Ultimate truth vis à vis stable truth

P.D. Welch

School of Mathematics, University of Bristol, England.

September 14th, 2006

Abstract. We show that the set of ultimately true sentences in Hartry Field’s Revenge-immune solution model to the semantic paradoxes is recursively isomorphic to the set of stably true sentences obtained in Hans Herzberger’s revision sequence starting from the null hypothesis. We further remark that this shows that a substantial subsystem of second order number theory is needed to establish the semantic values of sentences in Field’s relative consistency proof of his theory over the ground model of the standard natural numbers: Δ_{3}^{1}-CA_{0} (second order number theory with a Δ_{3}^{1}-Comprehension Axiom scheme) is insufficient.

We briefly consider his claim to have produced a “revenge-immune” solution to the semantic paradoxes by introducing this conditional. We remark that the notion of a “determinately true” operator can be introduced in other settings.

Keywords: theories of truth, fixed points, revision sequence, constructible sets, admissibility.

1. Introduction.

In (Field, 2003) Field constructs a system of a theory of truth over ground models M, with an additional conditional operator → in the language. We shall here use for illustrative purposes M = ℕ, the standard model of arithmetic, with L the appropriate arithmetical language using the connectives ∧, ∨, ¬ and the two quantifiers ∃, ∀; L^{+} will be, just as for him, this language augmented with the → connective, to be considered as an operator, and T r as a predicate.)

The construction in (Field, 2003) Section 2 is in ordinal stages, where each successor ordinal stage performs an entire construction of the “next strong Kleene fixed point” à la Kripke. Thus the transition of α ↣ α + 1 involves the usual Kripkean monotonic construction over an assignment of values amongst \{0, 1, \frac{1}{2}\} to the “conditionals” of the form |A → B|_{α}. (We are suppressing the secondary ordinal subscript of this latter intermediate construction, which Field calls the “mini-stages” - and also for the most part suppresses.) The latter assignments are determined by semantic values given at strictly previous (full) stages. We repeat his Clause 8 to illustrate:

|A → B|_{α} = \{ \begin{matrix}1 : & iff (∃β < α)(∀γ ∈ [β, α)(|A|_{γ} ≤ |B|_{γ}), \\ 0 : & iff (∃β < α)(∀γ ∈ [β, α)(|A|_{γ} > |B|_{γ}), \\ \frac{1}{2} : & otherwise. \end{matrix}

Although the motivations are entirely different (and this cannot be emphasised enough perhaps) the system here has striking structural similarities with the revision theory of truth using Herzberger style revision sequences (cf.(Herzberger, 1982b),(Herzberger, 1982a)). For a revision sequence, knowing all truth values at some stage α + 1, we can read off which truth value was assigned to B at stage α from Tr(B). In Field’s system we have the same effect: the “1 or non-1” semantic value of B at the previous stage α is imported for the next stage α + 1 via |⊤ → B|_{α+1}: knowing these semantic values means we know what happened before. There are similar, but differing, considerations for limit stages. For Herzberger, at limit stages the extension of T r is determined through similar looking rules to the first two clauses in the above definition, however there is no value of \frac{1}{2}, but in this third case an (arbitrary?) assignment of 0. In a single Herzberger sequence one can look at the set of stable sentences, that is, those that from some point on have a set truth-value. (We emphasise that we are not considering categorical truth sets in the style of Gupta and Belnap (Gupta and Belnap, 1993), where a global averaging is obtained by considering all possible starting hypotheses for the Herzberger revision process.)

The values |A → B|_{α} are continuous at limits (cf. his Continuity Lemma for Conditionals). It then makes sense to define ∥A∥ as the ultimate value of |A|_{α}:

||A|| = \{ \begin{matrix}1 : & iff (∃β)(∀γ ≥ β(|A|_{γ} = 1), \\ 0 : & iff (∃β)(∀γ ≥ β(|A|_{γ} = 0), \\ \frac{1}{2} : & otherwise \end{matrix}

In Section 4 we discuss his claim that his system, having successfully introduced a conditional operator, gives rise to a “revenge-immune” solution to the semantic paradoxes. We do not believe that the proposed system is any more immune to strengthened liar paradoxes than some already in existence.

Field asks if there are acceptable points, that is ordinals Δ so that for any A we have that ∥A∥ = |A|_{Δ}. Section 3 of (Field, 2003) is devoted to proving that there are such. For the paper it is sufficient for him to demonstrate the existence of such points but he adds: “I believe a more informative proof should be possible, which would show among other things that acceptable fixed points ... occur before Ω.” Here Ω is the next initial ordinal of cardinality greater than that of the underlying model M that is being discussed - so here for us this is ω_{1}. In fact his proof is simple and direct, but again adds that “the price of its simplicity is that the proof is less informative than one might like about the way the values of sentences change as the level increases towards an acceptable point.” The purpose of Section 2 of this note is to give a further proof (well, in fact 3 of them) for the existence of acceptable points (which are all below Ω). Probably none of them are “informative” in the latter sense he might like, and we are not sure that there could be anything very illuminating in general, meaning for the “general” sentence A, given the complexity of the notions involved. He shows that the least acceptable point is greater than a particular recursive ordinal λ_{0} that is used in his iterations of a “determinately true” operator D. Theorem 2 below shows that the process of calculating semantic values is necessarily complicated, and thus “how” a general sentence achieves its ultimate value involves looking at the theory of quasi-inductive definitions, or equivalently at the levels of the Gödel constructible hierarchy up to L_{ζ} - the first Σ_{2}-extendible level. We show (Theorem 3) that the least acceptable point is in fact exactly this ζ, and so far beyond the first nonrecursive ordinal ω_{1ck}. Remark (ii) below also indicates the strength of the proposed system: any system of analysis in which we can establish these eventual semantic values is highly impredicative.

DEFINITION 1.1. (i) The set of ultimate truths D =_{df} {⌜A⌝ : ∥A∥ = 1}. (ii) The set of (Herzberger) stable truths H =_{df} {⌜A⌝ : A is stably true in Herzberger’s revision sequence starting

from the null hypothesis}.

In the above by “the null hypothesis” we mean the hypothesis that all sentences are false, (or equivalently for our purposes this could be taken as “all sentences are true”: any recursive starting distribution of truth values would serve equally well). For the notion and properties of Herzberger’s revision sequence the reader may consult that author’s papers already cited, or the account in (McGee, 1991).

THEOREM 1.1. D is recursively isomorphic to the set H.

This set H has been noted to be equivalent to other sets of integers, defined independently. We shall comment on this further in Section 3, but briefly, if Q is the complete arithmetic quasi-inductive set, if Õ is the Σ_{2}-truth set of L_{ζ} (where ζ was named above), and if S is the set of (codes of) Turing programs that have eventually some constant value on their output tape, when allowed to run transfinitely (in the formalism of (Hamkins and Lewis, 2000)) then we have:

Fact 1 The following are recursively isomorphic:

(i) S ; (ii) Õ ; (iii) Q; (iv) H.

Our third method of demonstrating the existence of acceptable points, is in fact directed to adding a fifth set D to this list. The following theorem is what we shall actually prove:

THEOREM 1.2. Let D be the set of ultimate truths in Field’s system. Then D is recursively isomorphic to Õ.

