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All known instances of non-linearity and ill-foundedness have been discovered by defining theories in an ad-hoc manner using self-reference and other logical tricks. When one restricts one’s attention to the natural axiomatic theories—those that 2 arise in practice—ordered by + + 1 L(R) EA, EA , EA , IΣn, PA, ATR0, ΠnCA0, PAn, ZF, AD The well-ordering phenomenon persists, taking a very liberal view of what consti- tutes a “natural” theory. Note that the theories just cited run the gamut from weak fragments of arithmetic to subsystems of analysis and all the way to strong extensions of set theory. These theories come from different areas of mathemat- ics (e.g., arithmetic, analysis, set theory) and often codify different conceptions of mathematics. Indeed, many of the natural extensions of set theory that have been investigated are jointly inconsistent, yet comparable according to consistency strength. Explaining the contrast between natural axiomatic theories and axiomatic theories in general is widely regarded as a major outstanding conceptual problem in math- ematical logic. The following passage from a paper of S. Friedman, Rathjen, and Weiermann is representative: The fact that “natural” theories, i.e. theories which have something like an “idea” to them, are almost always linearly ordered with re- gard to logical strength has been called one of the great mysteries of the foundations of mathematics. (S. Friedman, Rathjen, Weier- mann [10] p. 382) If it is true that natural axiomatic theories are pre-well-ordered by consistency strength, and not an illusion engendered by a paucity of examples, then one would like to prove that it is true. However, the claim that natural axiomatic theories are pre-well-ordered by consistency strength is not a strictly mathematical claim. The problem is that we lack a definition of “natural axiomatic theory.” Without a precise definition of “natural,” it is not clear how to prove this claim, or even how to state it mathematically. Here is my plan for the rest of the paper. In §2 I will discuss the relevance of the pre-well-ordering phenomenon to the search for new axioms in set theory. In §3–5 I will discuss two research programs that emerged from the incompleteness theo- rems, namely, ordinal analysis and the study of iterated reflection principles. Both programs will play an important role in the approaches to the pre-well-ordering problem discussed in this paper. In §6 I will discuss a phenomenon from recursion theory, namely, the well-ordering of natural Turing degrees by Turing reducibility; this phenomenon is analogous in many ways to the pre-well-ordering of natural theories by consistency strength. In §7–10 I will discuss a few different approaches 2This is to say that the induced ordering on the equivalence classes modulo equiconsistency is a well-ordering. ONTHEHIERARCHYOFNATURALTHEORIES 3 to the pre-well-ordering problem that have emerged in recent work. Finally, in §11 I will discuss remaining questions and directions for future research. 2. The search for new axioms I would like to consider how the well-ordering phenomenon plays out in practice, namely, in set theory. Indeed, the well-ordering phenomenon has been a focal point of interest in the search for new axioms extending the standard ZFC axioms for set theory. Set theory has been developed in an explicitly axiomatic fashion, at least since Zermelo isolated the axioms that are the precursors to the standard ZFC axioms. The early progenitors of set theory discovered that various mathematical structures could be realized in set-theoretic terms and that their properties could be estab- lished on the basis of set-theoretic reasoning. Thus, set theory constitutes a highly general mathematical framework. On the other hand, many of the central problems of set theory (such as the Continuum Hypothesis, the Projective Measure problem, and Suslin’s Hypothesis) cannot be resolved on the basis of the ZFC axioms. This has motivated the search for new axioms for set theory. Set theorists have investigated many extensions of ZFC, including large cardinal ax- ioms, determinacy axioms, forcing axioms, and more. Is it possible to make rational judgments about these principles and their consequences? Maddy has promoted the maxim “maximize interpretability strength” as a guiding principle in the search for new axioms (see, e.g, [20, 21]). Let’s call this principle maddy’s maxim. For a large swathe of theories, including extensions of set theory, maximizing interpretability strength coincides with maximizing