Self-referential theories Samuel A. Alexander∗ Department of Mathematics, the Ohio State University August 11, 2020 Abstract We study the structure of families of theories in the language of arithmetic extended to allow these families to refer to one another and to themselves. If a theory contains schemata expressing its own truth and expressing a specific Turing index for itself, and contains some other mild axioms, then that theory is untrue. We exhibit some families of true self-referential theories that barely avoid this forbidden pattern. 1 Introduction This is a paper about families of r.e. theories, each capable of referring to itself and the others. Many of this paper's results first appeared in the author's dissertation [1]. There, they were stated in terms of families of interacting mechanical knowing agents. Here, we will speak instead of families of self-referential r.e. theories. We hope this will more directly expose the underlying mathematics. In epistemology, it is well-known that a (suitably idealized) truthful knowing machine capable of arithmetic, logic, and self-reflection, cannot know its own truth and its own code. This is due, in various guises, to authors such as Lucas [8], Benacerraf [3], Reinhardt [11], Penrose [9], and Putnam [10]. In terms of selfreferential theories, a true theory satisfying certain assumptions cannot contain schemata stating its own truth and its own Gödel number (if such a theory did exist, we could program a machine knower that knows precisely its consequences). Reinhardt conjectured, and Carlson proved [5], a truthful machine knower can know (in a local sense, i.e., expressed by infinite schemata rather than a single axiom) that it is truthful and has some code, without knowing which. A true self-referential theory can (in a local sense) state its own truth and recursive enumerability. We showed [2] that, alternatively, a truthful machine can (in a local sense) exactly know its own code, if not required to know its own truth. A true theory can state (in a local sense) its own Gödel number. Our goal is to generalize the above consistency results to multiple theories. The paper contains four main findings. In the following list of promises, except where otherwise stated, ≺ is an r.e. well-founded partial-order on ω, and expresses is meant in the local (infinite schema) sense. 1. There are true theories (Ti)i∈ω such that Ti expresses a Gödel number of Tj (all i, j) and Ti expresses the truth of Tj (all j ≺ i). 2. There are true theories (Ti)i∈ω such that Ti expresses a Gödel number of Tj (j ≺ i), the truth of Tj (j  i), and the fact that Tj has some Gödel number (all i, j). 3. If ≺ is ill-founded, and if we extend the base language to include a predicate for computable ordinals and require the theories to include rudimentary facts about them, then 1 and 2 fail. 4. Finally, if we do not extend the base language as in 3, then there do exist ill-founded r.e. partial orders ≺ such that 1 and 2 hold. Our proofs of 1 and 2 are constructive, but the proof of 4 is nonconstructive. In short, if 4 were false, either of 1 or 2 could be used to define the set WF of r.e. well-founded partial orders of ω using nothing but arithmetic and a truth predicate Tr for arithmetic. This is impossible since WF is Π11-complete and Tr is ∆11. ∗Email: alexander@math.ohio-state.edu 1 2 Preliminaries To us, theory and schema mean set of sentences (a sentence is a formula with no free variables). Definition 1. (Standard Definitions) 1. When a first-order structure is clear from context, an assignment is a function s mapping first-order variables into the universe of that structure. If x is a variable and u is an element of the universe, s(x|u) is the assignment that agrees with s except that it maps x to u. 2. We write M |= φ[s] to indicate that the first-order structure M satisfies the formula φ relative to the assignment s. We write M |= φ just in case M |= φ[s] for every assignment s. If T is a theory, M |= T means that M |= φ for every φ ∈ T . 3. We write FV(φ) for the set of free variables of φ. 4. We write φ(x|t) for the result of substituting term t for variable x in φ. 5. LPA is the language of Peano arithmetic, with constant symbol 0 and function symbols S, +, * with the usual arities. If L extends LPA, an L -structure has standard first-order part if it has universe N and interprets 0, S, + and * as intended. 6. We define LPA-terms n (n ∈ N), called numerals, so that 0 ≡ 0 and n+ 1 ≡ S(n). 7. We fix a computable bijection 〈•, •, •〉 : N3 → N. Being computable, this is LPA-definable, so we may freely act as if LPA contained a function symbol for this bijection. Similarly we may act as if LPA contained a binary predicate symbol • ∈W• for membership in the nth r.e. set Wn. 8. Whenever a computable language is clear from context, φ 7→ pφq denotes Gödel numbering. 9. A valid formula is one that is true in every structure. 10. A universal closure of φ is a sentence ∀x1 * * * ∀xnφ where FV(φ) ⊆ {x1, . . . , xn}. We write ucl(φ) to denote a generic universal closure of φ. Note that if M is a structure and ψ is a universal closure of φ, in order to prove M |= ψ it suffices to let s be an arbitrary assignment and show M |= φ[s]. To formalize self-referential theories, we employ an extension of first-order logic where languages may contain new unary connective symbols. This logic is borrowed from [5]. Definition 2. (The Base Logic) A language L of the base logic is a first-order language L0 together with a class of symbols called operators. Formulas of L are defined as usual, with the clause that Tiφ is a formula whenever φ is a formula and Ti is an operator. Syntactic parts of Definition 1 extend to the base logic in obvious ways (we define FV(Tiφ) = FV(φ)). An L -structure M is a first-order L0-structure M0 together with a function that takes one operator Ti, one L -formula φ, and one assignment s, and outputs either True or False-in which case we write M |= Tiφ[s] or M 6|= Tiφ[s], respectively-satisfying the following three requirements. 1. Whether or not M |= Tiφ[s] does not depend on s(x) if x 6∈ FV(φ). 2. If φ and ψ are alphabetic variants (meaning that one is obtained from the other by renaming bound variables so as to respect the binding of the quantifiers), then M |= Tiφ[s] if and only if M |= Tiψ[s]. 3. For variables x and y such that y is substitutable for x in Tiφ, M |= Tiφ(x|y)[s] if and only if M |= Tiφ[s(x|s(y))]. The definition of M |= φ[s] for arbitrary L -formulas is obtained from this by induction. Semantic parts of Definition 1 extend to the base logic in obvious ways. Traditionally the operator Ti would be written Ki, and the formula Kiφ would be read like "agent i knows φ". For the present paper, the added intuition would not be worth the philosophical distraction. 2 Theorem 3. (Completeness and compactness) Suppose L is an r.e. language in the base logic. 1. The set of valid L -formulas is r.e. 2. For any r.e. L -theory Σ, {φ : Σ |= φ} is r.e. 3. There is an effective procedure, given (a Gödel number of) an r.e. L -theory Σ, to find (a Gödel number of) {φ : Σ |= φ}. 4. If Σ is an L -theory and Σ |= φ, there are σ1, . . . , σn ∈ Σ such that1 σ1 → * * * → σn → φ is valid. Proof. By interpreting the base logic within first-order logic (for details see [1]). Definition 4. If L is a first-order language and I is an index set, let L (I) be the language (in the base logic) consisting of L along with operators Ti for all i ∈ I. In case I is a singleton, LPA(I) is a form of Shapiro's [12] language of Epistemic Arithmetic. Definition 5. • For any LPA(I)-formula φ with FV(φ) = {x1, . . . , xn}, and for assignment s (into N), let φs be the sentence φs ≡ φ(x1|s(x1)) * * * (xn|s(xn)) obtained by replacing all free variables in φ by numerals for their s-values. • For any language L extending LPA, if M is an L -structure, then M is said to interpret formulas by substitution if M has standard first-order part and the following property holds: for every L -formula φ and assignment s, M |= φ[s] if and only if M |= φs. For example, if s(x) = 0 and s(y) = 2 then (∀z(x = y + z))s ≡ ∀z(0 = S(S(0)) + z). Definition 6. If T = (Ti)i∈I is an I-indexed family of LPA(I)-theories and N is an LPA(I)-structure, we say N |= T if N |= Ti for all i ∈ I. Definition 7. Suppose T = (Ti)i∈I is an I-indexed family of LPA(I)-theories. The intended structure for T is the LPA(I)-structure MT with standard first-order part, interpreting the operators Ti (i ∈ I) as follows: MT |= Tiφ[s] if and only if Ti |= φs. If MT |= T, we say T is true. Lemma 8. For any family T = (Ti)i∈I of LPA(I)-theories, MT interprets formulas by substitution. Proof. In other words, we must show that for every LPA(I)-formula φ and assignment s, MT |= φ[s] if and only if MT |= φs. The proof is a straightforward induction. Definition 9. By the axioms of Peano arithmetic for LPA(I) we mean the axioms of Peano arithmetic, with induction extended to LPA(I). Lemma 10. For any LPA(I)-structure M , if M interprets formulas by substitution, then M satisfies the axioms of Peano arithmetic for LPA(I). Proof. Let M be any LPA(I)-structure which interprets formulas by substitution. This means M has standard-first order part and for every formula φ and assignment s, M |= φ[s] if and only if M |= φs. Let σ be an axiom of Peano arithmetic for LPA(I). If σ is not an instance of induction, then M |= σ since M has standard first-order part. But suppose σ is ucl(φ(x|0)→ ∀x(φ→ φ(x|S(x)))→ ∀xφ). To see M |= σ, let s be an arbitrary assignment and assume M |= φ(x|0)[s] and M |= ∀x(φ→ φ(x|S(x)))[s]. By assumption, M |= φs(x|0) and ∀m ∈ N, if M |= φs(x|m) then M |= φ(x|S(x))s(x|m). Evidently φ(x|S(x))s(x|m) ≡ φs(x|m+1). By mathematical induction, ∀m ∈ N, M |= φs(x|m). By assumption, M |= ∀xφ[s]. 