ∀x∀y : x+ s(y) = s(x+ y)6. ∀x : x× 0 = 07. ∀x∀y : x× s(y) = x× y + x8. Axiom schema. For any m that is a factor of n, m6= 0.11Factors of n, or multiples of such factors, are divisors of zero modulo n, which fact preventsus proving the inequation m 6=0 from the axioms already given. We need the inequations toguarantee the isomorphism of all models of QnA Simple Theory Containing its Own Truth Predicate 123Properties of Qn(i) Qn ` ∀x∀y(s(x) = s(y)→ x = y)).(ii) Qn ` ∀x∃y(x = s(y)).(iii) m 6≡ k (mod n) iff Qn ` m 6= k.(iv) If j ≡ m+ k (mod n) then Qn ` j = m + k.(v) If j ≡ m× k (mod n) then Qn ` j = m× k.The intended model M of Qn is (D, [0], σ,⊕,⊗), defined by:1. dom(M) = {[0], [1], . . . , [n− 1]}2. for all k < n− 1, (σ[k]) = [k + 1]3. σ([n− 1]) = [0]4. [k]⊕ [l] = [k + l]5. [k]⊗ [l] = [k × l]Clearly, M|= Qn. M is just the ring Zn (or isomorphic to Zn if you prefer)since for 0 ≤ m < n, [m] is the equivalence class of m (mod n), and so it is alsothe relevant coset of the ideal nZ, and hence M = the quotient ring Z/nZ. Itis obvious from arithmetic that M satisfies R1 - 8. Straightforwardly we haveidentities, inverses, associativity of and closure under ⊕ and ⊗, commutativityof ⊕ and left and right distributivity of ⊗ over ⊕.Interpretations of Qn will have to use 0 to name [0]. However, for somen the numerals naming the other members of M will be permutable whilstpreserving truth. We shall make use of interpretations of Qn that use thenumeral m to name [m].Proposition 1.1 If M |= Qn, then the domain of M has n members.Proof. By the domain axiom, M has the form {0M, . . . , (n − 1)M} so thedomain has ≤ n members. There are n distinct elements by the inequations(property (iii)) so the domain has ≥ n members. Theorem 1.2 All models of Qn are isomorphic to M.124 N. ShackelProof.Let M∗ |= Qn. Let ρ be the function ρ : M 7→M∗ such that: ρ : mM 7→ mM∗ .ρ is a surjection. Suppose x ∈ M∗. Then x = mM∗ for some m in L. HencemM ∈M and ρ(mM) = mM∗ρ is an injection. Suppose for a contradiction that mM 6= kM but ρ(mM) =ρ(kM). Then Qn ` m 6= k (property (iii)) and hence, since Qn is sound,mM∗ 6= kM∗ . But if ρ(mM) = ρ(kM), then mM∗ = kM∗ , which is thecontradiction we sought.Preservation of successor. We need to show that ρ(σ(mM)) = σ∗(ρ(mM)).LHS of equation.σ(mM) = σ([m]) = [m+ 1] = (m+ 1)Msoρ(σ(mM)) = ρ((m+ 1)M) = (m+ 1)M∗RHS of equation. We haveσ∗(ρ(mM)) = σ∗(mM∗)and sinceQn ` s(m) = s(m+ 0) = m+ 1then soundness givesσ∗(mM∗) = σ∗((m+ 0)M∗) = (m+ 1)M∗whence LHS = RHS.Preservation of addition.Suppose [m] ⊕ [k] = [j]. Then Qn ` j = m + k (property (iv)), when bysoundnessmM∗+M∗kM∗= jM∗i.e. ρ(mM) +M∗ρ(kM) = ρ(jM)i.e. ρ([m]) +M∗ρ([k]) = ρ([m]⊕ [k])Preservation of multiplication. Similar proof to addition with ⊗ for ⊕, × for+ and the use of property (v) instead of property (iv). Theorem 1.3 Qn is not weaker than the theory of rings.Proof. Since M is a ring and all models of Qn are isomorphic, there cannotbe non-standard models (in which, e.g. ∀x∀y : x+ y = y+x is not true) whichcan be used to show Qn is weaker. A Simple Theory Containing its Own Truth Predicate 125Theorem 1.4 Qn is complete.Proof. Suppose Qn is not complete. Then there is a sentence φ such that φis true in at least one model and false in at least one model of Qn. But this isimpossible, since all models are isomorphic. Corollary 1.5 For any model, M,Qn ` φ iff M |= φ.Definition 1.6 The truth set Tn = {φ ∈ Sent(L) : M |= φ}Corollary 1.7 All models of Qn have the same truth set (which is just the setof theorems of Qn)2 Finite ModelsTheorem 2.1 For any finite model M the truth set is decidable.Proof. First of all, the truth value of any atomic formulae can be determinedby a finite calculation, as can any finite Boolean combinations of such formulae.Let φ be a formula of the form ∀x : χ(x), where χ has only