A Note on the Unprovability of Consistency in Formal Theories of Truth

Abstract

Why is it that even strong formal theories of truth fail to prove their own consistency? Although Field (Mind, 115, 459, 2006) has addressed this question for many theories of truth, I argue that there is an important and attractive class of theories of truth that he omitted in his analysis. Such theories cannot prove that all their axioms are true, though unlike many of the cases Field considers, they do not prove that any of their axioms are false or that any of their rules of inference are not truth preserving. I argue that it is the fact that such theories are not finitely axiomatizable that stops them from proving their own consistency.

Appendices

Appendix 1

For reference, it will be useful to write out explicitly the axioms of MT and MT^∘ defined in Section 2. The theory MT consists of the axioms:

(MT1_s):

s = s

(MT2$_{s_{1}, ..., t_{n}, f}$):

$(s_{1} = t_{1} \ \& \ ... \ \& \ s_{n} = t_{n} )\supset f(s_{1}, ..., s_{n})=f(t_{1}, ..., t_{n})$

(MT3$_{s_{1}, ..., t_{n}, R}$):

$(s_{1} = t_{1} \ \& \ ... \ \& \ s_{n} = t_{n} )\supset (R(s_{1}, ..., s_{n}) \supset R(t_{1}, ..., t_{n}))$

(MT4):

$x^{\prime } \neq 0$

(MT5):

$x^{\prime }=y^{\prime } \supset x=y$

(MT6):

axioms giving the definitions of further primitive recursive functions

(MT7_ϕ):

$(\phi [z/0] \ \& \ (\forall x)(\phi [z/x] \supset \phi [z/x^{\prime }])) \supset (\forall x)\phi [z/x]$

where in MT7_ϕ, we assume that x does not appear free in ϕ[z/0].

To make notation less cumbersome, we have presented (and will continue to present) axioms as open sentences. In the version of the Hilbert calculus we have been working with, any open sentence is interderivable with the corresponding sentence in which all free variables are universally quantified over.

In MT1-MT3, s, the s_i, and the t_i are arbitrary terms, f any function and R any relation in the signature of the language ${\mathscr{L}}$. Thus, MT1-MT3 are schemes and not single axioms. These axioms are sometimes regarded as logical axioms, but it will be convenient to include them here. In MT4-MT6 (and in the axioms that follow), x, y and z are arbitrary variables, which can of course be replaced with any other variable in our language. The definitions of primitive recursive functions in MT6 involve no connectives or quantifiers, and are purely equational. In axiom MT7, ϕ ranges over all formulae in the full language. In each case (and in what follows) if an axiom is a schema, this is shown by indicating the varying part of the axiom in the subscript s in MTa_s.

We furthermore have the axioms:

(MT8):

¬T(0 = 1)

(MT9_i):

$[\text {Form}(a_{1}) \ \& \ \text {Form}(a_{2}) \ \& \ ... \text {Var}(x_{1}) \ \& \ \text {Var}(x_{2}) \ \& \ ... \\ \text {Term}(t_{1}) \ \& \ \text {Term}(t_{2}) \ \& \ ... ] \supset \mathbf {T}(s_{i}(a_{1}, ... , x_{1}, ..., t_{1}, ... ))$

where in MT9, each s_i(a₁,...,x₁,...,t₁,...) is a primitive recursive function taking the Gödel numbers of the relevant formulae, variables and terms into the Gödel number of the larger formulae corresponding to the i th axiom of the Hilbert calculus, as spelt out in Section 2.

In addition, we also have the axioms:

(MT10):

$(\text {Form}(x_{1}) \ \& \ \text {Form}(x_{2})) \supset ([ \mathbf {T}(x_{1} \supset x_{2}) \ \& \ \mathbf {T}(x_{1})] \supset \mathbf {T}(x_{2}))$

(MT11):

$(\text {Form}(c) \ \& \ \text {Form}(a) \ \& \ \text {Var}(x) \ \& \ \text {Var}(z) \ \& \ \text {NotFree}(x, c)) \supset (\mathbf {T}(c \supset a(x)) \supset \mathbf {T}(c \supset (\forall z) a[x/z]))$

(MT12):

$(\text {Form}(c) \ \& \ \text {Form}(a) \ \& \ \text {Var}(x) \ \& \ \text {Var}(z) \ \& \ \text {NotFree}(x, c)) \supset (\mathbf {T}(a(x) \supset c) \supset \mathbf {T}((\exists z)a[z/x] \supset c))$

The axioms of MT are just MTa for a = 1,...12. For any axiom system U, the axioms of U^∘ are just the closure of the set of axioms of U under the operation that takes a formula ϕ to $\mathbf {T}(\ulcorner \phi \urcorner ).$ Thus, whenever ϕ is an axiom of U^∘ so too is $\mathbf {T}^{(n)}(\ulcorner \phi \urcorner )$, where for every sentence τ we define $\mathbf {T}^{(0)}(\ulcorner \tau \urcorner )$ to be τ, and $\mathbf {T}^{(n)}(\ulcorner \tau \urcorner )$ to be $\mathbf {T}(\ulcorner \mathbf {T}^{(n-1)}(\ulcorner \tau \urcorner ) \urcorner )$.

For each axiom (scheme) MTa and n ≥ 0, we let MTa⁽ⁿ⁾ denote $\mathbf {T}^{(n)}(\ulcorner \phi \urcorner )$, where ϕ is the axiom MTa (or an instance thereof, if MTa is a schema.) With this notation in place, the axioms of (MT)^∘ are just MTa⁽ⁿ⁾ for all n ≥ 0, and a = 1,..., 12.

Appendix 2

In this section, we prove:

Theorem 2

For any formula X, if MT^∘⊩T(c) where c is a numeral and $c=\ulcorner X \urcorner$, then MT^∘⊩ X.

We prove Theorem 2 proof-theoretically, by repurposing standard cut-elimination arguments of the sort found in [16] or [12]. In the version of the sequent calculus we will use, we treat the left and right side of sequents as finite multisets of formulae, eliminating the need for a rule of exchange. Amongst the permissible initial sequents of our system we will include any sequents of the form:

(S1):

A ⊩ A

for A any atomic formula in the language ${\mathscr{L}}$ of MT. (For X non-atomic, the sequent X ⊩ X will be easily derivable from this.) The structural rules of the system will consist of left and right weakening, left and right contraction, and cut:

$$ \begin{array}{c}\displaystyle\frac{{\Delta} \vdash {\Gamma}}{A, {\Delta} \vdash {\Gamma}} \text{LW} \qquad \frac{{\Delta} \vdash {\Gamma}}{{\Delta} \vdash {\Gamma}, A} \text{RW}\\ \displaystyle\frac{A, A, {\Delta} \vdash {\Gamma}}{A, {\Delta} \vdash {\Gamma}} \text{LC} \qquad \frac{{\Delta} \vdash {\Gamma}, A, A}{{\Delta} \vdash {\Gamma}, A} \text{RC}\\ \displaystyle\frac{{\Delta} \vdash {\Gamma}, A \qquad A, {\Delta}^{\prime} \vdash {\Gamma}^{\prime}}{{\Delta}, {\Delta}^{\prime} \vdash {\Gamma}, {\Gamma}^{\prime}} \text{Cut} \end{array} $$