1We write A→ B → C for A→ (B → C), and likewise for longer chains. 3 Definition 11. Suppose T = (Ti)i∈I is a family LPA(I)-theories. If T+ = (T + i )i∈I is another such family, we say T ⊆ T+ if Ti ⊆ T+i for every i ∈ I. If T is a single LPA(I)-theory, we say T ⊆ T if T ⊆ Ti for all i ∈ I. If T1 = (T 1i )i∈I and T2 = (T 2i )i∈I are families of LPA(I)-theories, T1 ∪T2 is the family T′ = (T ′i )i∈I where each T ′i = T 1 i ∪ T 2i . Arbitrary unions ⋃ n∈X T n are defined similarly. Definition 12. Suppose T = (Ti)i∈I is a family of LPA(I)-theories. For each i ∈ I, we say Ti is Ti-closed if Tiφ ∈ Ti whenever φ ∈ Ti. We say T is closed if each Ti is Ti-closed. Definition 13. If I is an r.e. index set, a family T = (Ti)i∈I is r.e. just in case {(φ, i) : φ ∈ Ti} is r.e. 3 Generic Axioms If T is a family of theories whose truth was in doubt, and if we state a theorem removing that doubt, we often state more: that T∪S is true, where S is some background theory of provability, including non-controversial things like Peano arithmetic or the schema ucl(Ti(φ→ ψ)→ Tiφ→ Tiψ). The choice of S is somewhat arbitrary, or at best based on tradition. We will avoid this arbitrary choice by stating results in the form: "T is true together with any background theory of provability such that. . . " Definition 14. A family T of LPA(ω)-theories is closed-r.e.-generic if T is r.e. and MU |= T for every closed r.e. family U ⊇ T of LPA(ω)-theories. Lemma 15. If T is a union of closed-r.e.-generic families and T is r.e., then T is closed-r.e.-generic. Proof. Straightforward. Definition 16. For i ∈ I and for T an LPA(I)-theory, we write [T ]i for the family T = (Tk)k∈I where Ti = T and Tk = ∅ for all k 6= i. 3.1 Closed-r.e.-generic Building Blocks In this subsection, we will exhibit some examples of closed-r.e.-generic families. They can be combined in diverse ways, via Lemma 15, to form background theories of provability. This will allow us to state Theorem 24 below in a generalized way, essentially saying that a certain doubted theory is consistent with any background theory of provability made up of closed-r.e.-generic building blocks. The alternative would be for us to arbitrarily choose one such background theory and build it directly into Theorem 24, which would cause the core details in the proof of Theorem 24 to get jumbled up with unimportant distractions. It is common for a theory to state its own closure under modus ponens. When there are multiple theories, it is less clear whether each individual theory should only state its own closure thereunder, or the closure of all the other theories, or of some subset thereof. With the following lemma, we avoid arbitrarily imposing a decision along these lines. Lemma 17. For any i, j ∈ ω, the following family is closed-r.e.-generic: • [S]i where S is: (j-Deduction) the schema ucl(Tj(φ→ ψ)→ Tjφ→ Tjψ). Proof. Let U = (Uk)k∈ω be any closed r.e. family of LPA(ω)-theories such that U ⊇ [S]i where S is jDeduction. We must show MU |= [S]i. In other words we must show MU |= ucl(Tj(φ → ψ) → Tjφ → Tjψ) for any φ, ψ. Let s be an assignment and assume MU |= Tj(φ→ ψ)[s] and MU |= Tjφ[s], we must show MU |= Tjψ[s]. By Definition of MU, Uj |= (φ→ ψ)s and Uj |= φs. Clearly (φ→ ψ)s ≡ φs → ψs so by modus ponens Uj |= ψs, that is, MU |= Tjψ[s]. It might not be controversial to require that a theory express its own ability to prove valid sentences, but in a multi-theory context, should we require each theory to express that much about all its fellow theories? The following lemma allows us to avoid arbitrarily declaring the right answer to that question. Part 2 of this lemma illustrates an interesting combinatorial property of closed-r.e.-generic building blocks. Some schemas would not be suitable building blocks by themselves, but when paired with other schemas, the combination can become a suitable building block. 4 Lemma 18. For any i, j ∈ ω, the following families are closed-r.e.-generic: 1. [S]i where S is: (Assigned Validity) the schema φ s (φ valid, s an assignment). 2. [Assigned Validity]i ∪ [S]j where S is: (i-Validity) ucl(Tiφ) for φ valid. Proof. Both (1) and (2) are r.e. by Theorem 3. (1) Let U = (Uk)k∈ω be a closed r.e. superset of [S]i where S is Assigned Validity. We must show MU |= [S]i. If φ ∈ [S]i then φ is φs0 for some valid φ0 and some assignment s. Since φ0 is valid, MU |= φ0[s]. By Lemma 8, MU |= φs0. (2) Let U = (Uk)k∈ω be any closed r.e. family of LPA(ω)-theories such that Ui contains Assigned Validity and Uj contains i-Validity. By (1), MU satisfies Assigned Validity. It remains to show MU satisfies i-Validity. Let φ be valid and s an assignment. Since Ui contains Assigned Validity, Ui |= φs, so by definition of MU, MU |= Tiφ[s]. In modal logic, some papers treat the so-called positive introspection axiom (also known as the KK axiom) as one of the fundamental axioms of knowledge, and some do not. Rather than join either side, we prefer instead to study the combinatorial structure of the axiom, asking: are there other schemas we can add to it to make the combination closed-r.e.-generic? Lemma 19. For any i, j ∈ ω, the following family is closed-r.e.-generic: • [Assigned Validity]i ∪ [i-Validity]i ∪ [i-Deduction]i ∪ [S]j where S is: (i-Introspection) the schema ucl(Tiφ→ TiTiφ). Proof. Recursive enumerability is by Theorem 3. Let U = (Uk)k∈ω be any closed r.e. family of LPA(ω)theories such that Ui contains Assigned Validity, i-Validity and i-Deduction, and Uj contains i-Introspection. Then MU satisfies Assigned Validity and i-Validity by Lemma 18. By Lemma 17, MU satisfies i-Deduction. For i-Introspection, let s be an assignment and assume MU |= Tiφ[s], we will show MU |= TiTiφ[s]. Since MU |= Tiφ[s], Ui |= φs. By Theorem 3, there are σ1, . . . , σn ∈ Ui such that σ1 → * * * → σn → φs is valid. Since Ui contains i-Validity, Ui |= Ti(σ1 → * * * → σn → φs). By repeated applications of i-Deduction contained in Ui, Ui |= Tiσ1 → * * * → Tiσn → Tiφs. Since U is closed, Ui is Ti-closed and so contains Tiσ1, . . . ,Tiσn. So Ui |= (Tiφ)s and MU |= TiTiφ[s]. The following lemma shows that arithmetic is generic, which will enable us to state a later result (Theorem 24) in such a way that it is clear that the result is neither contingent on the presence, nor the absense, of arithmetic in the theories in question. Lemma 20. For any i ∈ ω, [S]i is closed-r.e.-generic, where S is the set of axioms of Peano arithmetic for LPA(ω). Proof. By Lemmas 8 and 10. Carlson proved [5] that it is consistent for an idealized knowing machine to know "I am a machine" (without knowing which specific machine it is). The following lemma sheds additional light: not only is it consistent for a knowing machine to know "I am a machine", in fact that knowledge is generic: it does not depend heavily on specific arbitrary decisions about the background theory of provability. Lemma 21. For any i, j ∈ ω, the following family is closed-r.e.-generic. • [S]i where S is: (j-SMT) (See [5] and [11]) ucl(∃e∀x(Tjφ↔ x ∈We)), e 6∈ FV(φ). Proof. Suppose U = (Ui)i∈ω is a closed r.e. family of LPA(ω)-theories and U ⊇ [S]i where S is j-SMT. We must show MU |= [S]i. That is, given φ with e 6∈ FV(φ), we must show MU |= ucl(∃e∀x(Tjφ↔ x ∈We)). Let s be an assignment and let x1, . . . , xk = FV(φ)\{x}. Since Uj is r.e., by the S-m-n theorem there is some n such that Wn = {m : Uj |= φ(x|m)(x1|s(x1)) * * * (xk|s(xk))}. Since e 6∈ FV(φ), and MU has standard first-order part, it follows that MU |= ∀x(Tjφ↔ x ∈We)[s(e|n)]. 5 Finally, the following lemma offers a way to obtain new building blocks from old. This can be combined with Lemma 21 to advance from "I am a machine" to "I know I am a machine". Lemma 22. For any i, j ∈ ω and any closed-r.e.-generic family T = (Tk)k∈ω, T ∪ [S]i is closed-r.e.-generic, where S is the schema: Tjφ (φ ∈ Tj). Proof. Suppose U = (Ui)i∈ω ⊇ T ∪ [S]i is closed and r.e. Right away MU |= T because T is closedr.e.-generic. It remains to show that MU |= [S]i, i.e., that MU |= S. Fix φ ∈ Tj and let s be any assignment. Since φ is a sentence, φ ≡ φs and thus Tj |= φs. Since Uj ⊇ Tj , Uj |= φs. By definition of MU, MU |= Tjφ[s]. We gather Lemmas 17–22 together into the following summary. Corollary 23. For any i, j ∈ ω, each of the following families is closed-r.e.-generic. 1. [j-Deduction]i. 2. [Assigned Validity]i. 3. [Assigned Validity]i ∪ [i-Validity]j . 4. [Assigned Validity]i ∪ [i-Validity]i ∪ [i-Deduction]i ∪ [i-Introspection]j . 5. [S]i where S is the set of axioms of Peano arithmetic for LPA(ω). 6. [j-SMT]i. 7. T ∪ [S]i, for any closed-r.e.-generic T, where S is the schema: Tjφ (φ ∈ Tj). The above building blocks are not exhaustive. In choosing building blocks, a primary concern was to facilitate creation of background provability theories strong enough to make our consistency result (Theorem 24) generalize Carlson's consistency result [5]. If that were our lone motivation, we could restrict Corollary 23 to only those families where i = j, but a secondary motivation was to provide inter-theory versions of those restricted building blocks. It would be interesting to investigate questions about whether the above building-blocks are minimal. For example, in Lemma 19, is it really necessary to bundle j-Introspection with all three other schemas? For now, we will leave those questions open. 4 First Consistency Result: Prioritizing Exact Codes The following theorem fulfills the first promise from the introduction. Theorem 24. Suppose ≺ is an r.e. well-founded partial order on ω and T0 = (T 0i )i∈ω is closed-r.e.