x free. Then,M |= φ iff M |= χ(0)∧χ(1)∧· · ·∧χ(n− 1). Likewise, an existentially quantifiedformula ∃x : χ(x) can be expressed as a finite disjunction and M satisfies it iffM satisfies the finite disjunction. Hence, any singly quantified formula can bere-expressed as a finitely long Boolean combination.An example will illustrate the general proof. Let φ be a closed prenexformula ∀x1∃x2 : t(x1, x2) = u(x1, x2). We now inflate by use of the facts justmention to get a formula satisfied by M iff M |= φ. From the outside movingin we get∃x2 : t(0, x2) = u(0, x2) ∧ ∃x2 : t(1, x2) = u(1, x2) ∧ . . .· · · ∧ ∃x2 : t(n− 1, x2) = u(n− 1, x2)and then(t(0, 0) = u(0, 0) ∧ t(1, 0) = u(1, 0) ∧ · · · ∧ t(n− 1, 0) = u(n− 1, 0))∨ . . . . . .∨ . . . . . .∨ (t(0, n− 1) = u(0, n− 1) ∧ t(1, n− 1) = u(1, n− 1) ∧ . . .· · · ∧ t(n− 1, n− 1) = u(n− 1, n− 1))This is a Boolean combination of a finite number of atomic formulae so itstruth value can be worked out by a finite calculation.126 N. ShackelEvidently, any closed prenex formula with m quantifiers at the front can beinflated in stages in this way to a Boolean combination of nm atomic formulaeeach of the form t(x1, . . . , xm) = u(x1, . . . , xm) Once again, its truth value canbe worked out by a finite calculation.Therefore a Turing machine can determine whether a closed formula be-longs to Tn, and hence the characteristic function of Tn is recursive. Theorem 2.2 The truth set for models of Qn is decidable.Proof. Apply proposition 1.1, theorem 2.1 and corollary 1.7. Models of Qnare finite so have decidable truth sets, which are all identical. Theorem 2.3 Qn is decidable.Proof. For any model M of Qn, the truth set Tn of M is decidable. Qn ` φiff M |= φ so any decision procedure for Tn will yield a decision procedure forQn 3 The Truth Predicate for QnA truth predicate for Qn, satisfying Tarski’s material adequacy condition wouldbe a predicate Tn(x) such that for all φ,Qn ` Tn(pφq)↔ φLemma 3.1 pφqM = [#φ]Proof. pφq is the Go¨del numeral for #φ so the object in M named by it is[#φ] Definition 3.2 Let Θ be any theory in the language L, M be any model ofΘ. An acceptable M coding for a theory Θ is a Go¨del coding # suchthat for any closed formulas φ, ψ, if pφqM = pψqM then for any model M∗ ofΘ,M∗ |= φ iff M∗ |= ψWe are interested only in acceptable M codings for Qn, which from hereonwe call acceptable codings. Not all codings are acceptable for Qn. The simplestunacceptable coding can be got by an injection from expressions into the setof multiples of n (see proposition 4.1 below).A Simple Theory Containing its Own Truth Predicate 127Lemma 3.3 For acceptable codings #, for any closed formulas φ, ψ, any ofthe following equivalent propositions:1. (a) pφqM = pψqM(b) #φ ≡ #ψ (mod n)(c) [#φ] = [#ψ]implies any of the following equivalent propositions:2. (a) M |= φ iff M |= ψ(b) φ ∈ Tn iff ψ ∈ Tn(c) M |= φ↔ ψ(d) Qn ` φ↔ ψProof. The definition gives us that if 1(a) then 2(a) and so equivalents to1(a) imply equivalents to 2(a). Lemma 3.4 Let # be an acceptable coding and φ, ψ closed formulae. If #φ ≡#ψ (mod n) then φ ∈ Tn iff ψ ∈ Tn.Proof. Using the aforementioned equivalences Remarks 3.5 Essentially, this lemma is telling us that an acceptable codingis a Go¨del coding such that if the Go¨del numbers of two closed sentences inL are in the same equivalence class modulo n, then they have the same truthvalue.Lemma 3.6 For any Qn an acceptable Go¨del coding exists.Proof. Let P be the set of primes and R the set of primes in the primefactorisation of n. Use P\R as the basis for a powers of primes Go¨del codingg. Then define the following function # : L 7→ N#φ ={n× g(φ) if φ ∈ Tng(φ) if φ /∈ TnBy the unique prime factorisation theorem # is an injection and#φ ∈ Tn iff [#φ] = [0] (since for all φ, n is a factor of n× g(φ))#φ /∈ Tn iff [#φ] 