Our system includes the standard logical rules of inference:

$$\begin{array}{c} \displaystyle\frac{{\Delta}, A \vdash {\Gamma}}{{\Delta}, A \& B \vdash {\Gamma}} \text{L\&}\qquad \frac{{\Delta}, B \vdash {\Gamma}}{{\Delta}, A \& B \vdash {\Gamma}} \text{L\&}\\ \displaystyle\frac{{\Delta} \vdash {\Gamma}, A \qquad {\Delta} \vdash {\Gamma}, B}{{\Delta} \vdash {\Gamma}, A\& B} \text{R\&}\\ \displaystyle\frac{{\Delta}, A \vdash {\Gamma} \qquad {\Delta}, B \vdash {\Gamma}}{{\Delta}, A \vee B \vdash {\Gamma}} {\text{L}\vee}\\ \displaystyle\frac{{\Delta} \vdash {\Gamma}, A}{{\Delta} \vdash {\Gamma}, A\vee B} {\text{R}\vee} \qquad \frac{{\Delta} \vdash {\Gamma}, B}{{\Delta} \vdash {\Gamma}, A\vee B} {\text{R}\vee}\\ \displaystyle\frac{{\Delta} \vdash {\Gamma}, A \quad {\Delta}, B \vdash {\Gamma}}{{\Delta}, A \supset B \vdash {\Gamma}} {\text{L}\!\supset} \qquad \frac{{\Delta}, A \vdash {\Gamma}, B}{{\Delta} \vdash {\Gamma}, A \supset B} {\text{R}\supset}\end{array} $$

$$\begin{array}{c} \displaystyle\frac{{\Delta}, A[x/t] \vdash {\Gamma}}{{\Delta}, \forall x A \vdash {\Gamma}} {\text{L}\forall} \qquad \frac{{\Delta} \vdash {\Gamma}, A[x/a]}{{\Delta} \vdash {\Gamma}, \forall x A} {\text{R}\forall}\\ \displaystyle\frac{{\Delta}, A[x/a] \vdash {\Gamma}}{{\Delta}, \exists x A \vdash {\Gamma}} {\text{L}\exists} \qquad \frac{{\Delta} \vdash {\Gamma}, A[x/t]}{{\Delta} \vdash {\Gamma}, \exists x A} {\text{R}\exists}\\ \end{array} $$

where in the quantifier rules, x is an arbitrary variable, t is an arbitrary term (we count variables, constants, and anything composed from them using functions from the signature of the language as terms), and a is a variable - the so-called eigenvariable of the inference - and is assumed not to appear free in the bottom sequent in the rules R∀ and L∃. Here (as usual) A[z/s], where z is a variable and s a term, is the result of replacing every free occurrence of z in A with s.

In each of the rules of inference, ${\Delta }, {\Delta }^{\prime }, {\Gamma }, {\Gamma }^{\prime }$ are the side formulae. The remaining formula in the conclusion (i.e., the bottom sequent) is the principal formula, and the remaining formulae in the premises (i.e., the upper sequents) are the active formulae of the inference. This means that the rule of weakening has no active formula (though it has a principal formula), and the rule of cut no principal formula (though it has active formulae).

The introduction of the equality symbol requires the addition of the following initial sequents:

(S2_s):

⊩ s = s

(S3$_{s_{1}, ..., t_{n}, f}$):

s₁ = t₁,...,s_n = t_n ⊩ f(s₁,...,s_n) = f(t₁,...,t_n)

(S4$_{s_{1}, ..., t_{n}, R}$):

s₁ = t₁,...,s_n = t_n,R(s₁,...,s_n) ⊩ R(t₁,...,t_n)

where s, the s_i, and the t_i are arbitrary terms, f any function and R any relation in the signature of the language ${\mathscr{L}}$.

Because the systems in which we are interested extend arithmetic, we also need the following initial sequents:

(S5):

$x^{\prime } = 0 \vdash$

(S6):

$x^{\prime }=y^{\prime } \vdash x=y$

(S7_p):

axioms of definition for primitive recursive functions

Where here x and y are arbitrary variables. In saying that something is a permissible initial sequent (here and in what follows), we also allow as an initial sequent any variant that may be obtained by renaming variables, or by uniformly substituting terms for free variables. So for example, given S5, the sequent $y^{\prime } = 0 \vdash$, the sequent $7^{\prime } = 0 \vdash$, and the sequent $(x.y)^{\prime } = 0 \vdash$ will also all be permissible initial sequents. Note also that the formulae in the initial sequents in S7 are all atomic.

We will also have a rule of mathematical induction I:

$$ \frac{\phi [z/a], {\Delta} \vdash \phi[z/a^{\prime}], {\Gamma}}{\phi[z/0], {\Delta} \vdash \phi[z/s], {\Gamma}} \text{I} $$

where here the formula ϕ is an abritrary formula in the language of MT (and so may involve the truth predicate), z is an arbitrary variable, s is an arbitrary term, and a is an arbitrary variable referred to as the eigenvariable of the inference. We require that a not appear free in ϕ[z/0],Δ or Γ. In this rule the formulae ϕ[z/a] and $\phi [z/a^{\prime }]$ are the active formulae, ϕ[z/0] and ϕ[z/s] are the principal formulae, and the formulae of Δ,Γ are the side formulae.

Now we turn to initial sequents that explicitly involve the truth predicate. In choosing these initial sequents, the idea is simply to capture the content of the axioms of MT^∘ in such a way that all initial sequents contain only atomic formulae.

We begin with initial sequents that involve the truth predicate and the equality symbol. For all n ≥ 1 we have:

(S8_s):

$\vdash \mathbf {T}^{(n)}(\ulcorner s=s \urcorner )$

(S9$_{\vec {s}, \vec {t}}$):

$\vdash \mathbf {T}^{(n)}(\ulcorner \vec {s}=\vec {t} \supset f(\vec {s})=f(\vec {t}) \urcorner )$

(S10$_{\vec {s}, \vec {t}}$):

$\vdash \mathbf {T}^{(n)}(\ulcorner \vec {s}=\vec {t} \supset [R(\vec {s}) \leftrightarrow R(\vec {t}) ] \urcorner )$

where s, the s_i and t_i are arbitrary terms. (These are axiom schemes.)

For each of the axioms S5-7 there will be a corresponding axiom involving the truth predicate, with all formulae moved to the right side in the natural way. In particular, for n ≥ 1 we have:

(S11):

$\vdash \mathbf {T}^{(n)}(\ulcorner x^{\prime } \neq 0 \urcorner )$

(S12):

$\vdash \mathbf {T}^{(n)}(\ulcorner x^{\prime } \neq y^{\prime } \supset x=y \urcorner )$

(S13):

initial sequents of the form $\vdash \mathbf {T}^{(n)}(\ulcorner ... \urcorner )$ corresponding to initial sequents of the form S7_p.

Again, with initial sequents S11-13 and those that follow, we will assume that any variant that may be obtained by renaming variables or by substituting terms for free variables is also a permissible initial sequent.

We also have initial sequents concerning the truth of the principle of mathematical induction:

(S14_ϕ):

$\vdash \mathbf {T}^{(n)}(\ulcorner [\phi [z/0]\ \& \ (\forall x)(\phi [z/x] \supset \phi [z/x^{\prime }])] \supset (\forall x)\phi [z/x] \urcorner )$

where ϕ is a formula in the language of MT, and where we assume that x does not appear free in ϕ[z/0].

We will also add some further mathematical initial sequents exclusively concerning the nature of truth predicate. In particular, we have the following for all n ≥ 1:

(S15):

T(0 = 1) ⊩

(S16):

$\vdash \mathbf {T}^{(n)}(\ulcorner \neg \mathbf {T}(0=1) \urcorner )$

(S17_i):

Form(a₁),Form(a₂),...,Var(x₁),Var(x₂),...,Term(t₁),Term(t₂),... ⊩T(s_i(a₁,...,x₁,...,t₁,...))