-generic. For each n ∈ N, let T(n) = (Ti(n))i∈ω where each Ti(n) is the smallest Ti-closed theory containing the following: 1. The axioms in T 0i . 2. ∀x(Tjφ↔ 〈pφq, j, x〉 ∈Wn) whenever j ∈ ω, FV(φ) ⊆ {x}. 3. ucl(Tjφ→ φ) whenever j ≺ i. There is some n ∈ N such that T(n) is true. Proof. By the S-m-n Theorem, there is a total computable f : N→ N such that ∀n ∈ N, Wf(n) = {〈pφq, i,m〉 : FV(φ) ⊆ {x} and Ti(n) |= φ(x|m)}. Using the Recursion Theorem, fix n ∈ N such that Wf(n) = Wn. For brevity write T for T(n) and Ti for Ti(n). We will show MT |= T. This is a self-referential statement: to show Ti is true includes showing MT |= ucl(Tjφ→ φ), which is essentially the statement that Tj is true. Hence the restriction j ≺ i, which allows induction since ≺ is well founded. We will show, by ≺-induction on i, that MT |= Ti for every i ∈ ω. Fix i ∈ ω and assume MT |= Tj for all j ≺ i. Suppose σ ∈ Ti, we will show MT |= σ. 6 Case 1: σ ∈ T 0i . Then MT |= σ because T0 is closed-r.e.-generic and T ⊇ T0 is closed r.e. Case 2: σ is ∀x(Tjφ ↔ 〈pφq, j, x〉 ∈ Wn) for some j ∈ ω, FV(φ) ⊆ {x}. Let s be an assignment, m ∈ N. The following are equivalent. MT |= Tjφ[s(x|m)] Tj |= φs(x|m) (Definition of MT) Tj |= φ(x|m) (Since FV(φ) ⊆ {x}) 〈pφq, j,m〉 ∈Wn (By definition of n) MT |= 〈pφq, j,m〉 ∈Wn (MT has standard first-order part) MT |= 〈pφq, j, x〉 ∈Wn[s(x|m)]. (Lemma 8) Case 3: σ is ucl(Tjφ → φ) for some j ≺ i. Let s be an assignment and assume MT |= Tjφ[s]. This means Tj |= φs. By our ≺-induction hypothesis, MT |= Tj , so MT |= φs. By Lemma 8, MT |= φ[s]. Case 4: σ is only present in Ti because of the clause that Ti is Ti-closed. Then σ is Tiσ0 for some σ0 ∈ Ti. Being in Ti, σ0 is a sentence, so for any assignment s, σ0 ≡ σs0, Ti |= σs0, and finally MT |= Tiσ0[s]. By ≺-induction, MT |= Ti for all i ∈ ω. This shows MT |= T, that is, T is true. The first promise from the introduction is met: for any r.e. well-founded partial order ≺ on ω, there are theories (Tn)n∈ω such that ∀i, j, k ∈ ω with j ≺ i, Ti expresses the truth of Tj , and Ti expresses a Gödel number of Tk. In order to fulfill the second promise we will extend Carlson's notion of stratification to the case of multiple operators, and introduce stratifiers, a tool used to deal with subtleties that arise when multiple self-referential theories refer to one another. In [2] the technique behind Theorem 24 was used to exhibit a machine that knows its own code. 5 Stratification For the second promise from the introduction, we need to prove a result like Theorem 24 where Ti includes ucl(Tjφ → φ) for all j  i, not just j ≺ i. This rules out the direct ≺-induction of the type used above. Induction on formula complexity will not work either: we would need to show all of Ti consistent just to show MT |= Ti(1 = 0) → (1 = 0). Instead, we will use ordinal induction. But there are no ordinals anywhere in sight. To obtain ordinals to induct on, we will modify the theories we care about, in a process called stratification. We will start with some informal motivational remarks. Readers who would like to advance directly to the formal definitions can safely skip Subsection 5.1. 5.1 Motivation for Stratification As explained above, we would like to invoke ordinal induction, but there are no ordinals in sight. In order to make ordinal induction relevant, we will do the following. We will extend the background language to contain not only the operators Ti (i ∈ ω), but also operators Tαi  (i ∈ ω, α ∈ ε0 * ω). And instead of focusing directly on Ti, we will focus on a theory Ui such that the result U − i of erasing superscripts from Ui is U−i = Ti. The intended interpretation of T α i φ[s] will be Ui ∩α |= φs, where Ui ∩α is the set of axioms of Ui whose superscripts are < α. Thus, we may think of Ui as a version of Ti with extra information about the structure of Ti. We will show (Theorem 50), for certain formulas φ whose superscripts are positive multiples of ε0, that φ holds (in the intended interpretation) if and only if φ − holds. We will use this, after proving that Ui holds, to conclude that Ti also holds. Suppose we would like Ti to contain the axiom Ti(1 + 1 = 2). Then, as we carry out the procedure in the above paragraph, we would ensure that Ui contain all sentences of the form T α i (1 + 1 = 2). This would have the side effect that for any β > α, Ui ∩ β |= Tαi (1 + 1 = 2), so that T β i T α i (1 + 1 = 2) would hold in structures with the intended interpretation. Next, suppose that for every arithmetical sentence φ, we would like Ti to include Tiφ→ TiTiφ. 7 Then we would arrange that Ui contain Tαi φ→ T β i T α i φ (whenever β > α). The reason for the β is as follows. The intended interpretation of Tαi φ shall be Ui ∩ α |= φ. Thus, it would make no sense to put the axiom Tαi φ → Tαi Tαi φ into Ui: the fact that Ui ∩ α |= φ does not generally imply that Ui ∩ α |= Tαi φ, since Ui ∩ α is limited to formulas in which all superscripts are < α. At least Tαi φ→ T β i T α i φ is plausible. Again, suppose that for some j ≺ i, we would like for Ti to include Ti(Tj(1 = 0)→ (1 = 0)). We would arrange that Ui contain (for all α): Tαi (Tj(1 = 0)→ (1 = 0)). Note the lack of superscript on Tj. The intuition is that Ui is a version of Ti with extra information about the structure of Ti (namely, that said structure arises from an increasing family of theories), but without any additional information about the structure of Tj . Similarly, suppose we would like Ti to include Tj(Ti(1 = 0))→ Ti(1 = 0). We would arrange that Ui contain (for each α): Tj(Ti(1 = 0))→ Tαi (1 = 0). Note the lack of superscript on the Ti within the scope of Tj. As above, the intuition is that Ui is a version of Ti with extra information about the structure of Ti. It does not have any extra information about the structure of Tj-not even about what Tj says about Ti. This is important because, when j ≺ i, we would like Ti to contain axioms declaring, essentially, the Gödel number of Tj . This Gödel number would be hardcoded into such axioms, and thus there would be no hope of such axioms remaining true if Tj were changed. 5.2 Stratification Formal Details To get a foothold for induction, instead of considering a particular theory Ti, we will be considering copies of Ti with ordinal-number superscripts added. To recover information about the original Ti from these modified theories, we will need to use sophisticated results from [4] about the structure of the ordinals. Definition 25. We define a binary relation ≤1 on Ord by transfinite recursion so that for all α, β ∈ Ord, α ≤1 β if and only if α ≤ β and (α,≤,≤1) is a Σ1-elementary substructure of (β,≤,≤1). The following theorem is based on calculations from [4]. It was used by Carlson to prove Reinhardt's conjecture [5]. We state it here without proof. Theorem 26. 1. The binary relation ≤1 is a recursive partial ordering on ε0 * ω. 2. For all positive integers m ≤ n, ε0 *m ≤1 ε0 * n. 3. For any α ≤ β ∈ Ord, α ≤1 β if and only if the following statement is true. For every finite set X ⊆ α and every finite set Y ⊆ [α, β), there is a set X < Ỹ < α such that X ∪ Ỹ ∼=(≤,≤1) X ∪ Y . The usefulness of Theorem 26 will appear in Theorem 38, but first we need some machinery. Definition 27. Let I = ((ε0 * ω)× ω)t ω. Thus LPA(I) contains operators T(α,i) for all α ∈ ε0 * ω, i ∈ ω, along with operators Ti for all i ∈ ω. As abbreviation, we write Tαi  for T(α,i), and refer to α as its superscript. 8 Definition 28. For any LPA(I)-formula φ, On(φ) ⊆ ε0 * ω denotes the set of superscripts appearing in φ. Definition 29. Suppose i ∈ ω. The i-stratified formulas of LPA(I) are defined as follows (where φ ranges over LPA(I)-formulas). 1. If φ is Tjφ0 for some j 6= i, then φ is i-stratified if and only if φ is an LPA(ω)-formula. 2. If φ is Tαj φ0 for some j 6= i, then φ is not i-stratified. 3. If φ is Tiφ0, then φ is not i-stratified. 4. If φ is Tαi φ0, then φ is i-stratified if and only if φ0 is i-stratified and α > On(φ0). 5. If φ is ¬φ0, φ1 → φ2, or ∀xφ0, then φ is i-stratified if and only if its immediate subformula(s) are. 6. If φ is atomic, then φ is i-stratified. An LPA(I)-theory T is i-stratified if φ is i-stratified whenever φ ∈ T . An LPA(I)-formula φ is very i-stratified if φ is i-stratified and On(φ) ⊆ {ε0 * 1, ε0 * 2, . . .}. For example: • Tω7 T57(1 = 0)→ T8(1 = 0) is 7-stratified but not 6or 8-stratified. • T57Tω7 (1 = 0) is not 7-stratified, nor is T57T7(1 = 0). • T57T8T7(1 = 0) is 7-stratified but T57T8T47(1 = 0) is not. We will not make use of the following lemma, but we state it to further illuminate Definition 29. Lemma 30. Suppose φ is an LPA(I)-formula, i ∈ ω. Then φ is i-stratified if and only if all of the following conditions hold. 1. For all j ∈ ω and α ∈ ε0 * ω, if Tαj  occurs in φ, then j = i. 2. Every occurrence of Ti in φ is inside the scope of Tj for some j 6= i. 3. Tαi  never occurs in φ inside the scope of Tj, for any α ∈ ε0 * ω or any j ∈ ω. 4. For all α, β ∈ ε0 * ω, if Tαi  occurs in φ inside the scope of T β i , then β > α. Proof. Straightforward. Definition 31. Suppose X ⊆ ε0 * ω and h : X → ε0 * ω is order preserving. For each LPA(I)-formula φ, define an LPA(I)-formula h(φ) inductively as follows: 1. If φ is ¬φ0, φ1 → φ2, or ∀xφ0, then h(φ) is ¬h(φ0), h(φ1)→ h(φ2), or ∀xh(φ0), respectively. 2. If φ is atomic or Tiφ0, then h(φ) ≡ φ. 3. If φ is Tαi φ0 where α ∈ X, then h(φ) ≡ T h(α) i h(φ0). 