consistency strength. For a sentence ϕ that is independent of ZFC, one can imagine the following four possibilities: (1) ϕ increases interpretability strength but ¬ϕ does not. (2) ¬ϕ increases interpretability strength but ϕ does not. (3) Neither ϕ nor ¬ϕ increases interpretability strength. (4) Both ϕ and ¬ϕ increase interpretability strength. It turns out that all four possibilities are realized; in the fourth case we cannot follow maddy’s maxim or we would land ourselves in inconsistency. However, this is not typically taken as a refutation of maddy’s maxim. The key point here is that when one restricts oneself to natural theories, only the first three possibilities are realized, since natural theories are linearly ordered by consistency strength. See [15] for a discussion of this point. Consider once more the variety of extensions of ZFC that are investigated by set theorists: large cardinal axioms, axioms of definable determinacy, forcing axioms, and more. These axiom systems, which have different motivations and often codify different foundational conceptions of mathematics, are pre-well-ordered by consis- tency strength. Indeed, they are pre-well-ordered according to all the common 1 notions of proof-theoretic strength (such as 1-consistency strength, Π1 reflection strength, and so on). A consequence is that these axiom systems all converge on arithmetic statements and even analytic statements. As Steel writes: 4 JAMES WALSH Thus the well-ordering of natural consistency strengths corresponds to a well-ordering by inclusion of theories of the natural num- bers. There is no divergence at the arithmetic level, if one climbs the consistency strength hierarchy in any natural way we know of. . . Natural ways of climbing the consistency strength hierarchy do not diverge in their consequences for the reals...Let T,U be natural theories of consistency strength at least that of “there are 1 1 infinitely many Woodin cardinals”; then either (Πω)T ⊆ (Πω)U or 1 1 (Πω)U ⊆ (Πω)T . (Steel [31] p. 159) That is, at the level of statements about R, all natural theories converge; maddy’s maxim and its variants suggest that we ought to endorse the sentences on which they converge. Thus, the apparent pre-well-ordering of theories by consistency strength and other notions of proof-theoretic strength plays a central role in the search for new axioms in set theory. 3. Ordinal analysis At this point it will be helpful to discuss two research programs that deal with questions of systematically reducing incompleteness. One is Gentzen’s program of ordinal analysis and the other is Turing’s program of completeness via iterated reflection principles. Both programs can be understood as reactions to the incom- pleteness theorems. The practitioners of ordinal analysis have attempted to reduce incompleteness by proving consistency statements in a systematic way, namely, from ever stronger transfinite induction principles. Turing’s program, on the other hand, uses the second incompleteness theorem as the major engine for overcoming incompleteness; the practitioners of Turing’s program try to systematically effect reductions in incompleteness by successively adding consistency statements to the- ories as new axioms. Both programs will play a major role in the approach to the “great mystery” discussed here. Ordinal analysis was developed in the context of Hilbert’s Program, an early twen- tieth century research program pioneered by David Hilbert. To combat skepticism about the cogency of infinitary mathematics, Hilbert proposed to (i) axiomatize in- finitary mathematics and (ii) prove the consistency of the axioms by finitary means. In 1931, Hilbert’s program reached a major obstacle in the form of G¨odel’s [12] sec- ond incompleteness theorem. Indeed, it follows from G¨odel’s theorem that if the principles of finitistic mathematics are codifiable in a reasonable axiomatic theory, then they do not prove their own consistency, much less the consistency of stronger theories. Thus, it is generally agreed that Hilbert’s program failed. Gentzen was apparently undeterred, however. Not long after G¨odel proved the incompleteness theorems, Gentzen [11] produced a consistency proof of arithmetic. Theorem 3.1 (Gentzen). If ε0 is well-founded, then arithmetic is consistent. The principles invoked in Gentzen’s proof are nearly universally regarded as finitisti- cally acceptable, except for the principle that the ordinal number ε0 is well-founded. Gentzen’s consistency proof marked the beginning of ordinal analysis, a research program whereby similar consistency proofs have