The statement of the abstract, Theorem 1, then follows from the Fact quoted above. We note the following further consequences:

Remark (i) Using the above Fact 1 and Theorem 1.2 the set of ultimately true sentences over arithmetic is a complete quasi-inductive set,; hence Field’s logic ⊧_{LCC} is not axiomatisable. Remark (ii) The “conservativeness proof” of (Field, 2003) Sect.6 (there performed in ZF C) can be effected in a sufficiently strong fragment of second order number theory, but not in KP (Kripke-Platek set theory) nor Δ_{3}^{1}-CA_{0}. (For the latter theory see (Simpson, 1999);^{1} for notions concerning KP and admissible ordinals see (Barwise, 1975).)

Similar results will hold for countable ground models M that are acceptable (this time in the sense of (Moschovakis, 1974)), that is, the least (Fieldian) acceptable point for M will be the least Σ_{2}-extendible ordinal relative to M, and hence will be less than |M|^{+}, the least cardinal greater than that of M. We shall then have (in the appropriate language, and in the appropriate sense) that D_{M} is equivalent to H_{M}. The proof of Theorem 1.2 (in the hard direction showing that Õ is (1-1) reducible to D) is essentially a demonstration that Field’s system carries hidden within it a construction of the Gödel hierarchy of constructible sets up to the level ζ. We can show that at stage α in his construction we have a uniform way of recovering certain Σ_{2}-truth sets of the levels L_{α} of this hierarchy, whenever the latter is a model of a sufficient amount of set theory. For other levels we seemingly do not have a uniform way of doing this: although Field’s clauses are uniform for all limit ordinals, the recovery of the truth sets for the L_{α} can be obtained from {A : |A|_{α} = 1}, but perhaps not always uniformly so.^{2} Happily below ζ there are many α where L_{α} models this theory, including ζ itself; this allows us then to exploit the uniformity at ζ, in order to find the (1-1) function decoding Õ from D.

We make a brief remark on our result concerning recursive isomorphism (essentially a “computable translation” - this will be explicitly defined below). One might hope for some more perspicaceous, or ‘natural’ translation of Herzberger stable truths directly into Field’s ‘ultimate truths’, and hopefully vice versa. We think this unlikely. We do not really see one in either direction: this is to do with the fact that the process of revision according to revision sequences, has more in common with the supervaluation approach than the Kleene strong jump that Field employs in his ‘mini-stages’: direct translations seem to be ruled out by the incomparability of the methods. One should also remember however, that all we ever do with theories of truth employing a system of gödel numbering over a model such as the standard model of arithmetic, is to build a truth set using that particular coding scheme. There are infinitely many choices of (sensible) coding scheme, all of which only yield identical truth sets up to recursive isomorphism. However, even bearing this in mind, there seems a difficulty in finding a meaning preserving translation or mapping, between the stable truths and the ultimate truths.^{3}

In the penultimate section we discuss a little the nature of the “revenge immunity” claim, and the import of the methods used in Field’s consistency proof. We claim that our argument shows that an extraordinary amount of analysis - in the sense of a large fragment of second order number theory - is needed for Field’s construction to work - well beyond anything that mathematicians working in analysis, (i.e. the structure of the real continuum rather than simply the natural numbers). In the final section we just remark that “determinately true” operators could have been introduced in a Herzbergerian revision theoretic setting.

2. The existence of acceptable points

First demonstration: The construction of (Field, 2003) Section 2 is performed within ZF C using the ground model M. It is in fact a very absolute construction, and absolute for transitive models of ZF C^{−}, that is ZF C without the power set axiom (which is not used). Hence if H is a transitive set, α, M ∈ H and H ⊧ ZF C^{−} then H ⊧ “|A|_{α} = j” iff |A|_{α} = j. Suppose, without loss of generality, that M is countable. By the Collection Axiom we may assume that there is some ordinal μ_{0} so that for all A ∈ L^{+}, if |A|_{γ} = j, for all sufficiently large γ, then there is an δ_{A} < μ_{0} so that ∀β > δ_{A}|A|_{β} = |A|_{δ_{A}} = j. For the other A, ie those with ∥A∥ = \frac{1}{2}, which alternate value unboundedly in the ordinals, we can again appeal to Collection to find, for any ordinal ν ≥ μ_{0}, some cardinal μ ≥ ν so that if, say, A alternates its value unboundedly in the ordinals, it does so unboundedly below μ in particular. (This already shows that the class of acceptable points is unbounded in the class of all ordinals.) But further, if H_{μ} is the set of sets of hereditary cardinality less than μ, then H_{μ} is a ZF C^{−} model. Now by the Löwenheim-Skolem theorem let X ≺ H be a countable elementary submodel of H with M ∈ X. Let N be a transitive set isomorphic to X. (Such an N exists by the Mostowski collapsing Lemma.) Let Δ < ω_{1} be the ordinal height of N. Then Δ is an acceptable point: one may easily verify that |A|_{Δ} = |A|_{μ} for any A.

By modifying the argument one may show that there is in fact a closed and unbounded set E ⊆ ω_{1} of acceptable ordinals.

Second demonstration: One could directly show that the construction, and argument that there are acceptable points, can be performed not just in ZF C^{−} but in much weaker theories, for example, Kripke-Platek (KP) augmented with a Σ_{2}-Separation axiom, or even in second order number theory with Π_{3}^{1} Comprehension. (We aren’t advocating actually doing this; that it could be done will follow from the third demonstration). If this is granted, then we can find countable models of such theories for which the construction is absolute. That it cannot be done in much weaker theories is also a concomitant of the theorem.

Third demonstration: This contains the heart of this note. We shall define an ordinal ζ. To see what this is we make the following observations. The Σ_{2} nature of the defining clauses of ∥A∥, together with the absolute nature of the construction, hint not only (a) that the definition of the function ∥ · ∥ can be construed as a valuation coming from a quasi-inductive definition but (b) that the first repeat point, that is the first acceptable point, is ζ - this latter ordinal has been characterised as the first Σ_{2}-extendible ordinal ((Burgess, 1986) Sect.14), that is the first ordinal ζ so that L_{ζ} has a proper Σ_{2}-elementary end extension.^{4} Equivalently it is also the first place where all Infinite Time Turing machine (ITTM) computations have either halted or have entered an infinite loop ((Welch, 2000) Thm. 2.1). In the next theorem Σ is the periodicity, or ordinal length, of that loop, or, equivalently, the ordinal height of that smallest Σ_{2} end-extension of L_{ζ}. We shall prove:

THEOREM 2.1. Over the ground model M = ℕ of arithmetic, the first acceptable point Δ_{0} = ζ. The class of acceptable points above ζ is: {Σ.ρ | ρ ∈ On}.

To state explicitly at the outset, we shall reserve the word recursive to have its usual standard meaning throughout this note. The notion of P ≤_{T} Q “P is recursive in Q”, for sets of integers P, Q, says simply that χ_{P}, the characteristic function of P, is recursive in an oracle for Q; alternatively put, there is an index e ∈ ω so that χ_{P} is the e’th function recursive in Q: χ_{P} = {e}^{Q}. To say that two sets P, Q are recursively isomorphic (written “P ≡ Q”) is to say that there is a total recursive bijection F : ℕ ⟷ ℕ with ∀n n ∈ P ⟺ F (n) ∈ Q. For such P, Q membership questions about P can be converted by a pencil and paper algorithm to membership questions about Q and conversely. For most mathematical purposes then, these sets are seen to be isomorphic if not identical. A weaker notion, written P ≤_{1} Q, is that “P is (1-1) reducible to Q”, that is, there is a recursive injection F : ℕ ⟶ ℕ with ∀n n ∈ P ⟺ F (n) ∈ Q. Here we could only paraphrase by saying that membership questions ?n ∈ P? can be converted into the questions ?F (n) ∈ Q? again via such an algorithm. Hence in this situation, as a slogan: “Q is at least as complicated as P ”. An effective version of the proof of the classical Cantor-Schröder-Bernstein theorem due to Myhill ((Rogers, 1967) Theorem (VI)) shows that P ≡ Q iff P ≤_{1} Q & Q ≤_{1} P.