4. If φ is Tαi φ0 where α 6∈ X, then h(φ) ≡ Tαi h(φ0). In practice, we will mainly be interested in φ when φ is i-stratified for some i, in which case Tαj  cannot occur within the scope of Tk in φ for any k, j. For such φ, h(φ) is simply the result of applying h to every superscript in φ that is in X. For example if X = {1, ω}, h(1) = 0, and h(ω) = ω * 2 + 1, then h ( T0i (1 = 0)→ T1i (1 = 0)→ Tωi (1 = 0) ) ≡ T0i (1 = 0)→ T0i (1 = 0)→ Tω*2+1i (1 = 0). In practice, we will primarily be interested in applying Definition 31 in the case where On(φ) ⊆ X. 9 Definition 32. Suppose X ⊆ ε0 * ω and h : X → ε0 * ω is order preserving. For any LPA(I)-structure N , we define an LPA(I)-structure h(N ) that has the same universe as N , agrees with N on LPA(ω), and interprets LPA(I)\LPA(ω) so that h(N ) |= Tαi φ[s] if and only if N |= h(Tαi φ)[s]. Lemma 33. Suppose X ⊆ ε0 * ω, h : X → ε0 * ω is order preserving, and N is an LPA(I)-structure. For any LPA(I)-formula φ and assignment s, h(N ) |= φ[s] if and only if N |= h(φ)[s]. Proof. By induction. Corollary 34. Suppose X ⊆ ε0 *ω and h : X → ε0 *ω is order preserving. For any valid LPA(I)-formula φ, h(φ) is valid. Proof. For any LPA(I)-structure N and assignment s, h(N ) |= φ[s] by validity, so N |= h(φ)[s] by Lemma 33. Definition 35. If X ⊆ Ord and h : X → Ord, we call h a covering if h is order preserving and whenever x, y ∈ X and x ≤1 y, h(x) ≤1 h(y). Definition 36. Suppose i ∈ ω. An LPA(I)-theory T is i-unistratified if the following conditions hold: 1. T is i-stratified. 2. (Uniformity) Whenever φ ∈ T , X ⊆ ε0 *ω, On(φ) ⊆ X, and h : X → ε0 *ω is a covering, then h(φ) ∈ T . Definition 37. If T is an LPA(I)-theory and α ∈ ε0 * ω, let T ∩ α be the set {φ ∈ T : On(φ) ⊆ α} of sentences in T that do not contain any superscripts ≥ α. Theorem 38. (The Collapse Theorem) Suppose T is an i-unistratified LPA(I)-theory. 1. If n is a positive integer and On(φ) ⊆ ε0 * n, then T |= φ if and only if T ∩ (ε0 * n) |= φ. 2. If α ≤1 β and On(φ) ⊆ α, then T ∩ α |= φ if and only if T ∩ β |= φ. Proof. Note that since T is i-unistratified, in particular T is i-stratified. We will prove (1), the proof of (2) is similar. (⇐) Immediate since T ∩ (ε0 * n) ⊆ T . (⇒) Assume T |= φ. By Theorem 3 there are σ1, . . . , σk ∈ T such that Φ ≡ σ1 → * * * → σk → φ is valid. Let X = On(Φ) ∩ (ε0 * n), Y = On(Φ) ∩ [ε0 * n,∞), note |X|, |Y | <∞. Since Y is finite, there is some integer n′ > n such that Y ⊆ ε0 *n′. By Theorem 26 part 2, ε0 *n ≤1 ε0 *n′. By Theorem 26 part 3, there is some X < Ỹ < ε0 * n such that X ∪ Ỹ ∼=(≤,≤1) X ∪ Y . Let h : X ∪ Y → X ∪ Ỹ be a (≤,≤1)-isomorphism. Since On(φ) ⊆ ε0 * n, h(φ) = φ. By Corollary 34, h(Φ) ≡ h(σ1)→ * * * → h(σk)→ φ is valid. Since T is i-unistratified, h(σ1), . . . , h(σk) ∈ T . Finally since range(h) < ε0 * n, h(σ1), . . . , h(σk) ∈ T ∩ (ε0 * n), showing T ∩ (ε0 * n) |= φ. Loosely speaking, what we have done in Theorem 38 is we have taken a proof of φ and we have collapsed the proof, shrinking its ordinals by using Theorem 26 part 3. Definition 39. For every i ∈ ω we define the following LPA(I)-schema: • (i-Collapse) ucl(Tαi φ↔ T β i φ) whenever T α i φ is i-stratified and α ≤1 β. Definition 40. For any LPA(I)-formula φ, φ− is the result of erasing all superscripts from φ. If T is an LPA(I)-theory, T− = {σ− : σ ∈ T}. 10 For example, if φ is Tω5 (1 = 0)→ Tω+15 Tω5 (1 = 0), then φ− is T5(1 = 0)→ T5T5(1 = 0). Lemma 41. If T is i-unistratified then for every φ ∈ T there is some ψ ∈ T such that ψ is very i-stratified and ψ− ≡ φ−. Proof. Let X = On(φ) = {α1 < * * * < αn}, Y = {ε0 * 1, . . . , ε0 * n}, and define h : X → Y by h(αj) = ε0 * j. Clearly h is order preserving; by Theorem 26 part 2, h is a covering. Since T is i-unistratified, T contains ψ ≡ h(φ). Clearly ψ is very i-stratified and ψ− ≡ φ−. Definition 42. For any LPA(ω)-structure N , we define an LPA(I)-structure N − that has the same universe as N , agrees with N on LPA(ω), and interprets LPA(I)\LPA(ω) as follows. For any LPA(I)formula φ, α ∈ ε0 * ω, i ∈ N, and assignment s, N − |= Tαi φ[s] if and only if N |= (Tαi φ)−[s]. Lemma 43. Suppose N is an LPA(ω)-structure. For every LPA(I)-formula φ and assignment s, N − |= φ[s] if and only if N |= φ−[s]. Proof. By induction. Corollary 44. If φ is a valid LPA(I)-formula, then φ− is a valid LPA(ω)-formula. Proof. Similar to the proof of Corollary 34. A converse-like statement holds for Corollary 44 as well. Lemma 45. For any valid LPA(ω)-sentence φ and i ∈ ω, there is a valid very i-stratified LPA(I)-sentence ψ such that ψ− ≡ φ. Proof. Let ψ 7→ ψ+ be the function taking LPA(ω)-formulas to LPA(I)-formulas defined as follows. 1. If ψ is atomic, or of the form Tjψ0 with j 6= i, then ψ+ ≡ ψ. 2. If ψ is Tiψ0, then ψ + ≡ Tε0*ni ψ + 0 , where n = min{m ∈ N : ε0 *m > On(ψ + 0 )}. 3. If ψ is ¬ψ0, ψ0 → ψ1, or ∀xψ0, then ψ+ is ¬ψ+0 , ψ + 1 → ψ + 2 , or ∀xψ + 0 , respectively. It is straightforward to show φ+ is very i-stratified. We claim φ+ is valid. Let M be any LPA(I)-structure, we will show M |= φ+. Let M+ be the LPA(ω)-structure with the same universe as M , which agrees with M on the interpretation of arithmetic and of Tj for j 6= i, and which interprets Ti as follows: M+ |= Tiψ[s] if and only if M |= (Tiψ)+[s]. Since φ is valid, M+ |= φ. It follows that M |= φ+. Definition 46. Let i ∈ ω. We define the following LPA(I)-schemas. • (i-Strativalidity) ucl(Tαi φ) whenever φ is a valid LPA(I)-formula and Tαi φ is i-stratified. • (i-Stratideduction) ucl(Tαi (φ→ ψ)→ Tαi φ→ Tαi ψ) whenever this formula is i-stratified. Definition 47. An LPA(I)-theory T is i-straticlosed if the following conditions hold: 1. T is i-unistratified. 2. T includes i-Strativalidity, i-Stratideduction and i-Collapse. 3. For every φ ∈ T , if Tαi φ is i-stratified then Tαi φ ∈ T . A family T = (Ti)i∈ω is straticlosed if each Ti is i-straticlosed. The following theorem serves as an omnibus of results from Section 5 of [5]. 11 Theorem 48. (Proof Stratification) Suppose T is an i-straticlosed LPA(I)-theory. Then: 1. Whenever T ∩ α |= φ, Tαi φ is an i-stratified sentence, and β > α, then T ∩ β |= Tαi φ. 2. For any very i-stratified LPA(I)-sentences ρ and σ, if ρ− ≡ σ− then T |= ρ↔ σ. 3. For any very i-stratified LPA(I)-sentence φ, T |= φ if and only if T− |= φ−. Proof. Note that since T is i-straticlosed, in particular T is i-unistratified and hence, i-stratified. Claim 0: Any time T |= Tαi (ρ↔ σ) and this is i-stratified, T |= Tαi ρ↔ Tαi σ. Assume the hypotheses. By i-Strativalidity, T |= Tαi ((ρ↔ σ)→ (ρ→ σ)). By i-Stratideduction, T |= Tαi ((ρ↔ σ)→ (ρ→ σ))→ Tαi (ρ↔ σ)→ Tαi (ρ→ σ) and T |= Tαi (ρ→ σ)→ Tαi ρ→ Tαi σ. It follows that T |= Tαi ρ→ Tαi σ. The reverse implication is similar. Claim 1: If T ∩ α |= φ, Tαi φ is an i-stratified sentence, and β > α, then T ∩ β |= Tαi φ. Given T ∩ α |= φ, there are σ1, . . . , σn ∈ T ∩ α such that σ1 → * * * → σn → φ is valid. By instances of i-Strativalidity and i-Stratideduction contained in T ∩ β, T ∩ β |= Tαi φ. Claim 2: If ρ and σ are very i-stratified LPA(I)-sentences and ρ− ≡ σ−, then T |= ρ↔ σ. By induction on ρ. Note that ρ is not of the form Tαj ρ0 (with j 6= i), as that is not i-stratified. If ρ is Tjρ0 then ρ ≡ ρ− ≡ σ− ≡ σ and the claim is immediate. The only nontrivial remaining case is when ρ is Tαi ρ0. Since ρ is very i-stratified, this implies α = ε0 *n (some positive integer n) and ρ0 is very i-stratified. Since σ − ≡ ρ− and σ is very stratified, this implies σ ≡ Tε0*mi σ0 for some positive integer m and very i-stratified σ0 with σ − 0 ≡ ρ − 0 . Assume m ≤ n, the other case is similar. By induction, T |= ρ0 ↔ σ0. By compactness, there is a natural ` ≥ n such that T ∩ (ε0 * `) |= ρ0 ↔ σ0. By Claim 1, T |= Tε0*`i (ρ0 ↔ σ0); Claim 0 then gives T |= T ε0*` i ρ0 ↔ T ε0*` i σ0. The claim now follows since T contains i-Collapse and ε0 *m ≤1 ε0 * n ≤1 ε0 * ` (Theorem 26 part 2). Claim 3: If φ is an i-stratified LPA(I)-sentence and T |= φ, then T− |= φ−. By compactness, find σ1, . . . , σn ∈ T such that σ1 → * * * → σn → φ is valid. By Corollary 44, so is σ−1 → * * * → σ−n → φ−, witnessing T− |= φ−. Claim 4: If φ is a very i-stratified LPA(I)-sentence and T− |= φ−, then T |= φ. By compactness, there is a valid sentence Φ ≡ σ−1 → * * * → σ−n → φ− where each σj ∈ T . By Lemma 45, there is a valid very i-stratified LPA(I)-sentence Ψ such that Ψ− ≡ Φ. And because Ψ− ≡ Φ, this implies Ψ ≡ σ∗1 → * * * → σ∗n → φ∗ where each (σ∗j ) − ≡ σ−j , (φ∗)− ≡ φ−, and σ∗1 , . . . , σ∗n, φ∗ are very i-stratified. By Lemma 41, there are very i-stratified σ∗∗1 , . . . , σ ∗∗ n ∈ T with each (σ∗∗j )− ≡ σ − j ≡ (σ∗j )−. By Claim 2, T |= φ∗ ↔ φ, and for j = 1, . . . , n, T |= σ∗∗j ↔ σ∗j . Thus T |= (σ∗∗1 → * * * → σ∗∗n → φ)↔ Ψ, and since Ψ is valid and the σ∗∗j ∈ T , this shows T |= φ. Definition 49. If T = (Ti)i∈ω is a straticlosed family of LPA(I)-theories, its stratification, written Str(T), is the family Str(T) = (Si)i∈I , where for every i ∈ ω, Si = T−i and ∀α ∈ ε0 * ω, S(α,i) = Ti ∩ α. Theorem 50. (The Stratification Theorem) Suppose T = (Ti)i∈ω is a straticlosed family of LPA(I)-theories. For any i ∈ ω, any very i-stratified LPA(I)-formula φ, and any assignment s, MStr(T) |= φ[s] if and only if MStr(T) |= φ−[s]. 