been discovered for a wide range of axiomatic theories. Developing such a consistency proof for a theory T involves, ONTHEHIERARCHYOFNATURALTHEORIES 5 among other things, determining the proof-theoretic ordinal of T . Informally, the proof-theoretic ordinal of T is the least ordinal α such that induction along α suf- fices to prove the consistency of T . Making this informal definition precise is no easy task, as we shall see. The methods of ordinal analysis have been used to analyze many theories of in- terest, including subsystems of first-order arithmetic, second-order arithmetic, and set theory. Ordinal analysis has not yet reached the level of full second-order arith- metic. Nevertheless, it is reasonable to expect that the existing results are part of a general connection between ordinals and consistency. Do Gentzen-style methods suffice for proving the consistency of any axiomatic theory? The following result ≺ [16] might seem to suggest a positive answer (TIΠ1 is a sentence expressing the validity of induction for Π1 predicates along ≺): Theorem 3.2 (Kreisel–Shoenfield–Wang). For any reasonable theory T , there is ≺ 3 a presentation ≺ of a recursive ordinal such that PA + TIΠ1 ⊢ Con(T ). This theorem is proved by showing that, for any true Π1 sentence ϕ, one can encode the truth of ϕ into an ordinal notation system ≺ such that recognizing that ≺ has no Π1 definable descending sequences is equivalent to recognizing the truth of ϕ. One would presumably not recognize the validity of Π1 transfinite induction along such a notation system without knowing the truth of ϕ, so one could not use such a transfinite induction principle to prove ϕ. Thus, the epistemic value of this theorem is limited. As we will see, this is one version of a pervasive problem known as the canonicity problem. The ordinal notations that are devised to prove Theorem 3.2 are not natural notations. They form a notation system that one would introduce only in an ad-hoc manner to solve a problem in proof theory. Does the distinction between the pathological notation systems and the natural notation systems reflect some intrinsic mathematical properties of the notation systems? If one had a convincing definition of “natural,” one might conjecture that, for any reasonable theory T , ≺ there is a natural presentation ≺ of a recursive ordinal such that PA+TIΠ1 ⊢ Con(T ). However, at present there is no convincing evidence that it is possible to precisely define the “natural” notation systems. The canonicity problem also makes it difficult to define “proof-theoretic ordinal,” as suggested earlier. One might try to define the proof-theoretic ordinal of T as the least ordinal α such that induction along α (along with finitary methods) suffices to prove the consistency of T . The problem with this definition is that, in formalized theories, transfinite induction principles are stated in terms of presentations of ordinals, not the ordinals themselves. Kreisel has shown that it is always possible to prove the consistency of a reasonable theory by induction along a sufficiently pathological presentation of ω. Conversely, Beklemishev has shown that, for any reasonable theory T and recursive ordinal α, there is a sufficiently pathological presentation of α such that transfinite induction along that presentation (along with finitary methods) will not suffice to prove the consistency of T . When one 3Though I state this result in terms of PA, in the original paper it is stated in terms of the system Zµ from [14]. 6 JAMES WALSH restricts one’s attention to “natural” presentations of ordinals, the definition seems to work, but the current state of affairs is vexing and unsatisfactory. At present, many conflicting definitions of the “proof-theoretic ordinal” of a theory have been proposed; these definitions often coincide in crucial cases. Perhaps the most common definition is this: The proof-theoretic ordinal of a theory T is the supremum of the order-types of the primitive recursive well-orderings whose well- 1 foundedness is provable in T . This is sometimes called the Π1 ordinal of a theory. Of course, this notion is only useful for measuring the proof-theoretic strength of theories in which well-foundedness is expressible, so it does not apply, e.g., to fragments of first-order arithmetic. Moreover, it is a somewhat coarse notion of 1 1 strength, since Π1 ordinals are invariant under the addition of true Σ1 sentences 1 to the object theory. Nevertheless, the Π1 ordinal of a theory is an ordinal, not a presentation of an ordinal, so this is a somewhat robust notion. 