We introduce some further definitions.

DEFINITION 2.1. For x ∈ L, we let ρ_{L}(x) =_{df} the least α such that x ∈ L_{α+1}.

For E a class of ordinals, let E^{∗} denote E together with the set of its limit points. E^{∗} is thus the closure of E.

DEFINITION 2.2. (i) Let ADM = {α | ⟨L_{α}, ∈⟩ ⊧ KP }.

(ii) Let ⟨τ_{ι} | ι < ω_{1}⟩ enumerate ADM^{∗} ∩ ω_{1}.

Here “KP” denotes Kripke-Platek set theory in the language L ˙∈ . If α ∈ ADM it is called admissible. Then τ_{0} = ω, τ_{1} = ω_{1ck}, where the latter is the first non-recursive ordinal. Note that ADM is not closed: τ_{ω} is not admissible.

We shall make use of the following facts:

LEMMA 2.1. (i) ζ = τ_{ζ}; Σ = τ_{Σ}. (ii) ∀μ < Σ ¬∃ν < μ L_{ν} ≺_{Σ_{2}} L_{μ},

((i) that τ_{ξ} = ξ is true for any Σ_{2}-extendible ordinal ξ, as for any such ξ L_{ξ} can be shown to be a model of Σ_{2}-KP which is moreover a union of such; ξ is thus a limit of “Σ_{2}-admissible ordinals” even; this can be deduced from the reflection properties enjoyed by Σ_{2}-extendibles. (ii) uses, or rather is just a restatement of, the leastness assumption on ζ together with that of Σ being the height of the least Σ_{2} end-extension of L_{ζ}.

DEFINITION 2.3. Let C ⊆ ω. We let A^{1,C} =_{df} the complete Σ_{1}-theory of ⟨L_{ω_{1ck}^{C}}[C], ∈, C⟩.

A^{1,C} may thus be identified with a set of integers coding Σ_{1} sentences in the language L_{˙∈,Ċ}. This set has itself a Σ_{1}-definition over the structure ⟨L_{ω_{1ck}^{C}}[C], ∈, C⟩, which is uniform in C (that is, the defining formula is independent of the choice of set C). Here ω_{1ck}^{C} is the first ordinal not recursive in C (equivalently: not arithmetic in C) and is the first ordinal which is admissible in the predicate C. (We shall henceforth drop the “ck” when considering the first ordinal not recursive in some C; we shall keep it for the first non-recursive ordinal ω_{1ck}; the first uncountable cardinal is thus ω_{1,} to avoid any confusion.)

DEFINITION 2.4. For ι < ω_{1} let A_{ι}^{2} = A_{ι} =_{df} the complete Σ_{2}-theory of ⟨L_{τ_{ι}}, ∈⟩.

It is thus Õ =_{df} A_{ζ} which we wish to show is recursively isomorphic to D.

PROPOSITION 2.2. (i) ∀ι < Σ ρ_{L}(A_{ι}) = τ_{ι}; (ii) A_{Σ} = A_{ζ} and hence ρ_{L}(A_{Σ}) = ζ = τ_{ζ}; (iii) ∀ι < Σ ω_{1}^{A_{ι}} = τ_{ι+1}.

Proof: (i): Assume ι < Σ and let τ = τ_{ι}; as τ < Σ for no ξ do we have L_{ξ} ≺_{Σ_{2}} L_{τ}. We say that a set x ∈ L_{τ} is (implicitly) Σ_{2}-definable over L_{τ} if there is a Σ_{2} formula φ(v_{0}) so that in L_{τ} x is the unique set so that φ[x]_{L_{τ}}. We let λ =_{df} sup{γ | γ is Σ_{2}-definable over L_{τ} }. Then λ ≤ τ. We shall claim that λ = τ. But first we note

(1) Any γ < λ is Σ_{2}-definable over L_{τ}.

For, if δ < λ is Σ_{2}-definable over L_{τ}, then note that any γ < δ is actually Σ_{1}-definable from δ: this is because for any τ ≤ Σ we have that L_{τ} ⊧“∀α∃f : ω ⟶ α with f surjective”. (This latter assertion holds, since if L_{τ} ⊧ “η is uncountable” we should be able to prove that there are even μ < ν < η with L_{μ} ≺ L_{ν} even which would contradict our leastness assumption on Σ.) Hence, given a γ < δ, the global wellordering of L is Σ_{1}-definable and so the L-least such surjective map f onto δ is Σ_{1}-definable in L_{λ} from δ. Thus if f(n) = γ then in turn γ is Σ_{1}-definable from δ. Putting this together with the Σ_{2}-definition of δ yields a Σ_{2}-definition of γ. QED(1)

Let A_{λ} be the complete Σ_{2}-theory of ⟨L_{λ}, ∈⟩.

(2) A_{ι} ⊆ A_{λ} Suppose σ ∈ A_{ι}. As σ is of the form ∃u∀vψ(u, v) for some ψ with all quantifiers bounded, then for some γ we have that

L_{γ} ⊧ “σ ∧ ∀η∃η′ > η(L_{η′} ⊧ ¬σ)”& ∀γ′ ≥ γL_{γ′} ⊧ σ.

(γ is thus the least point on from which σ is seen to be true.) But now the above yields “g = γ” as Π_{1} over L_{τ}. Hence γ is Σ_{2}-definable over L_{τ} and thus γ < λ. We conclude by the form of the above statement, since λ > γ, that σ ∈ A_{λ}.

QED(2)

(3) L_{λ} ≺_{Σ_{1}} L_{τ}. This is trivial if λ = τ so assume otherwise. If ξ < λ and φ(ξ) is a Σ_{1} assertion about ξ that is true in L_{τ} but not in L_{λ} then there is first ordinal γ < τ which sees a witness for the existential quantifier of φ: thus L_{γ} ⊧ φ(ξ). We thus have (where ψ is the Σ_{2} defining formula for ξ):

“∃x(ψ(x) ∧ g is the least ordinal so that L_{g} ⊧ φ(x)”.

This is a Σ_{2}-definition over L_{τ} of γ. Hence γ < λ. Hence (3) holds by the upwards persistence of Σ_{1} formulae to any larger model. QED(3)

But (3) immediately implies (by the upwards persistence of Σ_{2} formulae between Σ_{1} elementary substructures):

(4) A_{ι} ⊇ A_{λ}. Hence with (2) we have equality here. But λ < τ will imply that L_{λ} ≺_{Σ_{2}} L_{τ} which contradicts the leastness of (ζ, Σ). To see this last point, let φ(ξ) ≡ ∃u∀vψ(u, v, ξ) be a Σ_{2} assertion about ξ < λ which holds in L_{τ}. As ξ < λ it has a uniquely defining Σ_{2} formula θ(v_{0}) which is only true in L_{τ} of ξ itself. Hence L_{τ} is a model of the following:

“∃x∃u[θ(x) ∧ ∀vψ(u, v, x)”

This is a Σ_{2} sentence and so is in A_{ι} = A_{λ}. It thus holds in L_{λ}. But note that θ(ξ) holds in L_{λ} uniquely also: (a) if L_{λ} ⊧ ∃ξ_{0}∃ξ_{1}(ξ_{0} ≠ ξ_{1} ∧ θ(ξ_{0}) ∧ θ(ξ_{1})) then this is a sentence in A_{λ} which fails to hold in A_{ι} - a contradiction; so (b) suppose L_{λ} ⊧ θ(ξ_{0}) ∧ ξ_{0} ≠ ξ. By (3) we should again have by the upwards persistence of the Σ_{2} formula θ(ξ_{0}) that L_{τ} ⊧ θ(ξ_{0}) also a contradiction.