12 Proof. By induction on φ. The only nontrivial case is when φ is Tαi ψ. Since φ is very i-stratified, ψ is very i-stratified and we may write α = ε0 * n for some positive integer n, On(ψ) ⊆ ε0 * n. The following are equivalent. MStr(T) |= Tε0*ni ψ[s] Ti ∩ (ε0 * n) |= ψs (Definition of MStr(T)) Ti |= ψs (Theorem 38) T−i |= (ψ s)− (Theorem 48) T−i |= (ψ −)s (Clearly (ψs)− ≡ (ψ−)s) MStr(T) |= Tiψ−[s]. (Definition of MStr(T)) 6 Stratifiers In order to apply theorems from the previous section, it is necessary to work with families T = (Ti)i∈ω where each Ti is i-stratified. If we want T − i to (locally) express the truthfulness of T − j , we cannot simply add a schema like ucl(Tjφ → φ) to Ti, because this is not necessarily i-stratified: for example, the particular instance TjTi(1 = 0)→ Ti(1 = 0) is not i-stratified. But neither is, say, TjTαi (1 = 0)→ Tαi (1 = 0), where Tαi  occurs within the scope of Tj. We will use a schema ucl(Tjφ → φ+), where •+ varies over what we call i-stratifiers. Definition 51. Suppose X ⊆ ε0 *ω, |X| =∞, and i ∈ ω. The i-stratifier given by X is the function φ 7→ φ+ taking LPA(ω)-formulas to LPA(I)-formulas as follows. 1. If φ is atomic or of the form Tjφ0 with j 6= i, then φ+ ≡ φ. 2. If φ is Tiφ0 then φ + ≡ Tαi φ + 0 where α = min{x ∈ X : x > On(φ + 0 )}. 3. If φ is ¬ψ, ψ → ρ, or ∀xψ, then φ+ is ¬ψ+, ψ+ → ρ+ or ∀xψ+, respectively. By an i-stratifier we mean an i-stratifier given by some X. By the i-veristratifier we mean the i-stratifier given by X = {ε0 * 1, ε0 * 2, . . .}. For example, if •+ is the i-veristratifier and j 6= i then (TjTi(1 = 0)→ TiTi(1 = 0))+ ≡ TjTi(1 = 0)→ Tε0*2i T ε0 i (1 = 0). Lemma 52. Suppose Z ⊆ ε0 * ω, h : Z → ε0 * ω is order preserving, i ∈ ω, and •+ is an i-stratifier. For any LPA(ω)-formula θ with On(θ+) ⊆ Z, there is a computable i-stratifier •∗ with θ∗ ≡ h(θ+). Proof. Let X0 = {h(α) : α ∈ On(θ+)}, let X = X0 ∪ {α ∈ ε0 * ω : α > X0}, and let •∗ be the i-stratifier given by X. By induction, for every subformula θ0 of θ, θ ∗ 0 ≡ h(θ+0 ). Definition 53. By a stratifier-set, we mean a finite set I = {•+1 , . . . , •+k} where each •+p is an ip-stratifier for some ip ∈ ω, and i1, . . . , ik are distinct. With I as above, we write Indices(I) for {i1, . . . , ik}. We say I is computable if each •+p is computable. For example, if •+1 is a 1-stratifier, •+2 is a 5-stratifier, and •+3 is a 2-stratifier, then I = {•+1 , •+2 , •+3} is a stratifier-set and Indices(I) = {1, 5, 2}. For a non-example, if •∗1 and •∗2 are distinct 1-stratifiers, then {•∗1 , •∗2} is not a stratifier-set, because it fails the distinctness condition. 13 Definition 54. 1. Suppose N is an LPA(I)-structure and I is a stratifier-set. We define an LPA(I)structure N I as follows. The universe and interpretation of arithmetic of N I agree with those of N , as do the interpretations of Ti (i 6∈ Indices(I)) and Tαi  (any α, i). For each i ∈ Indices(I), let •+ ∈ I be the corresponding i-stratifier, and let N I interpret Ti as follows. For any LPA(I)-formula φ and assignment s, we consider two cases. (a) If φ is an LPA(ω)-formula, then N I |= Tiφ[s] if and only if N |= (Tiφ)+[s]. (b) If φ is not an LPA(ω)-formula, then N I |= Tiφ[s] if and only if N |= Tiφ[s]. 2. For any i ∈ ω, any i-stratifier •+, and any LPA(I)-structure N , let N + = N I where I = {•+} is the stratifier-set containing only •+. Case 1b in Definition 54 is somewhat arbitrary. We will only ever really care about whether N I |= Tiφ[s] when Tiφ is j-stratified for some j. If φ is not an LPA(ω)-formula then Tiφ is not j-stratified for any j. Lemma 55. (Compare Lemma 43) Suppose N is an LPA(I)-structure, i ∈ ω, and •+ is an i-stratifier. For every LPA(ω)-formula φ and assignment s, N + |= φ[s] if and only if N |= φ+[s]. Proof. By induction. Lemma 56. For any LPA(ω)-formula φ, any i ∈ ω, and any i-stratifier •+, φ is valid if and only if φ+ is valid. Proof. (⇒) Assume φ is valid. For any LPA(I)-structure N and assignment s, N + |= φ[s] by validity, so N |= φ+[s] by Lemma 55. (⇐) By Corollary 44. Lemma 57. Suppose M is an LPA(I)-structure, I0 is a stratifier-set, i ∈ ω, i 6∈ Indices(I0), and •+ is an i-stratifier. Let I = I0∪{•+}. Then M I = (M I0)+. Furthermore, M+ and M I agree on the interpretation of Ti. Proof. Straightforward. Lemma 58. Suppose i ∈ ω and suppose M is an LPA(I)-structure with the property that for every very i-stratified LPA(I)-formula φ and assignment s, M |= φ[s] if and only if M |= φ−[s]. Suppose I is a stratifier-set such that i 6∈ Indices(I). Then for every very i-stratified LPA(I)-formula φ and assignment s, M I |= φ[s] if and only if M I |= φ−[s]. Proof. By induction on φ. Let s be an assignment. The only interesting cases are the following. Case 1: φ is Tjψ for some j. Then φ − ≡ φ and the claim is trivial. Case 2: φ has the form Tαj ψ for some j 6= i. Impossible, this is not i-stratified. Case 3: φ has the form Tαi ψ. The following are equivalent: M I |= Tαi ψ[s] M |= Tαi ψ[s] (M and M I agree on Tαi ) M |= (Tαi ψ)−[s] (By hypothesis) M I |= (Tαi ψ)−[s]. (Since i 6∈ Indices(I), M and M I agree on Ti) Lemma 59. Suppose LPA(I)-structure M is an instance of Definition 7, and suppose I is a stratifier-set. Then M I interprets formulas by substitution. 14 Proof. By induction on |I|. If |I| = 0, we are done by Lemma 8. Otherwise, we may decompose I as I = I0 ∪ {•+} where •+ is an i-stratifier. By induction, M I0 interprets formulas by substitution (∗). By Lemma 57, M I = (M I0)+. By definition of interpreting formulas by substitution, for every LPA(I)-formula φ and assignment s, M I0 |= φ[s] if and only if M I0 |= φs. We must show that for every such φ and s, (M I0)+ |= φ[s] if and only if (M I0)+ |= φs. We induct on φ. By Definition 54, (M I0) and (M I0)+ agree on all symbols except Ti, and they agree on Tiφ0 if φ0 is not an LPA(ω)-formula. Thus the only nontrivial case is when φ is of the form Tiφ0 for some LPA(ω)-formula φ0. Any such φ is itself an LPA(ω)-formula and thus susceptible to Lemma 55. The following are equivalent. (M I0)+ |= φ[s] M I0 |= φ+[s] (Lemma 55) M I0 |= (φ+)s (By (∗)) M I0 |= (φs)+ (Clearly (φ+)s ≡ (φs)+) (M I0)+ |= φs. (Lemma 55) 7 Generic Stratified Axioms We now have enough technical machinery to fulfill the second promise from the Introduction. We will fulfill it in a general way, essentially saying: "The theories in question, whose truth were in doubt, are true together with any background theory of provability such that..." Just like in Section 3, we do this by introducing a notion of genericness. Throughout this section, ≺ is an r.e. well-founded partial-order of ω. Definition 60. If i ∈ ω, we say that a stratifier-set I is above i if ∀j ∈ Indices(I), i ≺ j. We adopt the following convention: if I is above i then we will write I as I(i) in order to remind ourselves that I is above i. Definition 61. (Compare Definition 14) Suppose T = (Ti)i∈ω is an r.e. family of LPA(I)-theories and each Ti is i-unistratified. We say T is ≺-straticlosed-r.e.-generic (or straticlosed-r.e.-generic, if ≺ is clear from context) if for every straticlosed r.e. family U ⊇ T, every i ∈ ω, and every computable stratifier-set I(i) above i, M I(i) Str(U) |= Ti. Lemma 62. If the family T = (Ti)i∈ω of LPA(I)-sets is r.e. and is a union of straticlosed-r.e.-generic families, then T is straticlosed-r.e.-generic. Proof. Straightforward. 7.1 Straticlosed-r.e.-generic Building Blocks As in Section 3.1, we exhibit some examples of straticlosed-r.e.-generic families, which can be combined (via Lemma 62) to form background theories of provability. This will allow us to state Theorem 72 below in a generalized way, essentially saying that certain doubted theories are consistent with any background theory of provability built up from such blocks. This saves us from having to arbitrarily impose any particular background theory of provability. In the following lemma, for part 3, the intuition is that for the purpose of straticlosed-r.e.-genericness, what things Ti says about Tj need not merely be true, but must even remain true when a j-stratifier is applied to them. Tj(φ → ψ) → Tjφ → Tjψ lacks this property, because it could be that (Tjφ)+ ≡ Tαj φ+, (Tjψ) + ≡ Tβj ψ+, where β < α. For parts 1–2, the reason we cannot merge these parts into [j-Deduction]i (j  i) is because [i-Deduction]i is not i-stratified. Lemma 63. (Compare Lemma 17) For any i, j ∈ ω, each of the following families is straticlosed-r.e.-generic. 15 1. [i-Stratideduction]i. 2. [j-Deduction]i (if j ≺ i). 3. [S]i (if i ≺ j) where S is the following schema (φ, ψ range over LPA(ω)-formulas): (Modified j-Deduction) ucl(Tj(φ→ ψ)→ Tjφ→ Tj(ψ ∧ φ)). Proof. Clearly these families are unistratified. Recursive enumerability follows from the fact that ≺ is r.e. In each case below, let U = (Uk)k∈ω be a straticlosed r.e. family extending the family in question. For brevity, let M = MStr(U). (1) Let I(i) be any computable stratifier-set above i, we must show M I(i) |= ucl(Tαi (φ → ψ) → Tαi φ → Tαi ψ) assuming this formula is i-stratified. Let s be an assignment and assume M I(i) |= Tαi (φ → ψ)[s] and M I(i) |= Tαi φ[s]. By Definition 54, M I(i) and M agree on Tαi , so M |= Tαi (φ→ ψ)[s]. By definition of M = MStr(U), this means Ui ∩ α |= (φ → ψ)s. Clearly (φ → ψ)s ≡ φs → ψs, so Ui ∩ α |= φs → ψs. By similar reasoning, Ui ∩ α |= φs. By modus ponens, Ui ∩ α |= ψs, which means M |= Tαi ψ[s]. Since M and M I(i) agree on Tαi , M I(i) |= Tαi ψ[s], as desired. (2) Let I(i) be any computable stratifier-set above i, we must show M I(i) |= ucl(Tj(φ → ψ) → Tjφ → Tjψ). Let s be an assignment and assume M I(i) |= Tj(φ → ψ)[s] and M I(i) |= Tjφ[s]. Since I(i) is above i and j ≺ i, M I(i) and M agree on Tj, so M |= Tj(φ → ψ)[s] and M |= Tjφ[s]. By definition of M , U−j |= φs → ψs and U − j |= φs, thus U − j |= ψs, so M |= Tjψ[s] and thus so does M I(i). (3) Let I(i) be any computable stratifier-set above i, we must show M I(i) |= ucl(Tj(φ → ψ) → Tjφ → Tj(ψ ∧ φ)). Let s be an assignment and assume M I(i) |= Tj(φ → ψ)[s] and M I(i) |= Tjφ[s]. If j 6∈ Indices(I(i)), then M I(i) and M agree on Tj, so reason as in (2) above. If not, we can write I(i) = I0 ∪ {•+} where •+ is a computable j-stratifier, and Lemma 57 ensures that M I(i) and M+ agree on Tj. By definition of M+, M |= (Tj(φ → ψ))+[s] and M |= (Tjφ)+[s]. Let α, β ∈ ε0 * ω be such that (Tj(φ → ψ))+ ≡ Tαj (φ+ → ψ+) and (Tjφ)+ ≡ T β j φ +. Then M |= Tαj (φ+ → ψ+)[s] and M |= Tβj φ+[s]. This means Uj ∩ α |= (φ+ → ψ+)s and Uj ∩ β |= (φ+)s. Since φ is a subformula of φ→ ψ, it follows β ≤ α, thus Uj ∩ α |= (ψ+ ∧ φ+)s. So M |= Tαj (ψ+ ∧ φ+)[s]. By Definition 51, Tαj (ψ + ∧ φ+) ≡ (Tj(ψ ∧ φ))+ (this is the reason for the ψ ∧ φ clause) and finally M+ |= Tj(ψ ∧ φ)[s]. In Lemma 18, we introduced Assigned Validity as a single schema for inclusion in Ti for any i. In the following lemma, we need to break the stratified version of Assigned Validity into different ω-indexed families because the stratified version of Assigned Validity intended for inclusion in Ti (for any particular i) needs to be i-stratified. Lemma 64. (Compare Lemma 18) For any i, j ∈ ω, each of the following families is straticlosed-r.e.