(S18_i):

$\vdash \mathbf {T}^{(n)}(\ulcorner [\text {Form}(a_{1}) \ \& \ \text {Form}(a_{2}) \ \& \ ... \text {Var}(x_{1}) \ \& \ \text {Var}(x_{2}) \ \& \ ... \\ \text {Term}(t_{1}) \ \& \ \text {Term}(t_{2}) \ \& \ ... ] \supset \mathbf {T}(s_{i}(a_{1}, ..., x_{1}, ..., t_{1}, ...))\urcorner )$

where in S17 and S18, each s_i(x₁,x₂,...) is a primitive recursive function taking the Gödel numbers of the relevant formulae, variables and terms into the Gödel number of the larger formulae corresponding to the i th axiom of the Hilbert calculus, as spelt out in Section 2.

In addition, we have the following initial sequents for all n ≥ 1:

(S19):

$\text {Form}(x_{1}), \text {Form}(x_{2}), \mathbf {T}(x_{1} \supset x_{2}), \mathbf {T}(x_{1}) \vdash \mathbf {T}(x_{2})$

(S20):

$\vdash \mathbf {T}^{(n)}(\ulcorner [\text {Form}(x_{1}) \ \! \& \ \! \text {Form}(x_{2})] \supset [(\mathbf {T}(x_{1} \supset x_{2}) \ \& \ \mathbf {T}(x_{1})) \supset \mathbf {T}(x_{2})] \urcorner )$

(S21):

$\text {Form}(c), \text {Form}(a), \text {Var}(x), \text {Var}(z), \text {NotFree}(x, c), \mathbf {T}(c \supset a(x)) \vdash \mathbf {T}(c \supset (\forall z) a[x/z])$

(S22):

$\vdash \mathbf {T}^{(n)}(\ulcorner [\text {Form}(c) \ \! \& \ \! \text {Form}(a) \ \! \& \ \! \ \text {Var}(x) \ \! \& \ \! \ \text {Var}(z) \ \! \& \ \! \ \text {NotFree}(x, c)] \supset [\mathbf {T}(c \supset a(x)) \supset \mathbf {T}(c \supset (\forall z) a[x/z])] \urcorner )$

(S23):

$\text {F\hspace {-.5pt}o\hspace {-.5pt}r\hspace {-.5pt}m\hspace {-.5pt}}(c),\hspace {-.5pt} \text {F\hspace {-.5pt}o\hspace {-.5pt}r\hspace {-.5pt}m\hspace {-.5pt}}(a\hspace {-.5pt}),\hspace {-.5pt} \text {V\hspace {-.5pt}a\hspace {-.5pt}r\hspace {-.5pt}}(\hspace {-.5pt}x\hspace {-.5pt})\hspace {-.5pt},\hspace {-.5pt} \text {Var\hspace {-.5pt}}(\hspace {-.5pt}z\hspace {-.5pt})\hspace {-.5pt},\hspace {-.5pt} \text {N\hspace {-.5pt}o\hspace {-.5pt}t\hspace {-.5pt}F\hspace {-.5pt}r\hspace {-.5pt}e\hspace {-.5pt}e\hspace {-.5pt}}(\hspace {-.5pt}x\hspace {-.5pt},\hspace {-.5pt} c\hspace {-.5pt})\hspace {-.5pt},\hspace {-.5pt} \mathbf {\hspace {-.5pt}T\hspace {-.5pt}}\hspace {-.5pt}(\hspace {-.5pt}a\hspace {-.5pt}(\hspace {-.5pt}x\hspace {-.5pt})\hspace {-.5pt} \supset c) \vdash \mathbf {T}((\exists z)a[z/x]\break \supset c)$

(S24):

$\vdash \mathbf {T}^{(n)}(\ulcorner [\text {Form}(c) \ \! \& \ \! \text {Form}(a) \ \! \& \ \! \ \text {Var}(x) \ \! \& \ \! \ \text {Var}(z) \ \! \& \ \! \ \text {NotFree}(x, c)] \supset [\mathbf {T}(a(x) \supset c) \supset \mathbf {T}((\exists z)a[x/z] \supset c)] \urcorner )$

We call this system $\overline {\mathbf {MT}}$. For a formula X in the language MT, we say that X is a theorem of $\overline {\mathbf {MT}}$ just in case there is a proof of the sequent ⊩ X in $\overline {\mathbf {MT}}$. We then have the following:

Lemma 3

A formula is a theorem of $\overline {\mathbf {MT}}$ iff it is a theorem of MT^∘.

Proof

First, we argue that if a formula is a theorem of $\overline {\mathbf {MT}}$, then it is a theorem of MT^∘. Given a sequent μ of the form σ₁,...,σ_i ⊩ τ₁,...,τ_j, let S(μ) be the sentence:

$$\neg (\sigma_{1} \ \& ... \& \ \sigma_{i} ) \vee (\tau_{1} \vee ... \vee \tau_{j}).$$

It then suffices to show that (i) for every initial sequent μ of $\overline {\mathbf {MT}}$, S(μ) is a theorem of MT^∘, and (ii) if a rule of inference of $\overline {\mathbf {MT}}$ takes sequents μ₁,μ₂,... into ν, then if each S(μ_i) is a theorem of MT^∘, then S(ν) is also a theorem of MT^∘.

First we prove (i). (In proving (i), recall that for every permissible initial sequent, the sequent obtained by renaming variables or uniformly substituting terms for free variables is also a permissible initial sequent.) That (i) holds may be verified by simply by going through the initial sequents S1-S24 and checking that in each case the relevant sentence S(μ) is a theorem of MT^∘. For example, each instance of S(S1) is a logical truth, and so is a theorem of MT^∘. Each of S(S2) through S(S7) is either identical with or a trivial logical consequence of MT1 through MT6, and thus a theorem of MT^∘. Likewise each of S(S8) through S(S14) is either identical with or a trivial logical consequence of MT1⁽ⁿ⁾ through MT7⁽ⁿ⁾ for n ≥ 1, S(S15) of MT8, S(S16) of MT8⁽ⁿ⁾, S(S17) of MT9, S(S18) of MT9⁽ⁿ⁾, S(S19) of MT10, S(S20) of MT10⁽ⁿ⁾, S(S21) of MT11, S(S22) of MT11⁽ⁿ⁾, S(S23) of MT12, and S(S24) of MT12⁽ⁿ⁾, where n > 1.

The proof of (ii) is also easy. One can verify that for every structural and logical rule taking sequents μ₁,μ₂,... into ν, s(ν) is a logical consequence of s(μ₁),s(μ₂),..., and thus if s(μ₁),s(μ₂),... are provable in MT^∘, so is s(ν). Consider then the rule of induction. Suppose this rule takes the sequent $\phi [z/a], {\Delta } \vdash \phi [z/a^{\prime }], {\Gamma }$ to ϕ[z/0],Δ ⊩ ϕ[z/s],Γ, where a does not occur free in Δ,Γ or ϕ[z/0]. If we denote the upper sequent of this inference by μ and the lower sequent of this inference by ν, then the sentence s(μ) is logically equivalent to:

$$({\Delta} \& \neg {\Gamma}) \supset (\phi [z/a] \supset \phi [z/a^{\prime}]).$$

If this is a theorem of MT^∘, and a does not occur free in Δ or Γ (as we are assuming), then in the Hilbert calculus presented in Section 2 we may immediately infer:

$$({\Delta} \& \neg {\Gamma}) \supset (\forall a)(\phi [z/a] \supset \phi [z/a^{\prime}]).$$

From this and MT7, we may then infer:

$$({\Delta} \& \neg {\Gamma}) \supset (\phi [z/0] \supset (\forall a)\phi [z/a])$$

where we use the fact that a does not appear free in ϕ[z/0]. Using the fact that in the Hilbert calculus we have the logical axiom $(\forall a)\phi [z/a] \supset \phi [z/s]$, we then have:

$$({\Delta} \& \neg {\Gamma}) \supset (\phi [z/0] \supset \phi [z/s])$$

which is logically equivalent to s(ν). Thus, if s(μ) is a theorem of MT^∘, so too is s(ν), as desired.