Hence λ = τ = τ_{ι} and ρ(A_{ι}) ≥ τ. QED(i)

For (ii): here Σ is the very least point where there is some ζ < Σ with L_{ζ} ≺_{Σ_{2}} L_{Σ}. Further these two levels of the L-hierarchy have identical Σ_{2}- theories, i.e., A_{Σ} = A_{ζ}. Hence ρ_{L}(A_{Σ}) = τ_{ζ}. That τ_{ζ} = ζ is Lemma 2.1 (i) above.

For (iii): again let τ = τ_{ι}; we saw above in (i) that every x ∈ L_{τ} has a Σ_{2} definition. A_{ι} itself is thus a complete Σ_{2}-truth set for L_{τ} (i.e. it contains a record of all elementary ∈ and = facts about L_{τ}) and arithmetically in A_{ι} we can decode a wellordering w ⊆ ω × ω of order type τ. Hence ω_{1}^{A_{ι}} > τ_{ι}. However for any B ⊆ ℕ, B ∈ L_{τ_{ι+1}}, we have that ω_{1}^{B} ≤ τ_{ι+1}. In particular ω_{1}^{A_{ι}} ≤ τ_{ι+1}. QED

The Σ_{1}-Separation scheme (see, e.g. (Barwise, 1975) Sect. I.9) is the infinite set of axioms which are the universal closure of axioms of the form:

∃u∀v(v ∈ u ⟷ v ∈ w ∧ φ(v)) where φ(v) is any Σ_{1}-formula (possibly with other set variables). For us, what matters is the following relatively easily proven fact (again see (Barwise, 1975), V 6.3 & 7.12):

PROPOSITION 2.3. Suppose β > ω. L_{β} ⊧ Σ_{1}-Separation if and only if {γ < β | L_{γ} ≺_{Σ_{1}} L_{β}} is unbounded in β. If L_{β} ⊧ Σ_{1}-Separation then β ∈ ADM ∩ ADM^{∗}.

PROPOSITION 2.4. (cf.(Welch, 2000) Thm 2.1) L_{ζ} ⊧ Σ_{1}-Separation.^{5}

We shall use the following notation. Let C_{α} = (C_{α,T} , C_{α,F}) where C_{0,T} = C_{0,F} = ∅. Then for α > 0 set:

C_{α,T} = {⊤ ⟶ A : |⊤ ⟶ A|_{α} = 1};

C_{α,F} = {A ⟶ ⊥ : |A ⟶ ⊥|_{α} = 1}.

In the above we have omitted gödel corners, but we assume that C_{α} ⊆ ℕ, after some suitable recursive coding, and pairing, of C_{α,T} with C_{α,F}. Note that if we set, for α ≥ 0 :

C_{α}^{+} = (C_{α,T}^{+} , C_{α,F}^{+} ) where we have taken:

C_{α,T}^{+} = {A : |A|_{α} = 1};

C_{α,F}^{+} = {A : |A|_{α} = 0}, then, for example, C_{α+1,T} is little different from C_{α,T}^{+} , as the former is simply the set of members of the latter with “⊤ ⟶” inserted before them (and similarly for C_{α+1,F} and C_{α,F}^{+} m.m.). Note that if ξ is acceptable in Field’s sense, then C_{ξ,T} = {⊤ ⟶ A | A ∈ D} and C_{ξ,F} = {A ⟶ ⊥ | ¬A ∈ D}. Hence C_{ξ} is (1-1) reducible to D.

LEMMA 2.2. For ι < Σ (i) A_{ι} is arithmetic in C_{ι}; (ii) ω_{1}^{C_{ι}} = τ_{ι+1} ; (iii) ρ_{L}(C_{ι}) = τ_{ι}(ι > 0); (iv) if L_{τ_{ι}} ⊧ Σ_{1}-Separation, then A_{ι} ≤_{1} C_{ι}; moreover this latter clause is uniform in ι for such models of Σ_{1}-Separation.

By “uniform in ι” in (iv) above, we mean that the (1-1) recursive function f:ℕ → ℕ that effects the reduction of A_{ι} to C_{ι} is the same for each such ι < Σ where L_{τ_{ι}} ⊧ Σ_{1}-Separation. Taking ι = ζ shows that Õ = A_{ζ} ≤_{1} C_{ζ} ≤_{1} D. By Proposition 2.3 the hypothesis of (iv) is false if ι is a successor ordinal (and is also for many limit ordinals).

Proof of Lemma 2.2: We have mentioned the C_{α}^{+} versions decorated with the plus sign, as this allows the following formulation.

PROPOSITION 2.5. (Kripke) C_{α}^{+} is a complete Π_{1}^{1,C_{α}} set of integers, (and hence by the above remark, so is C_{α+1}).

This was proven by Kripke in the case when α = 0, for his original theory using strong Kleene. (A proof is in (Burgess, 1986).) By Field’s definition, each C_{α}^{+} is constructed as the Kripkean fixed point using this scheme, with the distribution of semantic values 0, 1/2, 1 assigned to the various sentences involving the ⟶ operator. The proposition above then, is simply the relativisation of Kripke’s result to the “starting distribution” coded into C_{α}. Once we have C_{α}^{+} is a complete Π_{1}^{1,C_{α}} set of integers, we may apply a result that identifies such with the Σ_{1}-theory of the least admissible set over C_{α} :

PROPOSITION 2.6. If D ⊆ ω, and D^{+} is a complete Π_{1}^{1,D} set of integers, then D^{+} ≡ Σ_{1}-Theory of ⟨L_{ω_{1}^{D}} [D], ∈, D⟩; in fact there is a recursive bijection g : ℕ ⟷ ℕ, so that uniformly for all D,

n ∈ D^{+} ⟺ g(n) ∈ Σ_{1} ∩ Sent ∧ ⟨L_{ω_{1}^{D}} [D], ∈, D⟩ ⊧ g(n).

In particular: C_{α}^{+} ≡ Σ_{1}-Theory of ⟨L_{ω_{1}^{Cα}} [C_{α}], ∈, C_{α}⟩.

The proposition is essentially obtained from the relativisations of the theorems of Kleene and Spector, (see, e.g., (Rogers, 1967), Theorems XLI & XLIV) that classify the Π_{1}^{1} predicates on integers as those Σ_{1}-definable over HY P (the latter the class of hyperarithmetic reals, together with the fact that the reals of L_{ω_{1ck}} are those of HY P).

We now proceed to the proof of the clauses (i)-(iv) of the lemma. First observe that (ii) follows from (i) and (iii) using Proposition 2.2(iii): first we note that in general, if B is arithmetic in C, then ω_{1}^{B} ≤ ω_{1}^{C}. Thus ω_{1}^{C_{ι}} ≥ τ_{ι+1} (using (i) and Prop 2.2 (iii)). Further, using (iii), as ρ(C_{ι}) = τ_{ι}, ω_{1}^{C_{ι}} ≤ τ_{ι+1}. Hence (ii) holds.