-generic. 1. [S]i where S is: (i-Assigned Strativalidity) the schema φ s (φ valid and i-stratified, s an assignment). 2. [i-Assigned Strativalidity]i ∪ [i-Strativalidity]i. 3. [i-Assigned Strativalidity]i ∪ [i-Validity]j (if j 6= i). Proof. For unistratifiedness, use Corollary 34. Recursive enumerability follows from the fact that ≺ is r.e. In each case below, let U = (Uk)k∈ω be a straticlosed r.e. family extending the family in question. For brevity, let M = MStr(U). (1) Let I(i) be any computable stratifier-set above i, let φ be any valid i-stratified formula, and let s be any assignment. Since φ is valid, M I(i) |= φ[s]. By Lemma 59, M I(i) |= φs, as desired. 16 (2) Let I(i) be any computable stratifier-set above i. By (1), M I(i) |= i-Assigned Strativalidity. We must show M I(i) |= ucl(Tαi φ), where φ is any valid LPA(I)-formula and α < ε0 *ω is any ordinal such that Tαi φ is i-stratified. Let s be any assignment. Since Ui contains i-Assigned Strativalidity, in particular Ui contains φs. Since Tαi φ is i-stratified, α exceeds all the superscripts in φ (hence in φ s), so Ui∩α |= φs. By definition of M , this means M |= Tαi φ[s]. By Definition 54, M and M I(i) agree on Tαi , so M I(i) |= Tαi φ[s], as desired. (3) By (1), M I(i) |= i-Assigned Strativalidity for every computable stratifier-set I(i) above i. Let J(j) be a computable stratifier-set above j, we must show M J(j) |= i-Validity. Let φ be a valid LPA(ω)-formula, s an assignment. Case 1: i 6∈ Indices(J(j)). Then M J(j) and M agree on Ti. Let •+ be an i-stratifier. Since φ is valid, so is φ+ (by Lemma 56), so (φ+)s ∈ Ui (since [i-Assigned Strativalidity]i is part of line 3). Clearly ((φ+)s)− ≡ φs, so φs ∈ U−i , thus M |= Tiφ[s], and so does M J(j). Case 2: i ∈ Indices(J(j)). Thus j ≺ i and we can write J(j) = J0 ∪ {•+} for some computable i-stratifier •+. By Lemma 57, M J(j) and M+ agree on Ti. Let α ∈ ε0 * ω be such that (Tiφ)+ ≡ Tαi φ+. As in Case 1, (φ+)s is an instance of i-Assigned Strativalidity, so (φ+)s ∈ Ui (since [i-Assigned Strativalidity]i is part of line 3). In fact by choice of α, (φ+)s ∈ Ui ∩ α, so M |= Tαi φ+[s], that is, M |= (Tiφ)+[s]. By Lemma 55, M+ |= Tiφ[s]. Since M J(j) and M+ agree on Ti, M J(j) |= Tiφ[s]. In Lemma 63 above, we had to modify what Ti says about j-Deduction for i ≺ j. No such modification is needed in the following lemma. This is interesting because in modal logic, positive introspection is generally considered much more controversial and demanding than basic deduction. Lemma 65. (Compare Lemma 19) For any i, j ∈ ω, each of the following families is straticlosed-r.e.-generic. 1. [i-Assigned Strativalidity]i ∪ [i-Strativalidity]i ∪ [i-Stratideduction]i ∪ [i-Introspection]j (j 6= i). 2. [i-Assigned Strativalidity]i ∪ [i-Strativalidity]i ∪ [i-Stratideduction]i ∪ [S]i where S is: (i-Stratrospection) ucl(Tαi φ→ T β i T α i φ) whenever this is i-stratified. Proof. For unistratifiedness, use Corollary 34. Recursive enumerability follows from the fact that ≺ is r.e. In each case below, let U = (Uk)k∈ω be a straticlosed r.e. family extending the family in question. For brevity, let M = MStr(U). (1) By Lemma 63 (part 1) and Lemma 64 (part 2), M I(i) |= (i-Assigned Strativalidity)∪ (i-Strativalidity)∪ (i-Stratideduction) for every computable stratifier-set I(i) above i. Let J(j) be a computable stratifier-set above j, we must show M J(j) |= i-Introspection. In other words, we must show M J(j) |= ucl(Tiφ → TiTiφ) for any LPA(ω)-formula φ. Let s be any assignment and assume M J(j) |= Tiφ[s]. Case 1: i 6∈ Indices(J(j)). Then M J(j) and M agree on Ti. Thus M |= Tiφ[s]. Let •+ be the iveristratifier. By Theorem 50, M |= (Tiφ)+[s]. Let α be such that (Tiφ)+ ≡ Tαi φ+, so M |= Tαi φ+[s]. By definition, this means Ui∩α |= (φ+)s. Let β be such that (TiTiφ)+ ≡ Tβi Tαi φ+, so β > α. By Part 1 of Theorem 48, Ui ∩ β |= Tαi (φ+)s. Thus M |= T β i T α i φ +[s]. By Theorem 50, M |= (Tβi Tαi φ+)−[s], that is, M |= TiTiφ[s]. Since M and M J(j) agree on Ti, M J(j) |= TiTiφ[s], as desired. Case 2: i ∈ Indices(J(j)). Thus j ≺ i and we can write J(j) = J0 ∪ {•+} for some computable i-stratifier •+. By Lemma 57, M J(j) and M+ agree on Ti. Thus M+ |= Tiφ[s]. By Lemma 55, M |= (Tiφ)+[s]. Let α be such that (Tiφ) + ≡ Tαi φ+, so M |= Tαi φ+[s]. By definition of M , this means Ui ∩ α |= (φ+)s. Let β be such that (TiTiφ) + ≡ Tβi Tαi φ+, so β > α. By Part 1 of Theorem 48, Ui ∩ β |= Tαi (φ+)s. Thus M |= Tβi Tαi φ+[s]. In other words, M |= (TiTiφ)+[s]. By Lemma 55, M+ |= TiTiφ[s]. Since M+ and M J(j) agree on Ti, M J(j) |= TiTiφ[s], as desired. (2) Let I(i) be any computable stratifier-set above i, we must show M I(i) |= ucl(Tαi φ → T β i T α i φ) assuming this is i-stratified (so β > α). Let s be any assignment and assume M I(i) |= Tαi φ[s]. By Definition 54, M I(i) and M agree on Tαi , so M |= Tαi φ[s]. By definition of M , this means Ui ∩ α |= φs. By Part 1 of Theorem 48, Ui∩β |= Tαi φs. Thus, M |= T β i T α i φ[s], and thus so does M I(i) since it agrees with M on Tβi . 17 For the next lemma, note that the proof shows more than is necessary, namely that the structures in question satisfy all the axioms of Peano arithmetic for LPA(I), not just the i-stratified ones. But of course, the full set of Peano axioms for LPA(I) is not i-stratified. Lemma 66. (Compare Lemma 20) For any i ∈ ω, [S]i is straticlosed-r.e.-generic, where S is the set of those axioms of Peano arithmetic for LPA(I) that are i-stratified. Proof. Unistratifiedness and recursive enumerability are clear. Let U = (Uk)k∈ω be a straticlosed r.e. family extending [S]i. By Lemma 59, M I(i) Str(U) interprets formulas by substitution. By Lemma 10, M I(i) Str(U) satisfies the axioms of Peano Arithmetic for LPA(I), as desired. Lemma 67. (Compare Lemma 21) For any i, j ∈ ω, each of the following families is straticlosed-r.e.-generic. 1. [j-SMT]i (j 6= i). 2. [S]i, where S is: (i-Strati-SMT) ucl(∃e∀x(Tαi φ↔ x ∈We)) when this is i-stratified, e 6∈ FV(φ). Proof. Unistratifiedness and recursive enumerability are clear. In each case below, let U = (Uk)k∈ω be a straticlosed r.e. family extending the family in question. For brevity, let M = MStr(U). (1) Let I(i) be any computable stratifier-set above i. We must show M I(i) |= ucl(∃e∀x(Tjφ ↔ x ∈ We)) for every LPA(ω)-formula φ with e 6∈ FV(φ). Let s be an assignment and let {x1, . . . , xk} = FV(φ)\{x}. Case 1: j 6∈ Indices(I(i)). Then M I(i) and M agree on Tj. Since U−j is r.e., by the S-m-n theorem there is some n such that Wn = {m : U−j |= φ(x|m)(x1|s(x1)) * * * (xk|s(xk))}. Since e 6∈ FV(φ) and M has standard first-order part, it follows that M |= ∀x(Tjφ ↔ x ∈ We)[s(e|n)]. By first-order semantics, M |= ∃e∀x(Tjφ ↔ x ∈ We)[s]. Since M and M I(i) agree on Tj, M I(i) |= ∃e∀x(Tjφ ↔ x ∈ We)[s], as desired. Case 2: j ∈ Indices(I(i)). Thus i ≺ j and we can write I(i) = I0∪{•+} for some computable j-stratifier •+. By Lemma 57, M I(i) and M+ agree on Tj. Let α be such that (Tjφ)+ ≡ Tαj φ+. Since Uj ∩ α is r.e., by the S-m-n theorem there is some n such that Wn = {m : Uj ∩ α |= φ+(x|m)(x1|s(x1)) * * * (xk|s(xk))}. Since e 6∈ FV(φ) (thus e 6∈ FV(φ+)), and since M has standard first-order part, it follows that M |= ∀x(Tαj φ+ ↔ x ∈ We)[s(e|n)]. By first-order semantics, M |= ∃e∀x(Tαj φ+ ↔ x ∈ We)[s]. In other words, M |= (∃e∀x(Tjφ↔ x ∈We))+[s]. By Lemma 55, M+ |= ∃e∀x(Tjφ↔ x ∈We)[s]. Since M+ and M I(i) agree on Tj, M I(i) |= ∃e∀x(Tjφ↔ x ∈We)[s], as desired. (2) Let I(i) be any computable stratifier-set above i, we must show M I(i) |= ucl(∃e∀x(Tαi φ ↔ x ∈ We)) for every LPA(I)-formula φ such that this is i-stratified and e 6∈ FV(φ). Let s be any assignment and let {x1, . . . , xk} = FV(φ)\{x}. Since Ui ∩ α is r.e., by the S-m-n theorem there is some n such that Wn = {m : Ui ∩ α |= φ(x|m)(x1|s(x1)) * * * (xk|s(xk))}. Since e 6∈ FV(φ), and since M has standard firstorder part, it follows that M |= ∃e∀x(Tαi φ ↔ x ∈ We)[s]. By Definition 54, M and M I(i) agree on Tαi , so M I(i) |= ∃e∀x(Tαi φ↔ x ∈We)[s], as desired. If T = (Tk)k∈ω is straticlosed-r.e.-generic, we cannot simply take an axiom φ from Tj and insert Tjφ into Ti without violating straticlosed-r.e.-genericness, because such a φ is not necessarily i-stratified. Thus, the following lemma has a somewhat more complicated structure than Lemma 22. Lemma 68. (Compare Lemma 22) Let i, j ∈ ω and suppose T = (Tk)k∈ω is straticlosed-r.e.