4. Turing progressions The second incompleteness theorem suggests a method for dealing with the first incompleteness theorem. According to Turing [32], if we endorse the axioms of a reasonable axiomatic theory T , then there is a principled way of extending T , namely, by adopting T ’s consistency statement as an axiom. Of course, if one adopts the statement Con(T ) as an axiom, the statement Con(T + Con(T )) remains unprovable. However, there is a principled way of resolving this problem, namely, by adopting Con(T +Con(T )) as an axiom. Of course, this engenders a new problem, but it just as easily engenders a new solution, whence this process can iterated ad infinitum. Given presentations of recursive ordinals in the language of arithmetic, one can even extend this process into the effective transfinite. Naively, we might try to define the iterations of consistency over a theory T as follows: • T0 := T • Tα+1 := Tα + Con(Tα) • Tλ := α<λ Tα for λ a limit. However, suchS a definition does not even pin down the theory Tω+1. According to the definition, Tω+1 is just Tω + Con(Tω), but for Con(Tω) to be a statement of arithmetic we must have some effective presentation of Tω, and there are many choices for the latter. Accordingly, when one defines iterated consistency state- ments, one must first fix an ordinal notation system ≺. One can then define the iterations of consistency along ≺ within arithmetic via G¨odel’s fixed point lemma. Let me briefly indicate how this is done; thenceforth I will only give the intuitive “definitions” by transfinite recursion, trusting the reader to fill in the details.4 We fix a formula ConFormula(U, x) that formalizes the claim that x is the formula expressing the consistency of U. We also fix a formula Ax(x, T ) which formalizes that x is an axiom of T . We now want to define a formula Ax(x, α, T ) that defines 4For a discussion of this technique and comparisons with other techniques see the proof of Theorem 3.8 in [5]. ONTHEHIERARCHYOFNATURALTHEORIES 7 the axioms of the theory Conα(T ). We define the formula as the following fixed point: PA ⊢ Ax(x, α, T ) ↔ Ax(x, T ) ∨ ∃β ≺ αConFormula(Conβ(T ), x) Does iteratively endorsing consistency statements in this manner effect a significant reduction in incompleteness? The following result suggests a positive answer: Theorem 4.1 (Turing). For any true Π1 sentence ϕ, there is a presentation ≺ of ω +1 such that PA≺ ⊢ ϕ. This result is known as Turing’s Completeness Theorem. At first glance it may seem that this theorem is epistemically significant: we can come to know the truth of any Π1 statement ϕ simply by iterating consistency statements. How are the consistency statements used in the proof of ϕ ? The disappointing response is that they are not used at all. Instead, the truth of ϕ is encoded into a non- standard description of the base theory PA, which becomes available at iteration ω + 1. Given this non-standard description of PA, discerning what the theory PA≺ is committed to requires knowing the truth-value of ϕ. This is another appearance of the canonicity problem. As far as I know, this was the initial appearance of the problem, and Turing was the first to identify it. Turing was careful not to overstate the epistemic significance of his theorem. In- deed, he wrote: This completeness theorem as usual is of no value. Although it shows, for instance, that it is possible to prove Fermat’s last the- orem. . . (if it is true) yet the truth of the theorem would really be assumed by taking a certain formula as an ordinal formula. (Turing [32] p. 213) Nevertheless, one might hope that iterating Con along natural ordinal notations would also suffice to prove any true Π1 statement. Once again, if we had a precise definition of “natural” ordinal notation systems, we would be able to formulate a precise conjecture; however, no such characterization of “natural” ordinal notation systems is currently available. Turing discussed the possibility of using iterated consistency progressions to mea- sure the strength of Π1 statements. He wrote: We might also expect to obtain an interesting classification of number- theoretic theorems according to “depth”. A theorem which required an ordinal α to prove it would be deeper than one which could be proved by the use of an ordinal β less than α. However, this pre- supposes more than is justified. (Turing [32] p. 200) Naively, one approach would be to assign to each true Π1 sentence ϕ the least ordinal α such that PAα ⊢ ϕ as its measure of complexity. One could then extend this measure of complexity to theories: the complexity of T is the least