We proceed to prove (i),(iii),(iv) by induction on ι < Σ. For ι = 0 they either hold vacuously or are trivial. Suppose now ι = η + 1, and (i) to (iv) hold with η in place of ι. A_{ι} is the complete Σ_{2}-theory of ⟨L_{τ_{ι}}, ∈⟩.

But note that τ_{η+1} = τ_{ι} is the next admissible ordinal above τ_{η} and hence is the height of the smallest admissible set containing C_{η} as an element: τ_{ι} = ω_{1}^{C_{η}} (using for (ii) the inductive hypothesis.) As sets, L_{ω_{1}^{Cη}} [C_{η}] = L_{ω_{1}^{Cη}} (because A_{η} is definable over L_{τ_{η}} and C_{η} is arithmetic in A_{η}). Using our nomenclature above:

A^{1,C_{η}} = Σ_{1}-Th(⟨L_{ω_{1}^{Cη}} [C_{η}], ∈, C_{η}⟩)= Σ_{1}-Th(⟨L_{τ_{ι}}, ∈, C_{η}⟩). However the Σ_{2}-theory of this structure is simply obtained: it is (recursively isomorphic to) the Turing jump of A^{1,C_{η}}. Thus:

(A^{1,C_{η}})′ ≡ Σ_{2}-Th(⟨L_{τ_{ι}}, ∈, C_{η}⟩. However A_{ι} is the “pure Ċ-free” part of this theory, namely:

A_{ι} = Σ_{2}-Th(⟨L_{τ_{ι}}, ∈, C_{η}⟩ ∩ L ˙∈. Putting this together one gets: A_{ι} ≤_{T} (A^{1,C_{η}})′, and the latter is arithmetic in A^{1,C_{η}} ≡ C_{η}^{+} ≡ C_{η+1} = C_{ι} (where the last sentence of Proposition 2.6 is used for the first (1-1) equivalence here). Hence (i) holds for ι = η + 1, and we even have that A_{ι} is recursive in the Turing jump of C_{ι}. We have seen that A^{1,C_{η}} is definable over ⟨L_{τ_{ι}}, ∈⟩ (using the parameter C_{η}) which is a member of L_{τ_{ι}}; hence ρ(A^{1,C_{η}}) ≤ τ_{ι}. But we have just asserted that A^{1,C_{η}} ≡ C_{ι}. Hence ρ(C_{ι}) ≤ τ_{ι} as well. But ι < Σ, so we cannot have then ρ(C_{ι}) < τ_{ι}, as otherwise we should also have ρ(A_{ι}) < τ_{ι} , and this would contradict Proposition 2.2(i). This proves (iii) for ι. (iv) is vacuous for successor ι.

We now suppose that ι < Σ is a limit, and the inductive hypotheses (i)- (iv) hold for η < ι. As ι < Σ we shall have that L_{τ_{ι}} is the Σ_{2}-skolem hull of ∅ inside L_{τ_{ι}} (by using the argument of Proposition 2.2 again); that is, ∀x ∈ L_{τ_{ι}}∃kL_{τ_{ι}} ⊧“φ_{k}^{2}[x]” where ⟨φ_{k}^{2}|k ∈ ω⟩ is the above recursive enumeration of all Σ_{2}-formulae of the language L ˙∈ with v_{0} the single free variable. Thus we have that any x ∈ L_{τ_{ι}} is implicitly Σ_{2}-definable for some k as φ_{k}^{2}[x]. We shall assume that the form of such formulae can be expressed as :

(∗) φ_{k}^{2}(v_{0}) ⟺ ∃u∀vψ_{k}(u, v, v_{0}) for some Σ_{0} ψ_{k}.

We divide into cases depending on whether L_{τ_{ι}} satisfies the Σ_{1}-Separation scheme or not. In the former we shall have a uniform (1-1) recursive way of recovering A_{ι} from C_{ι}; in the latter case it is the presence of parameters in the argument that seemingly prevent a uniform reduction.

Case 1 ι is a limit ordinal, but L_{τ_{ι}} ⊭ Σ_{1}-Separation.

Let ⟨φ_{i}|i < ω⟩ be a recursive enumeration of all Σ_{1} formulae of one free variable of the language L ˙∈. The case hypothesis assures us there is a “Σ_{1}-parameter” p ∈ L_{τ_{ι}}, so that L_{τ_{ι}} is the Σ_{1}-skolem hull of the singleton set {p}^{6}. By the comment on Σ_{2}-skolem hulls above, there is k so that φ_{k}^{2} uniquely defines p. Moreover, by the form at (∗) of the formulae φ_{n}^{2}, there is α_{0} so that for all α < ι, α_{0} < α ⟶ L_{τ_{α}} ⊧“φ_{k}^{2}[p]”; that is p is named in the same way in the structures L_{τ_{α}}.^{7} We then have the following chain of equivalences. l ∈ A_{τ_{ι}}^{1} ⟺_{df} L_{τ_{ι}} ⊧ φ_{l}[p]

⟺ ∃β_{0} > α_{0}∀β > β_{0} L_{τ_{β+1}} ⊧“L_{τ_{β}} ⊧ φ_{l}[p]”

⟺ ∃β_{0} > α_{0}∀β > β_{0} ⟨L_{τ_{β+1}}[C_{β}], ∈, C_{β}⟩ ⊧“∃\overline{p} (φ_{k}^{2}[\overline{p}] ∧ L_{ρ(C_{β})} ⊧ φ_{l}[\overline{p}])” (using in the last equivalence the inductive hypothesis (iii)); the statement in the last line’s quotation marks here is actually a Σ_{1} sentence in L_{˙∈,Ċ} about k, l; so let us call this σ(k, l). Note that the map (k, l) ↦ σ(k, l) can be assumed recursive. Using the Proposition 2.6 the last equivalence becomes:

⟺ ∃β_{0} > α_{0}∀β > β_{0} g^{−1}(σ(k, l)) ∈ C_{β}^{+}

⟺ g^{−1}(σ(k, l)) ∈ C_{ι}.

We thus have A_{τ_{ι}}^{1} ≤_{1} C_{ι}. This implies that A_{ι} is recursive in (A_{τ_{ι}}^{1})′ ≤_{T} C′_{ι}. Hence (i) holds: A_{ι} is arithmetic in C_{ι}. This leaves (iii): ρ(C_{ι}) ≮ τ_{ι} (for otherwise ρ(C_{ι}) < τ_{α} for an α < ι, hence A_{ι} ∈ L_{τ_{ι}}; this is a contradiction as A_{ι}^{1} ∉ L_{τ_{ι}}). However C_{ι} is Σ_{2}(L_{τ_{ι}}). Hence ρ(C_{ι}) = τ_{ι} as required.

Case 2 ι is a limit ordinal, and L_{τ_{ι}} ⊧ Σ_{1}-Separation.