-generic. Then each of the following families is straticlosed-r.e.-generic. 1. T ∪ [S]i where S is the schema Tαi φ (φ ∈ Ti such that this is i-stratified). 2. T ∪ [S]i where S is the schema Tjφ− (φ ∈ Tj , j ≺ i). Proof. Unistratifiedness and recursive enumerability are clear. In each case below, let U = (Uk)k∈ω be a straticlosed r.e. family extending the family in question. For brevity, let M = MStr(U). 18 (1) Since T is straticlosed-r.e.-generic and U ⊇ T is straticlosed and r.e., immediately M J(j) |= T (by Definition 61) for all j ∈ ω and any computable stratifier-set J(j) above j. Let I(i) be any computable stratifier-set above i. Suppose φ ∈ Ti and α ∈ ε0 * ω are such that Tαi φ is i-stratified, and let s be any assignment. Since Ui ⊇ Ti, φ ∈ Ui, in fact since Tαi φ is i-stratified, it follows that φ ∈ Ui ∩ α. Since φ is a sentence, φ ≡ φs, and so Ui ∩ α |= φs, and so M |= Tαi φ[s]. By Definition 54, M I(i) agrees with M on Tαi , so M I(i) |= Tαi φ[s], as desired. (2) Since T is straticlosed-r.e.-generic and U ⊇ T is straticlosed and r.e., immediately MK(k) |= T (by Definition 61) for all k ∈ ω and any computable stratifier-set K(k) above k. Let I(i) be any computable stratifier-set above i. Suppose φ ∈ Tj where j ≺ i. Let s be any assignment. Since Uj ⊇ Tj , φ ∈ Uj . By Lemma 41, there is some very j-stratified ψ ∈ Uj such that ψ− ≡ φ−. Clearly since φ is a sentence, so is ψ. By compactness, there is some positive integer multiple α of ε0 such that Uj ∩α |= ψ. Since ψ is a sentence, ψ ≡ ψs and thus Uj ∩ α |= ψs. Thus, M |= Tαj ψ[s]. By Theorem 50, M |= Tjψ−[s], so by choice of ψ, M |= Tjφ−[s]. Since I(i) is above i and j 6 i, M and M I(i) agree on Tj, so M I(i) |= Tjφ−[s], as desired. 7.2 Stratifiable-r.e.-generic Building Blocks We have established some straticlosed-r.e.-generic building blocks, but the goal of this paper is to better understand the structure of non-stratified theories-stratification is only a means to an end. Therefore, we introduce a corresponding non-stratified building-block notion. Definition 69. If T0 = (T 0i )i∈ω where each T 0 i is an LPA(ω)-theory, we say T 0 is ≺-stratifiable-r.e.-generic (or stratifiable-r.e.-generic if ≺ is clear from context) if there is some ≺-straticlosed-r.e.-generic family T = (Ti)i∈ω of LPA(I)-theories such that each T−i = T 0i . Lemma 70. If T = (Ti)i∈ω is any straticlosed-r.e.-generic family of LPA(I)-theories, then T− = (T−i )i∈ω is a stratifiable-r.e.-generic family of LPA(ω)-theories. Proof. Straightforward. Corollary 71. (Compare Corollary 23) For all i, j ∈ ω, each of the following families of LPA(ω)-theories is stratifiable-r.e.-generic. 1. [j-Deduction]i (if j  i). 2. [Modified j-Deduction]i (if i ≺ j). 3. [Assigned Validity]i. 4. [Assigned Validity]i ∪ [i-Validity]j . 5. [Assigned Validity]i ∪ [i-Validity]i ∪ [i-Deduction]i ∪ [i-Introspection]j . 6. [S]i where S is the axioms of Peano Arithmetic for LPA(ω). 7. [j-SMT]i. 8. (If j  i) T∪ [S]i, for any stratifiable-r.e.-generic T = (Tk)k∈ω, where S is the schema: Tjφ (φ ∈ Tj). Proof. By combining Lemma 70 with Lemmas 63–68. For parts involving validity, Lemma 56 can be used to provide valid stratified counterparts of valid non-stratified formulas. Comparing the stratifiable-r.e.-generic families we exhibited (Corollary 71) with the closed-r.e.-generic families we exhibited (Corollary 23), we see that the stratifiable-r.e.-generic families are weaker in exactly two ways: 1. They do not allow Ti to state j-Deduction for Tj when i ≺ j, instead allowing what we called Modified j-Deduction. 19 2. Their closure property is more restricted: if T1 = (T 1k )k∈ω is closed-r.e.-generic and T 2 = (T 2k )k∈ω is stratifible-r.e.-generic, and if S1 is the schema Tjφ (φ ∈ T 1j ), and if S2 is the schema Tjφ (φ ∈ T 2j ), then Corollary 23 says T1 ∪ [S1]i is closed-r.e.-generic with no restrictions on j, whereas Corollary 71 only says that T2 ∪ [S2]i is stratifiable-r.e.-generic if j ≺ i. We leave it an open question to what extent Corollary 71 could be further strengthened. Our primary motivation in choosing building blocks was to facilitate creation of background provability theories at least strong enough to make our own consistency result (Theorem 72 below) generalize Carlson's consistency result [5]. If that were our lone motivation, we could restrict Corollary 71 to only those families where i = j, but a secondary motivation was to provide inter-theory versions of those restricted building blocks. 8 Second Consistency Result: Prioritizing Self-Truth In this section, we continue to fix an r.e. well-founded partial-order ≺ of ω. The following theorem will satisfy the second promise from the introduction: it will exhibit true theories (Ti)i∈ω such that Ti expresses a Gödel number of Tj (j ≺ i) and the truth of Tj (j  i). These theories can further be taken so that Ti expresses the fact that Tj has some Gödel number (all i, j), by Lemma 67. Theorem 72. Let T0 = (T 0i )i∈ω be any stratifiable-r.e.-generic family of LPA(ω)-theories. For every i ∈ ω and n ∈ N, let Ti(n) be the smallest Ti-closed LPA(ω)-theory containing the following axioms. 1. The axioms contained in T 0i . 2. Assigned Validity, i-Validity and i-Deduction. 3. ucl(Tjφ→ φ) whenever j  i. 4. ∀x(Tjφ↔ 〈pφq, j, x〉 ∈Wn) whenever j ≺ i, FV(φ) ⊆ {x}. Let each T(n) = (Ti(n))i∈ω. There is some n ∈ N such that T(n) is true. Proof. By the S-m-n Theorem, there is a total computable f : N→ N such that ∀n ∈ N, Wf(n) = {〈pφq, j,m〉 ∈ N : φ is an LPA(ω)-formula, FV(φ) ⊆ {x}, and Tj(n) |= φ(x|m)}. By the Recursion Theorem, there is an n ∈ N such that Wn = Wf(n). We will show T(n) is true. For the rest of the proof, we write T for T(n), Ti for Ti(n). The structure of the proof is as follows. • ("Definition of U" below) First, we will define a certain carefully-chosen family U = (Ui)i∈ω of LPA(I)theories (with each U−i = Ti) and the LPA(I)-structure M = MStr(U). • ("Preliminary Result" below) Next, we will show that ∀i ∈ ω, M |= Ui ∪ Ti. In order to deal with the difficulty mentioned at the beginning of Section 6, we will prove more than necessary, to obtain a strong ≺-induction hypothesis. Namely, we will prove, by ≺-induction, that ∀i ∈ ω, for every computable stratifier-set I(i) above i, M I(i) |= Ui ∪ Ti. – (Claim 1 below) In order to prove M I(i) |= Ui, we will use induction on α to show that M I(i) |= Ui ∩ α for all α ∈ ε0 * ω. – (Case 3 below) Part of proving M I(i) |= Ui∩α will be proving M I(i) |= ucl(Tα0i φ→ φ) whenever this is i-stratified, α0 < α. This is where we will use the α-induction hypothesis. – (Case 4 below) Part of proving M I(i) |= Ui∩α will be proving M I(i) |= ucl(Tjφ→ φ+) whenever j ≺ i, φ is an LPA(ω)-formula, and •+ is an i-stratifier. This is where we will take advantage of our strong ≺-induction hypothesis. • (Claims 2–3 below) Once we've established M I(i) |= Ui, we will essentially be able to conclude M I(i) |= Ti using the Stratification Theorem (Theorem 50). 20 • At the very end of the proof, having established that ∀i ∈ ω, M |= Ui ∪ Ti, we will use that to prove that MT |= T, i.e., that T is true. Definition of U. Since T0 is stratifiable-r.e.-generic, there is a straticlosed-r.e-generic family V = (Vi)i∈ω of LPA(I)-theories such that each V −i = T 0i . For every i ∈ N, let Ui be the smallest i-stratified LPA(I)-theory such that the following hold. 1. Ui contains Vi. 2. Ui contains i-Assigned Strativalidity, i-Strativalidity, i-Stratideduction and i-Collapse. 3. Ui contains ucl(T α i φ→ φ) whenever Tαi φ is i-stratified. 4. Ui contains ucl(Tjφ→ φ+) for every LPA(ω)-formula φ, j ≺ i, and i-stratifier •+. 5. Ui contains ∀x(Tjφ↔ 〈pφq, j, x〉 ∈Wn) whenever j ≺ i, FV(φ) ⊆ {x} and φ is an LPA(ω)-formula. 6. Whenever φ ∈ Ui and Tαi φ is i-stratified, Tαi φ ∈ Ui. Let U = (Ui)i∈ω. Observe that U is straticlosed and r.e. (to see Ui is i-unistratified, use Lemma 52; to see U is r.e., use Theorem 26 part 1); U ⊇ V; and for each i ∈ ω, U−i = Ti. Let M = MStr(U). Recall that Str(U) is the LPA(I)-family (Si)i∈I where ∀i ∈ ω and α ∈ ε0 * ω, Si = U − i = Ti and S(α,i) = Ui ∩ α. For the reader's convenience, here is how (by definition) M interprets Ti and T α i  for all i ∈ ω, α ∈ ε0 * ω: M |= Tiφ[s] iff Ti |= φs, M |= Tαi φ[s] iff Ui ∩ α |= φs. Preliminary Result. We would like to prove the following preliminary result: ∀i ∈ ω, M |= Ui ∪ Ti. For the sake of a stronger induction hypothesis, we will prove that ∀i ∈ ω, for every computable stratifier-set I(i) above i, M I(i) |= Ui ∪ Ti. This is more than enough because M I(i) = M when I(i) = ∅. Fix i ∈ ω. By ≺-induction, we have the following: (∗) For every j ≺ i, for every computable stratifier-set J(j) above j, M J(j) |= Uj ∪ Tj . Let I(i) be any computable stratifier-set above i. We must show M I(i) |= Ui ∪ Ti. Claim 1: ∀α ∈ ε0 * ω, M I(i) |= Ui ∩ α. By induction on α. Let σ ∈ Ui ∩ α. Case 1: σ ∈ Vi. Then M I(i) |= σ because V is straticlosed-r.e.