We now argue that if a sentence is a theorem of MT^∘, then it is a theorem of $\overline {\mathbf {MT}}$. It suffices to verify that for n ≥ 0, each MTi⁽ⁿ⁾ is a theorem of $\overline {\mathbf {MT}}$. This is largely routine. The following derived rule is easily proved and freely used in what follows:

$$ \frac{{}A_{1}, ... A_{n} \vdash B_{1}, ..., B_{m}}{\qquad\qquad\quad\vdash (A_{1} \& ... \& A_{n}) \supset (B_{1} \vee ... \vee B_{m})} \text{Der} $$

Proofs in $\overline {\mathbf {MT}}$ of the sequent ⊩ ψ where ψ is MTi⁽⁰⁾ for i = 1,...6 may be obtained trivially or by applying this derived rule to the initial sequents S2 through S7. If ψ is MTi⁽ⁿ⁾ for i = 1,..., 6 and n > 0, ⊩ ψ may be proved trivially from the initial sequents S8 through S13. In similar ways, if ψ is MTi⁽ⁿ⁾ for i = 8,..., 12 and n = 0 or n ≥ 0, ⊩ ψ may be derived from the initial sequents S15 through S24.

The only slightly non-trivial case to consider is if ψ is MT7⁽ⁿ⁾. If n > 0, a proof of ⊩ ψ is easily obtained from S14. So the only case left to consider is if ψ is MT7⁽⁰⁾, i.e.:

$$(\phi [z/0] \ \& \ (\forall x)(\phi [z/x] \supset \phi [z/x^{\prime}])) \supset (\forall x)\phi [z/x]$$

where x is not free in ϕ[z/0]. This may be proved easily in $\overline {\mathbf {MT}}$:

$$ \frac{\displaystyle\frac{\phi[z/x] \vdash \phi[z/x]}{\phi[z/x] \vdash \phi[z/x], \phi[z/x^{\prime}]} \text{RW}\qquad\frac{\phi[z/x^{\prime}] \vdash \phi[z/x^{\prime}]}{\phi[z/x], \phi[z/x^{\prime}] \vdash \phi[z/x^{\prime}]} \text{LW}}{\displaystyle\frac{\displaystyle\frac{\displaystyle\frac{\phi[z/x], \phi [z/x] \supset \phi[z/x^{\prime}] \vdash \phi[z/x^{\prime}]}{\phi[z/x], (\forall x)(\phi[z/x]\supset \phi[z/x^{\prime}]) \vdash \phi[z/x^{\prime}]}{\text{L} \forall }}{\phi[z/0], (\forall x)(\phi[z/x]\supset \phi[z/x^{\prime}]) \vdash \phi[z/x]}\text{I}}{\phi[z/0], (\forall x)(\phi[z/x]\supset \phi[z/x^{\prime}]) \vdash (\forall x) \phi[z/x]}{\text{R} \forall}} {\text{L}\!\supset} $$

Even if ϕ is not atomic, the sequents ϕ[z/x] ⊩ ϕ[z/x] and $\phi [z/x^{\prime }] \vdash \phi [z/x^{\prime }]$ may be easily derived. We may apply the rule I as shown, because x is not free in $(\forall x)(\phi [z/x]\supset \phi [z/x^{\prime }])$ (as the variable x is bound in this expression), and nor is x free in ϕ[z/0] by assumption. For the same reasons, the subsequent R∀ is also permissible. An application of the Der rule to the endsequent then gives a proof of MT7⁽⁰⁾.

It follows that $\overline {\mathbf {MT}}$ and MT^∘ have the same theorems. □

In light of Lemma 3, to prove Theorem 2 it suffices to prove:

Lemma 4

For any formula X, if $\overline {\mathbf {MT}}$ proves ⊩T(c) where c is a numeral and $c=\ulcorner X \urcorner$, then $\overline {\mathbf {MT}}$ proves ⊩ X.

Given any sequent Δ ⊩Γ, for any variable x and term t we denote the sequent obtained from Δ ⊩Γ by replacing all free occurrences of x with t by Δ[x/t] ⊩Γ[x/t]. Say the the set of permissible initial sequents of some system are closed under substitution iff for any permissible initial sequent Δ ⊩Γ, variable x and term t, if t is free for x in Δ and Γ,¹Δ[x/t] ⊩Γ[x/t] is also a permissible initial sequent. The permissible initial sequents of $\overline {\mathbf {MT}}$ are then closed under substitution.

The main result we will use to prove Lemma 4 is:

Lemma 5

Let V be a theory formulated in the sequent calculus in a language ${\mathscr{L}}$ containing at least a constant 0 and a unary function $^{\prime }$, and having the following properties:

(1).:

All formulae in the initial sequents of V are atomic.

(2).:

If A is an atomic formula of ${\mathscr{L}}$, A ⊩ A (i.e., (S1)) is an initial sequent of V, as are all the initial sequents (S2)-(S4) of equality.

(3).:

The initial sequents of V are closed under substitution.

(4).:

The rules of inference of V are precisely the usual structural rules, logical rules, and the rule of mathematical induction (as spelt out in the beginning of Appendix 2.)

(5).:

If τ is a closed term in the language of ${\mathscr{L}}$, then for some numeral n (i.e., term of the form $0^{\prime \prime }{}^{\dots } {}^{\prime }$) there is a proof of ⊩ τ = n that uses no induction or non-atomic cuts.

Suppose that in V there a proof of a sequent μ such that every formula in μ is atomic. Then there is a proof of μ in V such that every formula in the proof is atomic, and the only rules used are cut, contraction and weakening. Moreover, if every formula in μ is closed, then every formula in the proof may also be taken to be closed.

Proof

The proof involves standard proof-theoretic techniques of the sort found in [16] and [12], for example. To start, it is convenient to work with a larger infinitary sequent system V^ω to be defined into which V can be embedded. The formulae that appear in the sequents of V^ω will be formulae in the same language ${\mathscr{L}}$ of V.

To define V^ω, we first define the rank of a formula ϕ of ${\mathscr{L}}$ to be the number of connectives or quantifiers that occur in ϕ. The rank of an atomic formula is therefore 0. In the system V^ω the turnstile is decorated with indices $\vdash ^{\alpha }_{n}$. Intuitively, α will be an ordinal which gives an upper bound on the ‘height’ of the proof, and n will be a natural number such that every cut formula appearing above the sequent in question has rank < n. If n = 0, this means that no cuts occur above the sequent in question. In the system V^ω, whenever we have ${\Delta } \vdash ^{\alpha }_{n} {\Lambda }$, we will also have ${\Delta } \vdash ^{\alpha ^{\prime }}_{n^{\prime }} {\Lambda }$ for $\alpha ^{\prime } \geq \alpha$ and $n^{\prime } \geq n$.