Now there is no such parameter p as in Case 1. Let ⟨φ_{l}^{2} | l ∈ ω⟩ enumerate all Σ_{2}-sentences of L ˙∈.

l ∈ A_{ι} ⟺_{df} L_{τ_{ι}} ⊧ φ_{l}^{2}

⟺ ∃α_{0}∀β ≥ α_{0} L_{τ_{β}} ⊧ φ_{l}^{2}

We should justify this last equivalence: suppose φ_{l}^{2} is ∃u∀vψ_{l}(u, v). (⟹) If z is such that L_{τ_{ι}} ⊧ ∀vψ_{l}(z, v), then we may choose α_{0} < ι with z ∈ L_{τ_{α_{0}}} . Then ∀β ≥ α_{0} L_{τ_{β}} ⊧ ∀vψ_{l}(z, v). (⟸) Let α_{0} be as hypothesized. By Proposition 2.3, find γ < ι with α_{0} ≤ γ and L_{τ_{γ}} ≺_{Σ_{1}} L_{τ_{ι}}. Now if L_{τ_{γ}} ⊧ ∀vψ_{l}(z, v), this is a Π_{1} sentence about z that persists upwards to L_{τ_{ι}}.^{8}

Now, using similar reasoning:

⟺ ∃α_{0}∀β ≥ α_{0} L_{τ_{β+1}} ⊧“L_{τ_{β}} ⊧ φ_{l}^{2}”

⟺ ∃α_{0}∀β ≥ α_{0} ⟨L_{τ_{β+1}}[C_{β}], ∈, C_{β}⟩ ⊧“L_{ρ(C_{β})} ⊧ φ_{l}^{2}”

(again using in the last equivalence the inductive hypothesis (iii) as before); now the sentence in quotes depends only on l, so let us call it σ(l), again with l ↦ σ(l) assumed recursive. For the same reasons, we have the last equivalence:

⟺ ∃α_{0}∀β > α_{0} g^{−1}(σ(l)) ∈ C_{β}^{+}
⟺ g^{−1}(σ(l)) ∈ C_{ι}.

Thus A_{ι} ≤_{1} C_{ι} and the reduction here (g^{−1} ∘ σ) is independent of ι. QED (Lemma 2)

Proof of Theorem 3. As stated at the beginning of this “third demonstration”, we could consider running Field’s construction inside L, indeed inside L_{Σ}. The defining clauses of his semantic value assignments form a Σ_{2}-recursion. We note then:

(1) |A|_{ζ}=|A|_{Σ} for any sentence A of L^{+}. Hence C_{ζ} = C_{Σ}.

Proof: Immediate from the fact that there is a uniform Σ_{2}-recursion in the sentence variable A that defines these values over any L_{λ}, if λ = τ_{λ}, say, and then |A|_{λ} will be Σ_{2}-definable over L_{λ}. Focussing on L_{ζ}, and L_{Σ} respectively, we have that L_{ζ} ≺_{Σ_{2}} L_{Σ}, and the result then follows.

QED(1)

(2) If |A|_{ζ}= 0 (resp. 1) then ∀α ∈ (ζ, Σ) |A|_{α}= 0 (resp. 1).

Proof: Suppose this were false for some particular A with, say, |A|_{ζ}= 0, but |A|_{α} ≠ 0. Suppose β_{0} < ζ is such that ∀β ∈ (β_{0}, ζ)|A|_{β}= 0. By supposition L_{Σ} ⊧ ∃α > β_{0}|A|_{α} ≠ 0. The latter is a Σ_{1} sentence about A and β_{0} and will reflect down to L_{ζ} (as all Σ_{2}-sentences about objects in L_{ζ} do). Hence L_{ζ} ⊧ ∃α > β_{0}|A|_{α} ≠ 0. However this contradicts our choice of β_{0.} QED(2)

However (1) implies that the distribution of semantic values according to Field’s clauses are exactly the same at stage ζ and at stage Σ. Hence they will be identical at stage ζ + 1 and Σ + 1 and then at stages Σ and Σ + Σ, etc., etc. We thus shall have a “periodicity” of length Σ in these “snapshots” of semantic values. In short for any sentence A we have that |A|_{ζ} = |A|_{Σ.ρ} for any ρ > 0 (note that Σ is additively closed, so ζ + Σ = Σ). The force of (2) is that any 0,1 semantic value assigned at stage ζ to a sentence A will remain assigned at all later stages. As (1) shows any sentence which has an unstable value at stage ζ will also do so at Σ and indeed by repetition as above, at any stage Σ.ρ > ζ; we deduce that ζ is an acceptable point.

(3) ζ is the least acceptable point.

Proof: To summarise, we have shown that there is a (1-1) recursive function h : ℕ ⟹ ℕ so that A ∈ D iff h(⌜A⌝) ∈ C_{ζ} iff h(⌜A⌝) ∈ C_{ξ} where ξ is any acceptable point. Suppose for a contradiction that ξ < ζ was the least acceptable point.

We claim there is a θ < ζ which is an acceptable point and further L_{θ} ⊧ Σ_{1}-Separation. This is because L_{Σ} ⊧“There exists θ > ξ with C_{θ} = C_{ξ} and L_{θ} ⊧ Σ_{1}-Separation”. (ζ is such). Hence there is, by Σ_{1}-reflection such a θ below ζ. However then we have that C_{θ} = C_{ζ}. This is a contradiction to Proposition 2.2 (i), since, as we have seen, A_{ζ} is recursively obtainable from C_{θ} = C_{ζ}, we should have that ρ(A_{ζ}) < ζ. QED(3)

The sequence of sets of assignment values will then reappear with a periodicity Σ, as required for the theorem. QED(Theorem 3)

Proof of Theorem 2 Lemma 2 shows that Õ ≤_{1} D. However the argument at (1) above shows that there is a Σ_{2} formula ψ(v_{0}) ∈ L ˙∈ so that A ∈ D if and only if ψ(A) is true at L_{ζ} (equivalently L_{Σ}). Hence D ≤_{1} Õ. The result follows by Myhill’s Theorem - the effective Cantor-Schröder-Bernstein Theorem mentioned above.

QED(Theorem 2)

3. Further equivalences.

We discuss some further equivalences. We state the result first before going on to discuss the sets S, Q being introduced.

Fact 1^{9} The following are recursively isomorphic:

(i) S ; (ii) A_{ζ} ; (iii) Q; (iv) H.

We introduce some nomenclature (taken from (Hamkins and Lewis, 2000)) in order to talk about the class of infinite time turing machine (ITTM) computations. This will be the straightforward generalisation of that from ordinary recursion theory. We let ⟨P_{e}|e ∈ ω⟩ be an enumeration of all ITTM programs (which are no different from ordinary Turing programs, apart from the addition of a single new “limit state”, q_{L}, to which the machine enters at limit stages of time). We use the notation that “P_{e}(k) ↓ l” to mean the computation halts with output tape containing the string l ∈ 2^{ω}. “P_{e}(k) | 0” means that the computation halts, or eventually enters an infinite loop, but that 0 (the infinite string of zeros) remains on the output tape (even if the machine has not formally halted). Note there is, as expected, a third class of computations: it may be that the e’th program on input k loops without any stable contents on its output tape, we denote this as usual : “P_{e}(k) ↑”. The definition of S to follow is a generalisation of Turing jump.

DEFINITION 3.1. S =_{df} {e : P_{e}(0) | 0}

A perhaps more understandable version of any of the sets we have been considering is the set Q, the complete arithmetic quasi-inductive set of integers. To see what Q is we follow Burgess (Burgess, 1986) (who distilled this definition from that of a Herzberger revision sequence), and consider the following.

