Abstract
We sketch a broadening of the Gupta-Belnap notion of a circular or revision theoretic definition into that of a more generalized form incorporating ideas of Kleene’s generalized or higher type recursion. This thereby connects the philosophically motivated, and derived, notion of a circular definition with an older form of definition by recursion using functionals, that is functions of functions, as oracles. We note that Gupta and Belnap’s notion of ‘categorical in L’ can be formulated in at least one of these schemes.
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Appendix
Appendix
Proof of Lemma 3
Let . The class of relations over
that are\(\mathcal {A}\)-revision-theoretically definable form a Spector class.
We first return to considerations of why the a.q.i. relations form a Spector class (Theorem 3 above). The proof of the lemma (which the reader may omit) would start by mentioning Burgess’s argument ([2], Lemma 14.1) that the a.q.i. subsets of are those (1-1) reducible to those relations that are Σ2-definable over Lζ, where ζ is the first point where a repeating extension of any Herzberger revision theoretic sequence occurs. The class of such relations defined in such a manner is well-known to form a Spector class, but to check the latter one has to check the conditions in Definition 1 above. The main points are to argue first that there is for each n, a universal a.q.i. relation Un of n + 1 variables that parametrizes via its first cordinate the n-ary a.q.i. relations. This follows since there is a universal Σ2 formula for Σ2-satisfaction in the language of set theory over such an Lζ. The most significant feature of a class of sets Γ being a Spector class, is the Prewellordering property (v) on the list above: that for any relation (say
to take the unary case) in Γ there should be relations S, T in Γ that satisfy (4) above for the norm σ derived from a prewellordering.
For us here, a natural norm on such an R (which is a.q.i. as a binary relation) is given by the ordering that for n, m ∈ R we set n ≼ m⇔n is stably in R in the revision sequence from no later a stage than that of m.
One has to check that there are corresponding relations S, T which are also Σ2-definable and so themselves a.q.i., but this follows as ‘Formula ⌜σ⌝ stabilizes’ can itself be expressed in a Σ2-fashion over Lζ. Or one may argue this directly from Corollary 1 to the Uniform Definability Lemma. In more detail we may set
and
The Lemma itself follows from relativising all such concepts to the oracle \(\mathcal {A}\). The ordinal ζ is replaced by \(\zeta (\mathcal {A})\) the least ordinal \(\bar \zeta \) so that \(L_{\bar \zeta }[\mathcal {A}]\) has a Σ2-elementary end-extension. This ordinal \(\zeta (\mathcal {A})\) is then the least ordinal so that any \(\mathcal {A}\)-revision-theoretically definable relation has stabilized in its revision sequence by stage \(\zeta (\mathcal {A})\). The \(\mathcal {A}\)-a.q.i. relations are then Σ2-definable over \(L_{\zeta (\mathcal {A})}[\mathcal {A}]\), and one proceeds to argue that they form a Spector class as before.
For a recursive or hyperarithmetic starting distribution of truth values we let the Herzberger sequence commencing with \({H^{h}_{0}} =h\) be: \(\langle H^{h}_{\alpha } \mid \alpha \in {On}\rangle\), with eventual stability set \(H^{h}_{\infty }\). If h = ∅, where all sentences are initially assigned falsity, then it is omitted. The following lemma sketches why starting with different hypotheses leads to stability the sets that, “up to isomorphism” are the ‘same’. Sameness here is not identity, but in computational terms they are equivalent: there is a pencil and paper algorithm that given a number from one sets computes a corresponding value in the other, and vice versa.
Lemma 5
\(H_{\infty }^{h}\)isrecursively isomorphic to the stability set obtained by takingh = ∅. That is\(H^{h}_{\infty } \equiv _{1} H_{\infty } \).
Proving that \(H^{h}_{\infty } \equiv _{1} H_{\infty }\) holds directly appears difficult. So we go via Σ2-truth sets of the Gödel L-hierarchy.