ordinal such that PAα ⊢ ϕ for each Π1 theorem ϕ of T . Clearly, Turing’s completeness theorem shows that iterations of consistency depend on presentations of ordinals 8 JAMES WALSH and not just on ordinals themselves. The possibility remains, however, that one could prove informative results of this sort for large swathes of interesting theories by antecedently fixing some natural notation system; Beklemishev has recently pursued this possibility, as we will discuss in §5. Turing’s work was later pursued and greatly extended by Feferman [5]. Feferman shifted the focus from iterated consistency to iterated reflection principles of other sorts. Recall that RFN(T ) is the uniform reflection schema for T , that is, RFN(T ) := {∀~x PrT (ϕ(~x)) → ϕ(~x)) : ϕ(~x) ∈ LA}. After fixing a presentation ≺ of a recursive ordinal, we can then define iterations of reflection over a theory T so as to satisfy the following conditions: RFN • T0 := T RFN RFN RFN • Tα+1 := Tα + RFN(Tα ) RFN RFN • Tλ := α<λ Tα for λ a limit. Feferman provedS that, merely by iterating the uniform reflection schema along presentations of recursive ordinals, one can accrue resources sufficient, not only for proving any true Π1 statement, but for proving any true arithmetical statement. Theorem 4.2 (Feferman). For any true arithmetical sentence ϕ, there is a pre- ω ω +1 RFN sentation ≺ of an ordinal α<ω such that PA≺ ⊢ ϕ. This result is known as Feferman’s Completeness Theorem. In the proof of this theorem, Feferman used Turing’s technique of encoding the truth of statements into presentations of recursive ordinals, among other things. Feferman also proved that there are paths P through Kleene’s O such that iterating uniform reflection 1 along P is arithmetically complete [5]. However, no such path is even Σ1 definable [6]. That is, identifying such a path is more difficult than identifying the set of arithmetical truths. Thus, the epistemic significance of these results—as Feferman emphasized—is limited. 5. Ordinal analysis via iterated reflection In recent years there has been interest in the interface between ordinal analysis and Turing progressions. This research has been motivated by a number of the drawbacks of the research discussed in the previous two sections. 1 As discussed in §3, the standard notion of the Π1 ordinal of a theory is (i) only applicable to theories in which “well-foundedness” is expressible and (ii) insensitive 1 to the true Σ1 consequences of a theory. Building on earlier work of Schmerl [26], Beklemishev [2] introduced the notion of the Π1 ordinal of a theory; this notion of “ordinal analysis” is both (i) suitable for subsystems of first-order arithmetic and (ii) sensitive to the Π1 consequences of theories. Iterations of consistency in the style of Turing play a central role in the definition of Π1 ordinals. First we fix a base theory: Beklemishev uses EA+, a weak subsystem of arith- metic. The theory EA is a subsystem of arithmetic that proves the totality of exponentiation but not superexponentiation; EA is just strong enough to carry out ONTHEHIERARCHYOFNATURALTHEORIES 9 arithmetization of syntax in the standard way. EA+ extends EA with an axiom stat- ing the totality of superexponentiation; unlike EA it can prove the cut-elimination theorem and its standard corollaries, e.g., Herbrand’s Theorem. We then fix some natural ordinal notation system and define the iterations of EA+ so as to satisfy the following conditions: + + (1) EA0 := EA + + + (2) EAα+1 := EAα + Con(EAα ) + + (3) EAλ := α<λ EAα for λ a limit. + Given a target theoryS T , the Π1 ordinal of T (relative to the base theory EA and notation system ≺) is defined as follows: + |T |Π1 := sup{α : EAα ⊆ T } This definition yields interesting information only if T is conservatively approxi- + + mated by iterations of consistency over EA ; that is, only if T ≡Π1 EAα where α = |T |Π1 . Knowing that T is so approximated is useful, because there are elegant equations, first discovered by Schmerl [26], spelling out conservation relations be- tween iterated reflection principles. In what follows RFNΠn is the restriction of the 5 uniform reflection principle to Πn formulas: Theorem 5.1 (Schmerl). For any n,m ∈ N, RFNα EA+ RFNωm(α) EA+ Πn+1 ( ) is Πn-conservative over Πn ( ). One can use these equations to calculate the Π1 proof-theoretic ordinals of theo- ries that are conservatively approximated by iterated consistency statements in a uniform manner. Clearly, Π1 ordinals are well-defined for theories of first-order arithmetic in which well-foundedness is not directly expressible. The definition is also