Let Γ : P(ω) → P(ω) be any arithmetic operator (that is “n ∈ Γ(X)” is arithmetic; we emphasise that Γ need be neither monotone nor progressive). We define the following iterates of Γ : Γ_{0}(X) = Γ(X); Γ_{α+1}(X) = Γ(Γ_{α}(X)); Γ_{λ}(X) = lim inf_{α→λ} Γ_{α}(X) = ∪_{α<λ} ∩_{λ>β>α} Γ_{β}(X). We say that Y ⊆ ω is arithmetically quasi-inductive (AQI) if for some such Γ, Y is (1-1) reducible via a recursive function to Γ_{On}(∅). The complete arithmetically quasi-inductive set Q, to which all others are similarly reducible, has then also the various recursively isomorphic characterisations stated in Fact 1 above. The list of recursively isomorphic sets displayed in the Fact above all arise from definitions or procedures that can be seen to fall under the scheme of being arithmetically quasi-inductive. It can be shown that the set of stable truths of a Herzberger revision sequence, and Field’s set of ultimate truths, can be obtained via an arithmetic quasi-inductive definition.^{10} Each ITTM program can be thought of as producing a set of 0/1 values on an infinite tape according to a recursive quasi-inductive definition. Moreover, by consideration of a universal ITTM program, one may see that the complete arithmetical quasi-inductive set is in fact recursively quasi-inductive (and hence so are D, H … etc.) A further example of an occurrence of this scheme appears in computer science: see (Kreutzer, 2002) where the author seeks to separate out various fixed point logics on finite structures, by considering essentially the same kind of infinite quasi-inductive process.

COROLLARY 3.1. Let ⟨A_{n}|n ∈ ω⟩ be a recursive enumeration of all sentences of L^{+}. Let B ⊆ ℕ be any arithmetical quasi-inductive set. Then there is a recursive (1-1) function f so that so that B = {n : A_{f(n)} ∈ D}.

Perhaps unsurprisingly:

COROLLARY 3.2. ⊧_{LCC} is not axiomatisable.

The last part of the Remark (ii) - that certain theories are insufficient for the conservativeness result of Field’s Sect.6 - is only an observation that is a consequence of the fact that such theories cannot prove the existence of acceptable points, in particular of ζ. The least β-model of Δ_{3}^{1}-CA_{0} occurs as the set of the reals of some level L_{γ} of the constructible hierarchy, for an ordinal γ that is much smaller than ζ. Hence we can find such a model containing no code for a wellordering along which we can perform an arithmetic quasi-inductive definition and reach a point in the ordering where we have a repetition in the values produced. In the terms here, such orderings have no “acceptable” points.

4. Revenge-immunity

We discuss here some aspects of Field’s claim to have produced a “revengeimmune” solution to the semantic paradoxes.^{11} This claim itself seems at first sight paradoxical: surely any system based on a sufficiently strong theory to enable Gödel coding of the syntax and diagonalisation arguments to be performed, will have some semantic predicate of the system that allows one to “diagonalise out” of the system and thus either obtain a self-referential sentence concerning that semantic notion leading to a contradiction, or else one concludes that the semantic notion is not “representable” within the system?

Indeed Field’s system is no different (indeed cannot be) from others in this regard if we maintain a classical metatheory. The relevant semantic notion here is that of ultimate truth. Indeed he points out in Section 6 that (again assuming a classical metatheory) we may formulate a predicate D^{∗}(A) which is defined to have value 1 if ∥A∥ = 1, and 0 otherwise, then this cannot be semantically represented in his system via a formula in the language, for otherwise we should be able to produce a self-referential sentence L_{∗} equivalent to ¬D^{∗}(True(⟨L_{∗}⟩)) in the usual way, and thus a contradiction.

There is demonstrably within his system a hierarchy of ‘determinately true’ operators. Moreover such can be used to give an account within the system of how the traditional Liar fails to gain a truth value. Indeed he gives a hierarchy of such operators ⟨D^{α}|α < λ_{0}⟩ that can deal with more and more complicated super-liars.

Third, and probably most important, the new conditional can be used to show that the theory is not subject to “revenge problems.”

More fully, the addition of the new conditional operator to the language allows for the definition of a natural “determinately operator”, so that we can consistently handle “extended paradoxes”, such as sentences that assert of themselves that they are not determinately true.” (Field, 2003) p140.

The determinately true operators do go some way to helping Kripke out from his dilemma that Liar sentences are not classified as untrue in his minimal fixed points, whilst at the same time we are precluded from expressing this in the object language. Field with his operator D effortlessly can form such object language expressions. We note below in the next subsection that it is not the new conditional that allows for a notion of “determinately true”

- this can be captured using the older notions of a Herzbergerian Revision sequence (although in such sequences there is no “conditional” which allows for a naive theory of truth to be stated using them).

He then argues that the reason D^{∗} is not semantically representable by any formula in the language, is not because of any limitation of expression on the part of the object language (which includes arithmetic) or that of the metalanguage and theory in which his theorems are proven (that of set theory, ZF C). We take no issue then with the claim on p166 (op.cit) which states “there is no need to use a broader classical metalanguage to do the semantics; we can use L itself”. In general it seems then, to be a wrong turn down a blind alley, to argue that since the predicate indicating “has semantic value 1” cannot be representable in such a semantic theory of truth, that therefore one must broaden the metalanguage. Of course we can beef up the metatheory and use a construction dependent on a longer initial segment of the ordinals to capture a more complicated notion of ultimate truth by proceeding in a way that goes further up the L-hierarchy. Just as Kripke’s least fixed point theories (supervaluational, strong Kleene or otherwise) can be established over a model, say that of the standard natural numbers, using a relatively weak classical metatheory, so we might regard this particular construction of Field’s as pursued in a much stronger classical metatheory (and necessarily so as we have shown).

(Indeed one may analyse precisely what predicates or operators are so representable in this particular semantical construction over the standard model of arithmetic, and one obtains a precise answer: those predicates that are both AQI and co-AQI. As D^{∗} is a complete AQI set in effect, it is universal for all AQI sets, and by a standard diagonalisation argument cannot itself be both AQI and co-AQI. Hence, as we already knew, it cannot be representable.)

I emphasize ‘this construction’ because I take it that Field only wishes to show that his logical system is consistent. The question remains open as to whether a consistency proof could be established using a weaker classical metatheory. However it would appear that any argument for “revenge immunity” can not be done within a classical metatheory for the reasons outlined at the beginning of this section. If it could be shown that a consistency proof required the use of a strong theory - and I think showing this might be rather difficult - then this could be construed as a criticism of the theory, in the way that, say, principles that can ‘only’ be shown consistent relative to large cardinals in set theory, get criticized. In the context of ‘truth’ for the standard model of the natural numbers, the second order theory Δ_{3}^{1}-CA_{0} might well be said to have the status of a ‘large cardinal theory’. It is a very strong fragment of second order number theory, well beyond anything one might call “predicative” in Feferman’s sense, and beyond current proof theoretical techniques of ordinal analysis. (Another way to put this is to say that the least acceptable point Δ - here called ζ, is extremely large.) It is hard to think of any theorem in mathematics, in particular in analysis other than Borel Determinacy results, that require this strength in the background theory. If no simpler consistency result can be found for the theory LCC (using a classical metatheory) then one is left wondering as to the status of “truth for the natural numbers” if one wishes to adopt this system. That is so independently of the stance one takes on the “revenge-immunity” part of the theory. On the other hand it may well be that by abandoning classical reasoning in the metatheory some other argument may allow a simpler justification for LCC, for those brave enough to take that leap. Then one may have Field’s “revenge immunity” if one is prepared to have such a weak non-classical meta-theory.