The system V^ω will have the same initial sequents as V, and we will have ${\Delta } {\vdash ^{0}_{0}} {\Lambda }$ in V^ω iff Δ ⊩Λ is an initial sequent of V. The system V^ω will also have all the same logical and structural rules that V has, though without a rule of induction. For any logical rule, if the rule has premises ${\Delta }_{i} \vdash ^{\alpha _{i}}_{n_{i}} {\Lambda }_{i}$ (with i = 1 or 2) that are derivable in V^ω, then the corresponding conclusion ${\Delta } \vdash ^{\alpha }_{n} {\Lambda }$ will be derivable in V^ω, where ${\max \limits } \{ \alpha _{1}, \alpha _{2} \} < \alpha$ and ${\max \limits } \{ n_{1}, n_{2} \} \leq n.$

For the rules of contraction and weakening, if the premise ${\Delta } \vdash ^{\alpha }_{n} {\Lambda }$ is derivable in V^ω, then the corresponding conclusion ${\Delta }^{\prime } \vdash ^{\beta }_{n^{\prime }} {\Lambda }^{\prime }$ will be derivable in V^ω, where α ≤ β and $n^{\prime } \leq n$. In this way, contractions and weakenings do not contribute to the ‘height’ of the proof.

In the case of cut, if we have ${\Delta } \vdash ^{\alpha _{1}}_{n_{1}} {\Lambda }, A$ and $A, {\Delta }^{\prime } \vdash ^{\alpha _{2}}_{n_{2}} {\Lambda }^{\prime }$, then we will have ${\Delta }, {\Delta }^{\prime } \vdash ^{\alpha }_{n} {\Lambda }, {\Lambda }^{\prime }$ for any α and n such that ${\max \limits } \{ \alpha _{1}, \alpha _{2} \} < \alpha$, ${\max \limits } \{ n_{1}, n_{2} \} \leq n$, and rank(A) < n.

Instead of a rule of induction, the system V^ω will have an ω-rule that allows us to infer (∀x)ϕ(x) when each ϕ(1),ϕ(2),... is provable, and there is some finite upper bound on the ranks of the cut formulae that appear in the proofs of the ϕ(i). More specifically, if for each natural number i we have ${\Delta } \vdash ^{\alpha _{i}}_{n_{i}} \phi (i), {\Gamma }$, and for some n we have n_i ≤ n for all i, then we will have ${\Delta } \vdash ^{\alpha }_{n} (\forall x)\phi (x), {\Gamma }$ for any α such that for all i, α_i ≤ α. (In such an inference, we say that the formula (∀x)ϕ(x) in the lower sequent is the principal formula, and the formulae ϕ(i) in the upper sequents are the active formulae.) □

We then prove Lemma 5 using the following sequence of lemmas.

Lemma 5.1

If the sequent ${\Delta } \vdash ^{\alpha }_{n} {\Gamma }$ is provable in V^ω, then for any variable x and term t such that t is free for x in the sequent Δ ⊩Γ, ${\Delta }[x/t] \vdash ^{\alpha }_{n} {\Gamma }[x/t]$ is also provable in V^ω.

Proof

The proof is a simple induction on the construction of proofs in V^ω. The base case uses condition 3 of the Lemma, and the inductive step is straightforward. □

Next we show that V can indeed be embedded in V^ω.

Lemma 5.2

If the sequent Δ ⊩Λ is provable in V, then for some α,n we have that ${\Delta } \vdash ^{\alpha }_{n} {\Lambda }$ is provable in V^ω.

Proof

The argument is a straightforward induction on the construction of proofs in V. Using condition 4, the only non-trivial case to be considered is the use of the rule of induction in V. Suppose then that by inductive hypothesis we have $\phi [z/a], {\Delta } \vdash ^{\alpha }_{n} \phi [z/a^{\prime }], {\Lambda }$ in V^ω, where a is not free in ϕ[z/0],Δ or Γ. Using Lemma 5.1, this means that we have $\phi [z/i], {\Delta } \vdash ^{\alpha }_{n} \phi [z/i^{\prime }], {\Lambda }$ for each i. Letting $n^{\prime }$ be any natural number such that $n, \text {rank}(\phi ) < n^{\prime }$, we can then prove $\phi [z/0], {\Delta } \vdash ^{\alpha +(i-1)}_{n^{\prime }} \phi [z/i], {\Lambda }$ in V^ω for each i > 0. For example, in the case i = 3 we have:

$$\displaystyle\frac{\begin{array}{lr}{\displaystyle\frac{\begin{array}{ccc}\vdots&&\qquad\vdots\\ \phi[z/0], {\Delta} \vdash^{\alpha}_n \ \phi[z/0^{\prime}], {\Gamma}&&\qquad \phi[z/0^{\prime}], {\Delta} \vdash^{\alpha}_n \ \phi[z/0^{\prime\prime}], {\Gamma} \end{array}}{\phi[z/0], {\Delta} \vdash^{\alpha+1}_{n^{\prime}} \ \phi[z/0^{\prime\prime}], {\Gamma}} \text{Cut}}&\begin{array}{c}\vdots\\\phi[z/0^{\prime\prime}], {\Delta} \vdash^{\alpha}_{n} \ \phi[z/0^{\prime\prime\prime}], {\Gamma}\end{array}\end{array}}{\phi[z/0], {\Delta} \vdash^{\alpha+2}_{n^{\prime}} \ \phi[z/0^{\prime\prime\prime}], {\Gamma}} \text{Cut} $$

A simple induction on the construction of ϕ (invoking condition 2 for the base case) gives $\phi [z/0] \vdash ^{\text {{rank}}(\phi )}_{0} \phi [z/0]$. We thus also have $\phi [z/0], {\Delta } \vdash ^{\text {{rank}}(\phi )}_{0} \phi [z/0], {\Lambda }$. Applying the ω-rule to this and the sequent $\phi [z/0], {\Delta } \vdash ^{\alpha +(i-1)}_{n^{\prime }} \phi [z/i], {\Lambda }$ in V^ω gives

$$\phi[z/0], {\Delta} \vdash^{\alpha+\omega}_{n^{\prime}} (\forall z) \phi[z], {\Lambda}.$$

For any term s, we have $\phi [z/s] \vdash ^{\text {{rank}}(\phi )}_{0} \phi [z/s]$ and thus also have $(\forall z)\phi [z] \vdash ^{\text {{rank}}(\phi )+1}_{0} \phi [z/s]$. Applying the cut rule to this and the sequent $\phi [z/0], {\Delta } \vdash ^{\alpha +\omega }_{n^{\prime }} (\forall z) \phi [z], {\Lambda }$, we can then derive the sequent $\phi [z/0], {\Delta } \vdash ^{\alpha +\omega }_{n^{\prime }+1} \phi [z/s], {\Lambda }$ in V^ω as desired. □

Next we have the main component of our cut elimination Lemma:

Lemma 5.3

If in V^ω, for some n ≥ 1 we have ${\Delta } \vdash ^{\alpha }_{n} {\Lambda }, A$ and $A, {\Delta } \vdash ^{\beta }_{n} {\Lambda }$ and rank(A) = n, then we also have ${\Delta } \vdash ^{\max \limits {(\alpha \# \beta , \omega )}}_{n} {\Lambda }$.

Proof

Here α#β is the natural ordinal sum of α and β. The proof is by induction on α#β, and is a standard cut-elimination argument. We will assume familiarity with ideas and notation of basic cut elimination arguments of the sort found in [16] and [12], for example.