6= [0] (since for all φ, n is a not factor of n× g(φ))If the characteristic functions of the conditions for a function defined by casesfrom recursive functions are themselves recursive then the function defined by128 N. Shackelcases is also recursive. Since Tn is decidable the characteristic function formembership of Tn is recursive. Both g and multiplication by n are recursivefunctions. Hence # is recursive. Therefore # is a Go¨del coding. For any φ, ψ, φand ψ will have Go¨del numbers congruent modulo n only if they have the sametruth value. Therefore # is an acceptable coding. Definition 3.7 We make use of an acceptable coding, #.Let Kn = {k1, . . . , ki} be the set such thatk ∈ Kn iff 0 ≤ k < n and there exists a φ such that φ ∈ Tn andk ≡ #φ (mod n)So Kn is the set of numbers between 0 and n − 1 which are congruent to theGo¨del numbers of the true sentences of Qn.Lemma 3.8 If # is acceptable then for any φ, φ ∈ Tn iff for some k ∈Kn, [#φ] = [k].Proof. Suppose that φ ∈ Tn. By the division algorithm for some r suchthat 0 ≤ r < n, #φ ≡ r (mod n) and hence r ∈ Kn. Suppose that forsome k ∈ Kn, k ≡ #φ (mod n). Then for some ψ ∈ Tn, k ≡ #ψ (mod n),so #φ ≡ #ψ (mod n), hence φ ∈ Tn iff ψ ∈ Tn (because # is acceptable),whence φ ∈ Tn Theorem 3.9 Let the open L-formula Tn(x) be the open formula (x = k1∨x =k2 ∨ · · · ∨ x = ki). Tn(x) defines truth in Qn (and hence in M).Proof. We make use of an acceptable coding, #.For any φ ∈ Sent(L), Qn ` Tn(pφq)iff M|= Tn(pφq) (soundness and completeness)iff M|= [(pφq = k1) ∨ (pφq = k2) ∨ · · · ∨ (pφq = ki)iff either [#φ] = [k1] or[#φ] = [k2] or ... or [#φ] = [ki]iff, for some k, [#φ] = [k] and k ∈ Kniff φ ∈ Tn (by lemma 3.8)Theorem 3.10 Tn(x) is a truth predicate satisfying Tarskis material adequacycondition, i.e. for all φ ∈ Sent(L), Qn ` Tn(pφq)↔ φA Simple Theory Containing its Own Truth Predicate 129Proof. Given any φ, M |= Tn(pφq) iff φ ∈ Tn iff M|= φ, hence M|=Tn(pφq)↔ φ, whence by completeness we have Qn ` Tn(pφq)↔ φ We have now proved our main result. The set of Go¨del numbers of sentencestrue in arithmetic modulo n is definable in arithmetic modulo n, Qn, by thepredicate Tn(x). Hence for all n, arithmetic modulo n is a theory containingits own truth predicate.4 Coding, representability and acceptability.Clearly, the set of Go¨del numbers for a set of expressions, and representabilityin general, is relative to coding: but in Q (Robinson arithmetic) and extensionsof Q all codings can be shown to be N acceptable, simply because Go¨del codingsmust be injections into N. For the same reason, for all the non-standard modelsNS of Q of which I am aware, in which the domain is got by adding extraelements to N, all codings are NS acceptable. So in general, relativity tocoding is irrelevant and not discussed for that reason.However, when using L for a theory of a finite model then the syntax mayno longer be properly represented in the theory. Although any coding willstill be injective into N, and although there will still be an injection from Nto the numerals of L , any theory which induces a partition on the numeralsis in danger of proving that for distinct φ, ψ, pφq = pψq. Of course, it is onlyin danger of doing this if at least one equivalence class has more than onemember. In Qn the relation inducing the partition can be defined by x ∼ yiff Qn ` x = y, and since there are infinite numerals and only n equivalenceclasses of numerals, Qn is in that danger. Indeed, there is a coding for Qnwhich ensures that for all φ, ψ,Qn ` pφq = pψq.Proposition 4.1 For any n > 1 there exists a Go¨del coding of a languageL which makes the code of every formula and its negation congruent to zeromodulo n.Proof. Take any Go¨del coding #. # maps L into N (is an injection). Let gbe a function g : N 7→ N, g : x 7→ nx. Then let f : L 7→ N, f : x 