sensitive to Π1 sentences. For instance, whereas the Π1 ordinal of PA is ε0, the Π1 ordinal of PA + Con(PA) is ε0 × 2. Note that the definition of Π1 ordinals achieves Turing’s goal of providing a metric of strength for theories in terms of the number of iterations of consistency required to capture their Π1 theorems. It turns out that this metric of strength is the same one provided by ordinal analysis (that is, the definition of Π1 ordinals coincides with other standard definitions of proof-theoretic ordinal in standard cases). To provide such a metric of strength, we must fix (in advance) a natural ordinal notation system. Moreover, the definition only works for theories whose Π1 fragments can be conservatively approximated by iterated consistency statements. This is apparently a feature of “natural theories” but not of all theories. This approach to ordinal analysis was a major inspiration for the approach to the main topic of this paper discussed in §8.3. Beklemishev [3, 4] has also advocated for an approach to the canonicity problem for ordinal notation systems that makes use of reflection principles. I will briefly 5This theorem is stated for all ordinal notations systems. For an appropriate treatment of ordinal notations that makes sense of this statement see [23]. 10 JAMES WALSH describe Beklemishev’s notation system for PA and its fragments. The terms are generated by the constant symbol ⊤ and the function symbols ConEA(·), 1ConEA(·), 2ConEA(·), etc. The ordering on the terms is given, roughly, by consistency strength over EA.6 This notation system is well-suited to ordinal analysis via iterated re- flection, since PA can be conservatively approximated over EA by the terms of the notation system, regarded as first-order theories. The connection with the canonicity problem is this: If these reflection principles may be regarded as a canonical means of specifying the theory PA, then the ordinal notation system (not just the ordinal) has been extracted from a canonical presen- tation of PA. Various results discussed in this paper will lend support to the notion that reflection principles are canonical, and to the notion that natural theories can be approximated by iterating reflection principles. However, the results discussed in this paper seem too coarse grained to isolate Beklemishev’s notation system as canonical in a way that certain notation systems that give “wrong” answers for ordinal analysis are not. 6. Turing degree theory The pre-well-ordering of natural theories is paralleled by a phenomenon in Turing degree theory. The Turing degrees are not linearly ordered by 6In fact, Beklemishev uses the ordering ≺ defined as follows: α ≺ β if GLP ⊢ β →♦0α, where GLP is a certain polymodal logic corresponding to provability over EA. ON THE HIERARCHY OF NATURAL THEORIES 11 from a cone. Moreover the intersection of countably many cones contains a cone. Thus, assuming AD, the function 1 if A contains a cone µ(A)= (0 if A is disjoint from a cone is a countably additive measure on the σ algebra of degree-invariant sets. This measure is called Martin Measure. In the statement of Martin’s Conjecture, almost everywhere means almost everywhere with respect to Martin Measure. Conjecture 6.1 (Martin). Assume ZF + DC + AD. Then I. If f : 2ω → 2ω is degree-invariant, and f is not increasing a.e. then f is constant a.e. II. ≤m pre-well-orders the set of degree-invariant functions that are increasing ′ ′ ′ a.e. If f has ≤m rank α, then f has ≤m rank α +1, where f (x)= f(x) for all x. One can view Martin’s Conjecture as a conjecture about the functions in L(R), since, assuming appropriate large cardinal axioms, L(R) satisfies AD. Thus, Mar- tin’s Conjecture roughly states that the only definable degree-invariant functions (up to almost everywhere equivalence) are constant functions, the identify function, and iterates of the Turing Jump. Though Martin’s Conjecture is presently open, many informative partial results and special cases are known (see [30, 27]). We remind the reader that a function f is uniformly degree-invariant if there is a recursive φ such that if A ≤T B via e then f(A) ≤T f(B) via φ(e). A function f is order-preserving if A ≤T B implies f(A) ≤T f(B). Finally, a function f is increasing if, for all A, A ≤T f(A). Theorem 6.2 (Slaman–Steel). Part I of Martin’s Conjecture holds for all uni- formly degree-invariant functions. Theorem 6.3 (Steel). Part II of Martin’s Conjecture holds for all uniformly degree- invariant functions. Theorem 6.4 (Slaman–Steel). If f is a Borel order-preserving function that is (α) increasing a.e., then