However, that aside, and returning to Field’s construction under discussion, we might imagine after following it through, that we no longer have the feeling, as Kripke seemingly does at the paragraph towards the end of his paper (Kripke, 1975), that the ‘ghost of the Tarski hierarchy is still with us.’ Kripke might have felt that Liar sentences are not true in the object language, ‘... but we are precluded from saying this is the object language...’ What the semantical notion of ultimate truth, or D^{∗}, does is transport us to a situation where the assertion ‘D^{∗}(A) ≠ 1’ is so far removed from ordinary ‘untruth’ about A that we may be uninclined to worry that, residually, L_{∗} ought to be, or is ‘nevertheless’, untrue. However exporting our possibly selfreferential notions away to these remoter regions, or complicating them up, so that they no longer enter the circle of our concerns about basic, immediate, common or garden ‘truth’ seems not to be a clear way through to a genuine revenge-immune solution. The ghost, although thinner and ghostlier, is still perceptible. We seem to need the failure of the Law of Excluded Middle within a non-classical metatheory, (so that we cannot make claims about any ultimate D^{∗}’s bivalency). In which case though, presumably much is up for grabs...

5. Endnote: ‘Determinately true’ predicates in a herzbergerian setting.

Whilst Field uses the ‘⟶’ operator in his consistency proof construction to capture some of the notions of a conditional, its most striking feature is that it allows one to capture some notion of the ‘history’ of the semantical values of a sentence within the object language. (i) ⊤ ⟶ A , when evaluated at a successor stage α = γ + 1, tells you whether A was true or not at stage γ; (ii) when at a limit α, whether or not A was true on a ‘tail’ of the ordinals below α. His operator D(A) which is (⊤ ⟶ A) ∧ A, if (i) it has value 1 at stage γ + 1 says “A was true at the previous stage and it is so now”; (ii) for limit stages this has the effect of “A has recently been always true and is so now” (with a little fanciful interpretation of stages as being that of time.)

However, note how we can get this effect in Herzberger’s two-valued setting: ⊤ ⟶ A is no more than “Tr(A)” and D(A) becomes simply “T r(A) ∧ A”. The effects of (i) and (ii) are thus there also in the Herzberger system. Field then iterates these definitions producing: D^{α+1}(A) as D(D^{α}(A)), and for suitable recursive limit ordinals, and in a suitable notation system, D^{λ}(A) as well. One can mimic this process exactly within Herzberger’s system also. In short: one does not need a new conditional in order to have a “determinately true” operator in order to handle “extended paradoxes” of this kind within the object language. In other words the description of the ‘ “singular” status of the Liar and the Truth-teller,’ as Field puts it (Field, 2003) p.157, as well as the extended variants, can be equally well handled within Herzberger.

Notes

1 In brief, this is second order number theory with a set induction scheme, but with comprehension restricted to formulae expressible in a Δ_{3}^{1} fashion.

2 In a nutshell: we are working in a part of the constructible hierarchy, where each level has a Σ_{1}-truth set that is definable using, in general, a necessary single infinite ordinal parameter. This parameter has a Σ_{2}-definition over L_{α} but not a uniform one independent of α. For those L_{α} which are models of Σ_{1}-Separation, there is in any case no such parameter from which all sets are Σ_{1}-definable. However for these latter type of L_{α}, we do in fact have a uniform recursive way of recovering a Σ_{2} truth set from the Fieldian set of locally stable truths C_{α}, as this is the “liminf” of the previous Σ_{2}-truth sets for the models L_{β} (β < α). Note: Since writing this, Sy Friedman has pointed out that we can uniformly treat both types of stages with a marginally more complicated reduction procedure; one then has for models L_{λ} for limit λ a way of finding the Σ_{2}-truth sets recursively in the Turing jump of the “liminf” truth sets. We have not re-written this into the above, as it would only involve further technicalities, without bringing forth any great perspiciences.

3 One might further add that a similar situation pertains to the least Kripkean fixed points under the various schemes, such as weak- and strong-Kleene, and the supervaluation scheme: such sets are all recursively isomorphic - without there being any obvious recursive “translation” mechanism.

4 This means that for some ξ > ζ we have that L_{ζ} ≺_{Σ_{2}} L_{ξ}, the subscript Σ_{2} indicating that Σ_{2} formulae with parameters allowed from L_{ζ} have the same truth value in both structures.

5 Actually the theorem cited shoes that L_{ζ} is a model of the Δ_{2}-Comprehension, and Σ_{2}- Collection Schemes.

6 The existence of such a parameter can be established as follows: suppose there were no such p. Then the Σ_{1}-skolem hull of ∅ in L_{τ_{ι}}, is not all of L_{τ_{ι}}; one can argue that this hull is transitive, and in fact is thus L_{α_{0}} for some α_{0} < τ_{ι}. Hence L_{α_{0}} ≺_{Σ_{1}} L_{τ_{ι}}. Now consider the Σ_{1}-skolem hull of {α_{0}} in L_{τ_{ι}}; again by supposition this is not all of L_{τ_{ι}}; it is transitive and hence is some L_{α_{1}} ≺_{Σ_{1}} L_{τ_{ι}}, with α_{0} < α_{1} < τ_{ι}. Continuing in this way we could create an infinite chain of such substructures of ordinal height α_{n} < τ_{ι}. This would contradict Proposition 2.3. In such a situation as here there is a “standard Σ_{1}-parameter” p = p_{τ_{ι}}^{1} ∈ L_{τ_{ι}}, so that L_{τ_{ι}} is the Σ_{1}-skolem hull of the singleton set {p}. Such a parameter is least in a particular wellordering on finite sequences of ordinals below τ_{ι}. It can be shown that in our situation this standard parameter is a single ordinal π < τ_{ι.}, (perhaps just 0). For a fuller account of such parameters and hulls see (Devlin, 1984) IV.

7 The point here is that if we suppose that L_{τ_{ι}} ⊧“φ_{k}^{2}[p] ” where φ_{k}^{2}[p] ⟺ ∃u∀vψ_{k}(u, v, p), then if L_{τ_{ι}} ⊧ ∀vψ_{k}(z, v, p) for some particular z we may take α_{0} sufficiently large so that z ∈ L_{τ_{α_{0}}} . Then for any α > α_{0} we shall have L_{τ_{α}} ⊧ ∀vψ_{k}(z, v, p) (as the latter is Π_{1}, i.e. a universal statement). Thus p is named in the same way by φ_{k}^{2} in all such structures. Note that p will have other, different, Σ_{2} definitions in each such structure, but they will not necessarily persist upwards to L_{τ_{ι}}.

8 Note how this argument can fail if L_{τ_{ι}} is not a model of Σ_{1}-Separation: take ι = ω, and φ_{l}^{2} as “there is a largest admissible ordinal”. Then for all n ∈ ω, L_{τ_{n}} ⊧ φ_{l}^{2}, but this sentence is false in L_{τ_{ω}} (and moreover g^{−1}(σ(l)) ∈ C_{ω}).

9 The equivalence of (i) and (ii) is (Welch, 2000) Thm 2.6; whilst working on this latter paper, we were unaware of the very concrete links with the much earlier work of Burgess in (Burgess, 1986). There he explicitly states the equivalence of (iii) and (iv) as Prop.13.1; but the equivalence of (ii) and (iii) is essentially his Theorem 14.1 there too. 10 This is done directly for the Herzberger sequence with a null starting hypothesis in (Burgess, 1986). 11 Since this article was written Field has produced an extended explication of the claim to have produced such a solution ((Field, 2008)) - hence we have not considered that here. A very detailed reply to Field on this topic is the recent (Leitgeb, 2006).

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