For the base case, it suffices to suppose that α = β = 0. But if ${\Delta } {\vdash ^{0}_{n}} {\Lambda }, A$ and $A, {\Delta } {\vdash ^{0}_{n}} {\Lambda }$, then these must be initial sequents, and so by condition 1 we must have rank(A) = 0 and thus n = 0, and so the theorem holds vacuously.

Suppose now that the theorem is true whenever α#β < γ, and that we have ${\Delta } \vdash ^{\bar {\alpha }}_{n} {\Lambda }, A$ and $A, {\Delta } \vdash ^{\bar {\beta }}_{n} \ {\Lambda }$ with rank(A) = n, $\bar {\alpha } \# \bar {\beta } = \gamma$, and n ≥ 1. We must show that ${\Delta } \vdash ^{\max \limits {(\bar {\alpha } \# \bar {\beta }, \omega )}}_{n} {\Lambda }$.

Because A is not atomic and all formuae in initial sequents are atomic, neither ${\Delta } \vdash ^{\bar {\alpha }}_{n} {\Lambda }, A$ nor $A, {\Delta } \vdash ^{\bar {\beta }}_{n} \ {\Lambda }$ are initial sequents. These sequents are therefore the conclusions of inferences.

Suppose first that A is not the principal formula in one of these inferences. Without loss of generality, we suppose that A is a side formula in the inference with conclusion ${\Delta } \vdash ^{\bar {\alpha }}_{n} {\Lambda }, A$. Then it is also a side formula in the sequent(s) that form the premises of the inference whose conclusion is ${\Delta } \vdash ^{\bar {\alpha }}_{n} {\Lambda }, A$, and the cut may be applied to the premises. For example, suppose ${\Delta } \vdash ^{\bar {\alpha }}_{n} {\Lambda }, A$ is the result of a R&:

$$ \frac{{\Delta} \vdash^{\xi_{1}}_{n} \ {\Lambda}^{\prime}, B, A \qquad {\Delta} \vdash^{\xi_{2}}_{n} \ {\Lambda}^{\prime}, C, A}{{\Delta} \vdash^{\bar{\alpha}}_{n} \ {\Lambda}^{\prime}, B \& C, A} \text{R\&} $$

where ${\Lambda } ={\Lambda }^{\prime }, B \& C$, and $\xi _{1}, \xi _{2} < \bar {\alpha }$. Then because $\xi _{1} < \bar {\alpha }$, we have $\xi _{1} \# \bar {\beta } < \bar {\alpha } \# \bar {\beta } = \gamma$, and thus we may apply our inductive hypothesis to ${\Delta } \vdash ^{\xi _{1}}_{n} \ {\Lambda }^{\prime }, B, A$ and $A, {\Delta } \vdash ^{\bar {\beta }}_{n} \ {\Lambda }$ to obtain ${\Delta } \vdash ^{\xi _{1} \# \bar {\beta }}_{n} \ {\Lambda }, {\Lambda }^{\prime }, B$ (after weakenings.) Likewise, we have ${\Delta } \vdash ^{\xi _{2} \# \bar {\beta }}_{n} \ {\Lambda }, {\Lambda }^{\prime }, C$. By R& we then have ${\Delta } \vdash ^{{\max \limits } (\xi _{1} \# \bar {\beta }, \xi _{2} \# \bar {\beta })+1}_{n} \ {\Lambda }, {\Lambda }^{\prime }, B \& C$, which by the fact that ${\Lambda } ={\Lambda }^{\prime }, B \& C$, the use of contraction, and the fact that ${\max \limits } (\xi _{1} \# \bar {\beta }, \xi _{2} \# \bar {\beta })+1 \leq \bar {\alpha } \# \bar {\beta }$ gives ${\Delta } \vdash ^{\bar {\alpha } \# \bar {\beta }}_{n} \ {\Lambda }$ and thus ${\Delta } \vdash ^{\max \limits {(\bar {\alpha } \# \bar {\beta }, \omega )}}_{n} {\Lambda }$ as desired.

Although we have only given the argument for the case in which ${\Delta } \vdash ^{\bar {\alpha }}_{n} {\Lambda }, A$ is the conclusion of an R& inference, the argument for all the other possible rules of inference from which ${\Delta } \vdash ^{\bar {\alpha }}_{n} {\Lambda }, A$ could have been derived are essentially identical.

To complete the induction, we may thus suppose that A is the principal formula in both ${\Delta } \vdash ^{\bar {\alpha }}_{n} {\Lambda }, A$ and $A, {\Delta } \vdash ^{\bar {\beta }}_{n} \ {\Lambda }$. We must consider the cases in which (i) these sequents are the conclusions of R∘ and L∘ inferences respectively for some connective or quantifer ∘, (ii) ${\Delta } \vdash ^{\bar {\alpha }}_{n} {\Lambda }, A$ is the conclusion of an application of the ω-rule and $A, {\Delta } \vdash ^{\bar {\beta }}_{n} \ {\Lambda }$ the result of a L∀, and (iii) one of these sequents is the conclusion of a contraction or weakening.

Case (iii) is easily seen to reduce to cases (i) or (ii) or the case in which A is just a side formula. Both cases (i) and (ii) may be dealt with by standard cut-elimination arguments. For example, suppose we are in case (i) and that A has the form B&C. Then ${\Delta } \vdash ^{\bar {\alpha }}_{n} {\Lambda }, A$ must be the result of an R& inference, and $A, {\Delta } \vdash ^{\bar {\beta }}_{n} \ {\Lambda }$ must be the result of a L& inference. That is to say, we must have something like:

$$ \begin{array}{c} \displaystyle\frac{B, {\Delta} \vdash^{\beta_{1}}_{n} \ {\Lambda}}{B \& C, {\Delta} \vdash^{\bar{\beta}}_{n} \ {\Lambda}}\text{L\&}\\ \displaystyle\frac{{\Delta} \vdash^{\alpha_{1}}_{n} \ {\Lambda}, B \qquad {\Delta} \vdash^{\alpha_{2}}_{n} \ {\Lambda}, C}{{\Delta} \vdash^{\bar{\alpha}}_{n} {\Lambda}, B \& C} \text{R\&} \end{array} $$

where $\beta _{1} < \bar {\beta }$ and $\alpha _{1}, \alpha _{2} < \bar {\alpha }$. Then because rank(B) < n, we may apply the cut rule to $B, {\Delta } \vdash ^{\beta _{1}}_{n} {\Lambda }$ and ${\Delta } \vdash ^{\alpha _{1}}_{n} {\Lambda }, B$ to obtain ${\Delta } \vdash ^{{\max \limits } (\alpha _{1}, \beta _{1})+1}_{n} {\Lambda }$. Because ${\max \limits } (\alpha _{1}, \beta _{1}) < {\max \limits } (\bar {\alpha }, \bar {\beta }) \leq \bar {\alpha } \# \bar {\beta }$, we have ${\max \limits } (\alpha _{1}, \beta _{1}) +1 \leq \bar {\alpha } \# \bar {\beta }$, and so ${\Delta } \vdash ^{\bar {\alpha } \# \bar {\beta }}_{n} {\Lambda }$ and thus ${\Delta } \vdash ^{\max \limits {(\bar {\alpha } \# \bar {\beta }, \omega )}}_{n} {\Lambda }$ as desired.