-generic and U ⊇ V is straticlosed and r.e. Case 2: σ is an instance of i-Assigned Strativalidity, i-Strativalidity, or i-Stratideduction. Then M I(i) |= σ by Lemma 63 or Lemma 64. Case 3: σ is ucl(Tα0i φ→ φ) for some i-stratified LPA(I)-formula φ such that T α0 i φ is i-stratified. Since σ ∈ Ui ∩ α, this forces α0 < α. Let s be an assignment and assume M I(i) |= Tα0i φ[s], then: M I(i) |= Tα0i φ[s] (Assumption) M |= Tα0i φ[s] (M and M I(i) agree on T α0 i  by Def. 54) Ui ∩ α0 |= φs (Definition of M ) M I(i) |= φs (By α-induction, M I(i) |= Ui ∩ α0) M I(i) |= φ[s]. (Lemma 59) Case 4: σ is ucl(Tjφ→ φ+) for some LPA(ω)-formula φ, j ≺ i, and i-stratifier •+. By Lemma 52 we may assume •+ is computable. Let J(j) be the computable stratifier-set J(j) = I(i) ∪ {•+}, which is above j 21 since I(i) is above i and j ≺ i. Let s be an assignment and assume M I(i) |= Tjφ[s], then: M I(i) |= Tjφ[s] (Assumption) M |= Tjφ[s] (Since j ≺ i and I(i) is above i, M I(i) and M agree on Tj) Tj |= φs (Definition of M ) M J(j) |= φs (Since M J(j) |= Tj by (∗)) (M I(i))+ |= φs (Lemma 57) M I(i) |= (φs)+ (Lemma 55) M I(i) |= (φ+)s (Clearly (φs)+ ≡ (φ+)s) M I(i) |= φ+[s]. (Lemma 59) Case 5: σ is ∀x(Tjφ↔ 〈pφq, j, x〉 ∈Wn) for some LPA(ω)-formula φ with FV(φ) ⊆ {x} and j ≺ i. Let s be any assignment, say s(x) = m. The following biconditionals are equivalent: M I(i) |= Tjφ↔ 〈pφq, j, x〉 ∈Wn[s] M |= Tjφ↔ 〈pφq, j, x〉 ∈Wn[s] (M I(i) and M agree on the symbols in question) M |= Tjφ[s] iff M |= 〈pφq, j,m〉 ∈Wn (Lemma 59) M |= Tjφ[s] iff 〈pφq, j,m〉 ∈Wn (M has standard first-order part) Tj |= φs iff 〈pφq, j,m〉 ∈Wn (Definition of M ) Tj |= φ(x|m) iff 〈pφq, j,m〉 ∈Wn. (Since FV(φ) ⊆ {x}) The latter is true by definition of n. Case 6: σ is an instance Tβi φ↔ T γ i φ of i-Collapse (so β ≤1 γ and T β i φ↔ T γ i φ is i-stratified). Let s be an assignment, since M I(i) and M agree on Tβi  and T γ i , we need only show M |= T β i φ↔ T γ i φ[s]. In other words we must show Ui ∩ β |= φs if and only if Ui ∩ γ |= φs. This is by Theorem 38. Case 7: σ is Tα0i φ for some LPA(I)-formula φ such that T α0 i φ is i-stratified and φ ∈ Ui. Since T α0 i φ is i-stratified, On(φ) ⊆ α0, so φ ∈ Ui ∩ α0. Thus M |= Tα0i φ, so M I(i) |= T α0 i φ since M I(i) and M agree on Tα0i . Cases 1–7 establish M I(i) |= Ui ∩ α. By arbitrariness of α, Claim 1 is proved. Claim 2: For any assignment s and any very i-stratified LPA(I)-formula φ, M I(i) |= φ[s] if and only if M I(i) |= φ−[s]. By Theorem 50, for all such s and φ, M |= φ[s] if and only if M |= φ−[s]. The claim now follows from Lemma 58 (i 6∈ Indices(I(i)) because I(i) is above i). Claim 3: M I(i) |= Ti. For any σ ∈ Ti, there is some τ ∈ Ui such that τ− ≡ σ; since Ui is i-unistratified, we may take τ to be very i-stratified (Lemma 41). By Claim 1, M I(i) |= Ui, so M I(i) |= τ . By Claim 2, M I(i) |= σ. For each i ∈ ω, letting I(i) = ∅, Claims 1–3 show that M |= Ui ∪Ti. It follows that M |= T. Now, for every i ∈ ω, MT interprets Ti as follows: MT |= Tiφ[s] iff Ti |= φs. This is exactly the same way that M interprets Ti. It follows that M and MT agree on LPA(ω)-formulas. Thus, since M |= T, MT |= T, i.e., T is true. 9 Well-Foundation and Ill-Foundation The following is a variation on Kleene's O. 22 Definition 73. Simultaneously define O ⊆ N and | • | : O → Ord so that O ⊆ N is the smallest set such that: 1. 0 ∈ O (it represents the ordinal |0| = 0). 2. ∀n ∈ O, 2n ∈ O (it represents the ordinal |2n| = |n|+ 1). 3. If φe (the eth partial recursive function) is total and range(φe) ⊆ O, then 3 * 5e ∈ O (it represents the ordinal |3 * 5e| = sup{|φe(0)|, |φe(1)|, . . .}). To avoid technical complications, we have differed from the usual Kleene's O in the following way: in the usual definition, in order for 3 * 5e to lie in O, it is also required that |φe(0)| < |φe(1)| < * * * . Definition 74. L OPA is the language of Peano arithmetic extended by a unary predicate O. The following notions are defined by analogy with Section 2: 1. For any assignment s and L OPA(I)-formula φ with FV(φ)={x1, . . . , xn}, φs ≡ φ(x1|s(x1)) * * * (xn|s(xn)). 2. If T = (Ti)i∈I is an I-indexed family of L OPA(I)-theories, the intended structure for T is the L O PA(I)structure MT with universe N, interpreting symbols of PA as usual and interpreting O as O, and interpreting Ti (i ∈ I) as in Definition 7. For any L OPA(I)-structure N , we write N |= T if ∀i ∈ I, N |= Ti. We say T is true if MT |= T. Definition 75. If I is an index set and T = (Ti)i∈I is a family of L OPA(I)-theories, then for any i ∈ I such that MT |= Ti, we define the ordinal ‖Ti‖ = sup{|m|+ 1 : Ti |= O(m)}. The above definition makes sense: since MT |= Ti and OMT = O, the supremands are defined. Definition 76. The basic axioms of O are the following L OPA-axioms. 1. O(0). 2. O(n)→ O(2n), for every n ∈ N. 3. ∀x(φn(x)↓ & O(φn(x)))→ O(3 * 5n), for every n ∈ N. We have written the last two lines using infinite schemata to strengthen the following result. Theorem 77. Let I be an index set, ≺ a binary relation on I. Suppose T = (Ti)i∈I is a family of L OPA(I)theories with the following properties: 1. ∀i ∈ I, Ti contains the axioms of Peano arithmetic. 2. ∀i ∈ I, Ti contains the basic axioms of O. 3. ∀i ∈ I, ∀j ≺ i, ∃n ∈ N such that Ti |= ∀x(TjO(x)↔ x ∈Wn). 4. ∀i ∈ I, ∀j ≺ i, Ti |= ∀x(TjO(x)→ O(x)). If MT |= Ti ∪ Tj (in particular if T is true) and j ≺ i, then ‖Tj‖ < ‖Ti‖. Proof. Assume MT |= Ti ∪ Tj and j ≺ i. By hypothesis there is some n ∈ N such that Ti |= ∀x(TjO(x)↔ x ∈Wn) and Ti |= ∀x(TjO(x)→ O(x)). From these, Ti |= ∀x(x ∈Wn → O(x)). Since MT |= Ti, in particular MT |= ∀x(TjO(x)↔ x ∈Wn). This meansWn = {m ∈ N : Tj |= O(m)}. Since Tj includes the axiom O(0), Wn 6= ∅. Since Wn 6= ∅, by computability theory there is some k ∈ N such that PA |= (domain(φk) = N) ∧ (range(φk) = Wn). Since Ti includes PA, Ti also implies as much. Combined with Ti |= ∀x(x ∈ Wn → O(x)), it follows that Ti |= ∀x(φk(x)↓ & O(φk(x))). Since Ti contains the basic axiom ∀x(φk(x)↓ & O(φk(x))) → O(3 * 5k), Ti |= O(3 * 5k). 23 To finish the proof, calculate ‖Tj‖ = sup{|m|+ 1 : Tj |= O(m)} = sup{|m| : Tj |= O(m)} (Since Tj contains O(n)→ O(2n) for all n ∈ N) = sup{|m| : m ∈Wn} (Since Wn = {m ∈ N : Tj |= O(m)}) = sup{|φk(0)|, |φk(1)|, . . .} (By choice of k) = |3 * 5k| (Definition 73) < sup{|m|+ 1 : Ti |= O(m)} (Since Ti |= O(3 * 5k)) = ‖Ti‖. Corollary 78. (Well-Foundedness of True Self-Referential Theories) Let I, T, ≺ be as in Theorem 77. If T is true then ≺ is well founded, by which we mean there is no infinite descending sequence i0 i1 * * * . In particular Corollary 78 says that if I, T, ≺ are as in Theorem 77 and T is true then ≺ is strict: there is no i with i ≺ i. This gives a new form (under the additional new assumption of containing/knowing basic rudiments of computable ordinals) of the Lucas–Penrose–Reinhardt argument that a truthful theory (or machine) cannot state (or know) its own truth and its own Gödel number. We could remove Peano arithmetic from Theorem 77 if we further departed from Kleene and changed line 3 of Definition 73 to read: 3. If We ⊆ O, then 3 * 5e ∈ O (and |3 * 5e| = sup{|n| : n ∈We}, or |3 * 5e| = 0 if We = ∅) (and altered Definition 76 accordingly). The previous paragraph would still stand, in fact giving a version of the Lucas–Penrose–Reinhardt argument in which the theory (machine) is not required to contain (know) arithmetic. We close the paper by showing that Corollary 78 fails without O. Let WF be the set of all r.e. wellfounded partial orders on ω and let Tr be the set of all true LPA-sentences. It is well-known that WF is computability theoretically Π11-complete and Tr is ∆ 1 1, so WF cannot be defined in LPA ∪ {Tr}. Theorem 79. (Ill-Foundedness of True Self-Referential Theories) 1. There exists an r.e., ill-founded partial order ≺ on ω such that for every closed-r.e.-generic T0 = (T 0i )i∈ω there is an n ∈ N such that T(n) is true, where T(n) is as in Theorem 24. 2. There exists an r.e., ill-founded partial order ≺ on ω such that for every ≺-stratifiable-r.e.-generic T0 = (T 0i )i∈ω there is an n ∈ N such that T(n) is true, where T(n) is as in Theorem 72. Proof. We prove (1), (2) is similar. Assume ¬(1). For each r.e. partial order≺ on ω, let S(≺) be the statement of Theorem 24 for ≺, minus the requirement that ≺ be well founded. Combining ¬(1) with Theorem 24, ≺ is well founded if and only if S(≺) is true. We will argue that S(≺) is expressible in LPA ∪ {Tr}, which is absurd because that would mean it is possible to define WF in LPA ∪ {Tr}. S(≺) is equivalent to the following: • For any (Gödel number of an) r.e. family T0 = (T 0i )i∈ω of LPA(ω)-theories, if T0 is closed-r.e.-generic (i.e., if MU |= T0 for every closed r.e. family U ⊇ T0 of LPA(ω)-theories), then there is some n ∈ N such that T(n) is true (i.e., such that MT(n) |= T(n)), where T(n) = (Ti(n))i∈ω, where each Ti(n) is the smallest Ti-closed theory containing the following: 1. The axioms in T 0i . 2. ∀x(Tjφ↔ 〈pφq, j, x〉 ∈Wn) whenever j ∈ ω, FV(φ) ⊆ {x}. 3. ucl(Tjφ→ φ) whenever j ≺ i. This is manifestly expressible in LPA except for the clauses MU |= T0 and MT(n) |= T(n). We will show that MU |= T0 is expressible in LPA ∪ {Tr}; the expressibility of MT(n) |= T(n) is similar. Define an operator FU which takes an LPA(ω)-formula φ and outputs an LPA-formula FU(φ) as follows: 24 • If φ is atomic, let FU(φ) ≡ φ. • If φ is ¬φ0, φ1 → φ2, or ∀xφ0, let FU(φ) be ¬FU(φ0), FU(φ1)→ FU(φ2), or ∀xFU(φ0), respectively. • Suppose φ is Tiψ and FV(ψ) = {x1, . . . , xk}. Let f : Nk → N be the computable function such that for all m1, . . . ,mk ∈ N, f(m1, . . . ,mk) = pψ(x1|m1) * * * (xk|mk)q. Let FU(φ) be: "Ui proves the sentence with Gödel number f(x1, . . . , xk)" (so FV(FU(φ)) = {x1, . . . , xk}). It is easy to check that for every LPA(ω)-formula φ and assignment s, MU |= φ[s] if and only if N |= FU(φ)[s]. In particular, for every LPA(ω)-sentence φ, MU |= φ if and only if N |= FU(φ). Thus, the clause MU |= T0 can be expressed in LPA ∪ {Tr} as follows: ∀i∀x(x ∈ T 0i → Tr(pFU(x)q)). The way we prove Theorem 79 by referring to the computability theoretical complexity of WF is similar to a recent argument by Kripke [7]. References [1] Alexander, S. (2013). 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