7→ g(#x).Both g and # are injections and hence f is an injection from L into N andas such is a Go¨del coding. Under f all formulas of L are congruent to 0 modulon, a fortiori , so is each formula and its negation. Hence the need for an acceptable coding if anything of significance is to begot out of a coding. But even with an acceptable coding, coding of the syntaxof L into any model of Qn will be many-one and consequently the capacity for130 N. Shackelrepresentation of relations of the models must be fairly crude. The proof aboveshows that being the code of a true sentence is definable. Adapting the Qnacceptable coding given above in a fairly obvious manner would create whatKetland calls a Frege coding.Definition 4.2 A Frege Coding Fc. Given a model M , pick any two ele-ments of the domain, say a and b. Then define the coding as follows, for anyφFc(φ) ={a if M |= φb if M |= ¬φIntuitively, a and b are just the two truth values, ‘The True’ and ‘The False’.Definition 4.3 A Frege Coding for Qn. In our case we can define Fc ,using the acceptable coding of lemma 3.6.Fc(φ) ={[1] if #φ ≡ 0 (mod n)[0] if #φ 6≡ 0 (mod n)It might be that the capacity for representation in Qn is so crude that verylittle else is representable or definable. I dont know. I havent investigatedrepresentability or definability in general for Qn. The following results may berelevant to the problems of definability and representability in Qn:Corollary 4.4 Given an acceptable coding, the diagonal function is not rep-resentable in Qn relative to that coding.Proof. If the diagonal function were representable, we could derive a contra-diction in the standard way. But Qn is consistent (it has a model). Corollary 4.5 Introduce a new predicate T ∗(x) and let DT ∗ be the set of allaxioms T ∗(pφq ↔ φ), with φ ∈ Sent(L). Then Qn ∪ DT ∗is a conservativeextension of Qn .Proof. Take any model M . Define an expansion (M,E) by interpretingthe new predicate T ∗(x) as the set E, where E is got from the set Kn definedabove as followsmM ∈ E iff for some k ∈ Kn,m ≡ k (mod n).Then (M,E) |= Qn ∪DT ∗. (Plainly E is definable in Qn). A Simple Theory Containing its Own Truth Predicate 1315 Final remarksFirst we note some further questions worth asking. 1) Are there codings bywhich we can show M to be an acceptable structure? 2) Could it be thatacceptability is not a problem in free logic?A reservation that might be raised is whether Tn(x) is properly called atruth predicate? Certainly we have seen that it satisfies Tarksi’s materialadequacy condition. Nevertheless, you might reject it because you deny thecoding we used being a Go¨del coding. However, since such coding can never beone-one into a finite model, it is the best coding we can get, and it is sufficientfor truth. So this reservation appears pedantic and we set it aside.In theorem 3.10 we proved our main result. Since for all n, the set ofGo¨del numbers of sentences true in arithmetic modulo n is definable by thepredicate Tn(x), arithmetic modulo n is a simple theory containing its owntruth predicate.2References[1] Boolos, G. and Jeffrey, R. C. Computability and Logic. 3rd ed. CambridgeUniversity Press, 1989.[2] Quine, W. V. From a Logical Point of View : 9 Logico-PhilosophicalEssays. 2nd ed. Harvard University Press, 1980.Nicholas ShackelPhilosophyCardiff UniversityGreat BritainOxford Uehiro Centre for Practical EthicsUniversity of OxfordGreat BritainE-mail: shackeln@cardiff.ac.uk2This paper has benefited from extensive discussion with Jeffrey Ketland.

https://orca.cardiff.ac.uk/id/eprint/120804/7/shackel%20A%20Simple%20Theory%20Containing%20its%20Own%20Truth%20Predicate%20SAJL%202018%20121-131.pdf

A simple theory containing its own truth predicate

Abstract

Similar works

Full text

Available Versions

Online Research @ Cardiff

PhilPapers