The case of quantifier rules involves some more details. We also consider case (ii) explicitly. In this case, A has the form (∀x)ϕ(x), ${\Delta } \vdash ^{\bar {\alpha }}_{n} {\Lambda }, A$ is the result of an application of the ω-rule, and $A, {\Delta } \vdash ^{\bar {\beta }}_{n} \ {\Lambda }$ the result of an application of L∀. In this case, we have:

$$ \begin{array}{c} \displaystyle\frac{\phi[x/t], {\Delta} \vdash^{\beta_{1}}_{n} \ {\Lambda}}{(\forall x)\phi(x), {\Delta} \vdash^{\bar{\beta}}_{n} \ {\Lambda}}{\text{L}\forall}\\ \displaystyle\frac{{\Delta} \vdash^{\alpha_{0}}_{n} \ {\Lambda}, \phi[x/0] \qquad {\Delta} \vdash^{\alpha_{1}}_{n} \ {\Lambda}, \phi[x/1] \qquad {...}}{{\Delta} \vdash^{\bar{\alpha}}_{n} \ {\Lambda}, (\forall x)\phi(x)} {\omega\text{-rule}} \end{array} $$

where $\beta _{1} < \bar {\beta }$ and each $\alpha _{i} < \bar {\alpha }$. Suppose the term t in the premise of the L∀ inference is an open term containing the free variables $\vec {v}$. Let $t^{\prime }$ be the closed term obtained by substituting 0 for all the $\vec {v}$. Then by Lemma 5.1 we will also have $\phi [x/t^{\prime }], {\Delta } \vdash ^{\beta _{1}}_{n} {\Lambda }$. (If t has no free variables then we let $t^{\prime }$ be t.) By condition 5 of the Lemma, there will be some numeral p such that $\vdash t^{\prime }=p$ is provable in V without induction or non-atomic cuts. It follows easily that ${\vdash ^{m}_{1}} t^{\prime }=p$ in V^ω for some natural number m. Using the fact that V^ω contains the initial sequents of equality (by condition 2), it is also easily shown that $t^{\prime }=p, \phi [x/p] \vdash ^{\bar {m}}_{1} \phi [x/t^{\prime }]$ for some natural number $\bar {m}$. Applying the cut and weakening rules to $t^{\prime }=p, \phi [x/p] \vdash ^{\bar {m}}_{1} \phi [x/t^{\prime }]$ and ${\vdash ^{m}_{1}} t^{\prime }=p$ gives $\phi [x/p] {\vdash ^{q}_{1}} \phi [x/t^{\prime }]$ for some natural number q. Applying weakening and cut rules to this and $\phi [x/t^{\prime }], {\Delta } \vdash ^{\beta _{1}}_{n} {\Lambda }$ and using the fact that rank$(\phi [x/t^{\prime }])<n$ gives $\phi [x/p], {\Delta } \vdash ^{\max \limits (\beta _{1}, q)+1}_{n} {\Lambda }$. Applying the cut rule to this sequent and ${\Delta } \vdash ^{\alpha _{p}}_{n} {\Lambda }, \phi [x/p]$, we then have ${\Delta } \vdash ^{\max \limits (\alpha _{p}, \max \limits (\beta _{1}, q)+1)+1}_{n} \ {\Lambda }$.

It is easily verified that if at least one of α or β are infinite, then $\max \limits (\alpha _{p}, \max \limits (\beta _{1}, q)+1)+1 \leq \max \limits (\beta _{1}+2, \alpha _{p}+1)$, the right hand side of which is in turn $\leq \max \limits (\bar {\beta }+1, \bar {\alpha })$. The only way $\max \limits (\bar {\beta }+1, \bar {\alpha })$ could be strictly greater than $\bar {\alpha } \# \bar {\beta }$ is if $\bar {\alpha }=0$, which would require α_p < 0, which is impossible. So if one of α or β are infinite, then ${\Delta } \vdash ^{\bar {\alpha } \# \bar {\beta }}_{n} \ {\Lambda }$. If both α and β are finite, then we have ${\Delta } \vdash ^{\omega }_{n} \ {\Lambda }$. So either way ${\Delta } \vdash ^{\max \limits {(\bar {\alpha } \# \bar {\beta }, \omega )}}_{n} {\Lambda }$, as desired. □

Finally we have the cut elimination Lemma itself:

Lemma 5.4

(Cut Elimination) If in V^ω we have ${\Delta } \vdash ^{\alpha }_{n+1} {\Lambda }$ for n ≥ 1, then we also have ${\Delta } \vdash ^{\omega ^{\alpha }}_{n} {\Lambda }$.

Proof

The proof is by induction on the construction of the derivation of ${\Delta } \vdash ^{\alpha }_{n+1} {\Lambda }$. If ${\Delta } \vdash ^{\alpha }_{n+1} {\Lambda }$ is an initial sequent then the result is immediate. If the last step of the derivation of ${\Delta } \vdash ^{\alpha }_{n+1} {\Lambda }$ is a weakening, contraction, cut of rank < n or logical rule then the inductive step follows immediately from the fact that ${\max \limits } (\omega ^{\alpha _{1}}, \omega ^{\alpha _{2}})+1 \leq \omega ^{{\max \limits } (\alpha _{1}, \alpha _{2})+1}$. If the last step of the derivation is an application of the ω-rule, then the inductive step follows immediately from the fact that $\sup (\omega ^{\alpha _{i}}) \leq \omega ^{\sup (\alpha _{i})}$. Suppose finally that the last step of the derivation of ${\Delta } \vdash ^{\alpha }_{n+1} {\Lambda }$ is a cut of rank n:

$$ \frac{{\Delta}_{1} \vdash^{\alpha_{1}}_{n+1} {\Gamma}_{1}, A \qquad A, {\Delta}_{2} \vdash^{\alpha_{2}}_{n+1} {\Gamma}_{2}}{{\Delta} \vdash^{\alpha}_{n+1} {\Gamma}}\text{Cut} $$

where Δ = Δ₁,Δ₂, and Γ = Γ₁,Γ₂, and rank(A) = n, and α₁,α₂ < α. By weakening we have ${\Delta } \vdash ^{\alpha _{1}}_{n+1} {\Gamma }, A$ and $A, {\Delta } \vdash ^{\alpha _{2}}_{n+1} {\Gamma }$ and by inductive hypothesis we have ${\Delta } \vdash ^{\omega ^{\alpha _{1}}}_{n} {\Gamma }, A$ and $A, {\Delta } \vdash ^{\omega ^{\alpha _{2}}}_{n} {\Gamma }$. By Lemma 5.3 we therefore have

$${\Delta} \vdash^{\max(\omega^{\alpha_{1}} \# \omega^{\alpha_{2}}, \omega)}_{n} {\Gamma}.$$

The only way we could have $\omega ^{\alpha _{1}} \# \omega ^{\alpha _{2}} < \omega$ is if α₁ = α₂ = 0, in which case α ≥ 1, the premises of the cut are initial sequents, and thus A is atomic, and so ${\Delta } \vdash ^{\omega ^{1}}_{1} {\Lambda }$ is immediate (in fact, ${\Delta } {\vdash ^{1}_{1}} {\Lambda }$.) So we may assume $\omega ^{\alpha _{1}} \# \omega ^{\alpha _{2}} \geq \omega$, in which case we have ${\Delta } \vdash ^{\omega ^{\alpha _{1}} \# \omega ^{\alpha _{2}}}_{n} {\Gamma }.$ Because $\omega ^{\alpha _{1}} \# \omega ^{\alpha _{2}} \leq \omega ^{\alpha }$, we then have ${\Delta } \vdash ^{\omega ^{\alpha }}_{n} {\Gamma }$ as desired. □

An immediate corollary of this lemma is that if in V^ω we have ${\Delta } \vdash ^{\alpha }_{n} {\Lambda }$ for any n and α, then we have ${\Delta } \vdash ^{\alpha ^{*}}_{1} {\Lambda }$ for some α^∗. Armed with this result, we may prove Lemma 5 easily. For suppose in V there is a proof of a sequent Δ ⊩Λ such that every formula in this sequent is atomic. From Lemma 5.2 we then know that in V^ω we have ${\Delta } \vdash ^{\alpha }_{n} {\Lambda }$ for some n and α, and so ${\Delta } \vdash ^{\alpha ^{*}}_{1} {\Lambda }$ for some α^∗. Now, in a proof of ${\Delta } \vdash ^{\alpha ^{*}}_{1} {\Lambda }$ no logical rule or the ω-rule can be used, because such a rule would introduce a non-atomic formula. Because there are no non-atomic cuts in the proof it would follow that there would be a non-atomic formula in the final sequent, contrary to assumption. Thus, in the proof of ${\Delta } \vdash ^{\alpha ^{*}}_{1} {\Lambda }$ there are only atomic cuts, contractions, and weakenings. Because there is no use of the ω-rule, this is in fact a proof in V.