Proof
Let 〈Hν ∣ ν ∈ On〉 be the Herzberger revision sequence commencing with H0 = ∅. Let \((\bar {\zeta }, \bar {\Sigma } )\) be the lexicographic least pair with \(L_{\bar \zeta } \prec _{{\Sigma }_{2}} L_{\bar \Sigma }\). By absoluteness of the definition of 〈Hν〉ν∈On we shall have that (〈Hν〉)LΣ = 〈Hν ∣ ν < Σ〉. Since Hζ(HΣ) is defined by the Σ2 liminf rule, it is thus Σ2-definable over Lζ(LΣ). We use the notation ‘n ∈ Tα’ to denote that n is (a code of) a sentence in the Σ2-truthset, or theory, of Lα. We shall then have that n ∈ Hζ, if it holds, is one Σ2 fact that holds about n over Lζ and so contributes to Tζ. There is thus a 1-1 recursive function so that n ∈ Hζ ↔ g(n) ∈ Tζ.
By the given Σ2-elementarity Hζ = HΣ and so this pair of ordinals gives the first pair of repeating extensions of the T-symbol in the revision sequence: it is the beginning of Herzberger’s “Grand Loop”. Thus \(H_{\zeta } =H_{\infty }\) and we have then \(H_{\infty } \leq _{1} T_{\zeta }\) (where ≤1 stands for “is 1-1 reducible to”).
However we can show that \(T_{\zeta } \leq _{1} H_{\infty }\). This is shown in [25] in the proof of Lemma 2.6. More precisely, it is shown there that there is a (1-1) recursive function \(\bar {g}\), so that for any limit ordinal λ < Σ that \(\widehat {T_{}}_{\lambda } = \bar {g}^{-1}\)“Hλ. Here \(\widehat {T_{}}_{\lambda }\) is the Liminf of the theories Tα for α < λ: ⌜σ⌝ ∈ T̂λ ↔ ∃α < λ∀β > α (β < λ → ⌜σ⌝ ∈ Tβ).
But for Lλ a model of Kripke-Platek set theory KP, it can be easily shown that this liminf theory is just the Σ2-theory Tλ. [For the ‘harder’ direction: as Lλ⊧KP it satisfies π2-Reflection, that is for any π2 sentence τ, if Lλ⊧τ then there are unboundedly many α < λ with Lα ⊧ τ. Hence ⌜τ⌝ ∈ could not be in \(\widehat {T}_{\lambda }\).] As Lζ has a Σ2-end extension, it can be reasoned that it is such a model of KP, and thus Tζ = \(\bar {g}^{-1}\)“Hζ. As \(H_{\zeta } =H_{\infty }\) we get \(T_{\zeta } \leq _{1} H_{\zeta } =H_{\infty }\). Together with the last paragraph this yields \(T_{\zeta } \equiv _{1} H_{\infty }\), that is recursive isomorphism, between these two sets.
All the above relatives uniformly to any starting hypothesis H0 = h for any h. We have then (with obvious adaptations of notation) that \(T^{h}_{\zeta } \equiv _{1} H^{h}_{\infty }\) with appropriate ordinals \(\zeta ^{h} , {\Sigma }^{h}\).
In particular if h is recursive, or indeed hyperarithmetic, we shall have that \(L_{\zeta ^{h}} [ h ]\) is just Lζ (and similarly for the Σ case). It can be shown that such h are outright Σ1-definable over Lζ (or LΣ) without the use of any further parameters. But this means that the Σ2-Theory of \(L_{\zeta ^{h}} [ h ]\), that is \(T^{h}_{\zeta }\), will be recursively obtainable via a (1-1) reduction procedure from Tζ alone. Thus we shall have: \(H^{h}_{\infty } \equiv _{1} T^{h}_{\zeta } \leq _{1} T_{\zeta } \equiv _{1} H_{\infty }\). But conversely \(T_{\zeta } \leq _{1} T^{h}_{\zeta }\) because the ‘pure’ L-part of the Σ2-theory \(T_{\zeta }^{h}\) is easy to extract: it is just those Σ2-sentences that do not mention the predicate for h anywhere. Thus \(H_{\infty } \!\leq _{1}\! H^{h}_{\infty }\), and we have the final recursive isomorphism: \(H^{h}_{\infty } \!\equiv _{1}\! H_{\infty }\).
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Welch, P.D. Rethinking Revision. J Philos Logic 48, 137–154 (2019). https://doi.org/10.1007/s10992-018-9482-1
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DOI: https://doi.org/10.1007/s10992-018-9482-1