Finally, suppose that every formulae in Δ ⊩Γ is closed. Then in the proof in V of Δ ⊩Γ just constructed, suppose there is a lowermost sequent ${\Delta }^{\prime } \vdash {\Gamma }^{\prime }$ in which a free variable x occurs. This cannot be the endsequent by assumption. Using the fact that the initial sequents of V are closed under substitution, we may replace all occurences of x by 0 in ${\Delta }^{\prime } \vdash {\Gamma }$ and above, with the result being a proof of Δ ⊩Γ in V in which the free variable in question has been eliminated. In this way, all free variables in the proof may be eliminated, so that every formula in the proof is closed.

We now prove Lemma 4.

Proof of Lemma 4

It is easily verified that the conditions of Lemma 5 are met by $\overline {\mathbf {MT}}$, and so we use the result of Lemma 5 freely.

Suppose that for some formula X we have that $\overline {\mathbf {MT}}$ proves the sequent ⊩T(c), where c is a numeral with $c=\ulcorner X \urcorner$. We must show that $\overline {\mathbf {MT}}$ also proves the sequent ⊩ X. Because T(c) is atomic and closed, we know by Lemma 5 that there is a proof π of ⊩T(c) in $\overline {\mathbf {MT}}$ in which every formula is closed and atomic, and the only rules used are cut, contraction and weakening.

For any closed, atomic formula S, if S has the form T(τ) where τ is a closed term whose value is the Godel number of a (possibly open) formula X, we let S^∗ = X. If S has any other form, we let S^∗ = S. For a multiset of formulae M, we let M^∗ be the multiset consisting of the S^∗ for each S in M, and for a sequent s of the form Δ ⊩Γ, we let s^∗ be the sequent Δ^∗⊩Γ^∗.

We prove that if a sequent s consisting only of closed, atomic formulae is provable in $\overline {\mathbf {MT}}$ using only cuts, weakenings and contractions, then s^∗ is also provable in $\overline {\mathbf {MT}}$. Applying this to the proof π and using the fact that T(c)^∗ is just X, it follows that the sequent ⊩ X is provable in $\overline {\mathbf {MT}}$, as desired.

It suffices then to prove (i) for each initial sequent s of the form S1 through S24, s^∗ is provable in $\overline {\mathbf {MT}}$, and (ii) for each of the rules of cut, weakening and contraction in which the upper sequent(s) are u_i (i = 1, 2) and the lower sequent l, if $\overline {\mathbf {MT}}$ proves each $u_{i}^{*}$, then $\overline {\mathbf {MT}}$ also proves l^∗.

To prove (i), we must go through each S1 −24 case by case. The case of S1 is trivial. With the exception of S4, initial sequents S2 −S7 do not involve the truth predicate, and thus the result is immediate for each of them. Consider the case of S4 in which R is the truth predicate, and in which s₁ and t₁ are closed terms. If s₁ and t₁ evaluate to different numerals, then s₁ = t₁ ⊩ will be provable in $\overline {\mathbf {MT}}$, and thus this instance of S4^∗ will also be provable by weakening. If s₁ and t₁ are equal to each other but not the Gödel number of a formula, then this instance of S4^∗ is just an instance of S4, and so provable in $\overline {\mathbf {MT}}$. If s₁ and t₁ are equal and the Gödel number of a formula X, then the corresponding instance of S4^∗ is s₁ = t₁,X ⊩ X, which is provable in $\overline {\mathbf {MT}}$ from S1 and weakening.

Each of the sequents S8^∗ through S13^∗ for n > 1 is of the form S8 through S13, and thus provable in $\overline {\mathbf {MT}}$. If n = 1, the sequents S8^∗ through S13^∗ all follow easily from the corresponding sequents S2 through S7 in $\overline {\mathbf {MT}}$ (using the Der rule, for example).

If n > 1 the result for S14^∗ is immediate, and if n = 1 the result for S14^∗ follows from the derivability of the sequent ⊩MT7 in $\overline {\mathbf {MT}}$ (as follows from Lemma 3). The arguments for S15^∗ and S16^∗ are similarly straightforward.

For S17, it suffices to to show that if a₁,a₂,...,x₁,x₂,...,t₁,t₂,... are in fact the Gödel numbers of formulae, variables and terms in ${\mathscr{L}}$, then $\overline {\mathbf {MT}}$ proves

$$\text{Form}(a_{1}), ... ,\text{Var}(x_{1}), ... ,\text{Term}(t_{1}), ... \vdash S_{i} (a_{1}, ..., x_{1}, ... , t_{1}, ...)$$

where S_i is an instance of the relevant logical axiom of the Hilbert calculus. But S_i is a tautology and so $\overline {\mathbf {MT}}$ proves ⊩ S_i, and thus by weakening, also proves the corresponding instance of S17^∗. The case of S18^∗ is dealt with similarly.

Similar arguments also apply for each of S19, S21 and S23. In these cases, if one of the auxiliary formulae (e.g., NotFree(x,c)) is false, then this fact is provable and the relevant Si^∗ follows by weakening. If all these auxiliary formulae are true, then the provability of the relevant Si^∗ follows from the easily shown provability in $\overline {\mathbf {MT}}$ of the relevant sequents of the form $X \supset Y, X \vdash Y$, or $C \supset A(x) \vdash C \supset (\forall z)A[x/z]$, or $A(x) \supset C \vdash (\exists z)A[z/x] \supset C$ (where x is not free in C.) Once the arguments for S19, S21 and S23 have been completed, the arguments for S20, S22 and S24 are straightforward as before.

This completes the proof of (i). We now prove (ii). The argument is straightforward. For example, given an instance of cut:

$$ \frac{{\Delta} \vdash {\Gamma}, A \qquad A, {\Delta}^{\prime} \vdash {\Gamma}^{\prime}}{{\Delta}, {\Delta}^{\prime} \vdash {\Gamma}, {\Gamma}^{\prime}}\text{Cut} $$

The following inference is also a legitimate application of cut:

$$ \frac{{\Delta}^{*} \vdash {\Gamma}^{*}, A^{*} \qquad A^{*}, {\Delta}^{\prime*} \vdash {\Gamma}^{\prime*}}{{\Delta}^{*}, {\Delta}^{\prime *} \vdash {\Gamma}^{*}, {\Gamma}^{\prime *}} \text{Cut} $$

and thus if the upper sequents of this later inference are provable in $\overline {\mathbf {MT}}$, then so is the lower sequent. Equally simple considerations apply in the case of contraction and weakening. □