there exists an α<ω1 such that f(x) ≡T x a.e. Theorem 6.5 (Lutz-Siskind). Part I of Martin’s Conjecture holds for all order- preserving functions. There are other results that speak to this phenomenon. For instance, Steel [29] has shown that there cannot be any “simple” descending sequences in the Turing jump hierarchy, in the following sense: Theorem 6.6 (Steel). Let P ⊂ R2 be arithmetic. Then there is no sequence (xn)n<ω such that for every n, ′ (i) xn ≥T xn+1 and 12 JAMES WALSH (ii) xn+1 is the unique y such that P (xn,y). We will explore the analogy between the recursion-theoretic and proof-theoretic well-ordering phenomena throughout this paper. On the one hand, we will describe proof-theoretic theorems from [22, 23, 24, 33] whose statements and proofs were in- spired by this recursion-theoretic research. On the other hand, we will also describe purely recursion-theoretic results from [19] that were inspired by the aforementioned proof-theoretic theorems. 7. Interlude My goal is to bring precision to the question of the pre-well-orderedness of natural theories and to offer (at least partial) solutions. Before summarizing this research, I would like to mention three themes that will be interwoven. The first theme is that many natural theories can be axiomatized by reflection principles over natural base theories. Indeed, the fragments of natural theories corresponding to different syntactic complexity classes can often be conservatively approximated by iterated reflection principles of the appropriate complexity class. The second theme is that ordinal analysis is an expression of the well-ordering phenomenon. For different notions of “proof-theoretic strength,” there are cor- responding notions of “proof-theoretic ordinals.” Insofar as the proof-theoretic ordinal afforded by some definition of “proof-theoretic ordinal” measures the proof- theoretic strength of theories, the attendant method of ordinal analysis well-orders the theories within its ken according to that notion of strength. The heuristic that we will try to vindicate in this paper is the following: to calculate the proof-theoretic ordinal of a theory T is to determine T ’s rank in the hierarchy of natural theories ordered by proof-theoretic strength. The third theme is that reflection principles play a role proof theory analogous to the role jumps play in recursion theory. Just as natural Turing degrees are apparently equivalent to ordinal iterates of the Turing jump, natural theories are apparently equivalent to ordinal iterates (along natural presentations of well-orderings) of re- flection principles. Just as natural Turing degrees can be obliquely studied in terms of jumps, natural axiomatic theories can be obliquely studied in terms of reflection principles. 8. Analogues of Martin’s Conjecture The well-ordering phenomenon is evident in ordinal analysis. Recall that Bek- lemishev’s method (see §5) for calculating the proof-theoretic ordinal of a theory T involves demonstrating that T can be approximated over a weak base theory by a class of formulas that are well understood. In particular, the Π1 fragments of natural theories are often proof-theoretically equivalent to iterated consistency statements over a weak base theory, whence these theories are amenable to or- dinal analysis. Why are the Π1 fragments of natural theories proof-theoretically equivalent to iterated consistency statements? Our approach to this problem is inspired by Martin’s Conjecture. In particular, we shift our focus from theories to well-behaved functions on theories. Specifically, we fix an appropriate base theory T and focus on functions on finite extensions of T ON THE HIERARCHY OF NATURAL THEORIES 13 (that is, on sentences). The goal of this approach is to show that the consistency operator and its iterates are canonical according to well-motivated but formally precise criteria. Since Martin’s Conjecture focuses on degree-invariant functions, one might be tempted to focus on functions that are extensional: Definition 8.1. A function g is extensional over a theory T if for all ϕ and ψ if T ⊢ ϕ ↔ ψ then T ⊢ g(ϕ) ↔ g(ψ). The identity function and constant functions are all extensional; so is the consis- tency operator ϕ 7→ ConT (ϕ). By analogy with Martin’s Conjecture, one might conjecture that the consistency operator and its iterates are canonical among recursive extensional functions. For instance, one might conjecture that the consistency operator is the weakest exten- sional operator that is stronger than the identity operator. In §8.1–8.2, we will describing some results that are “negative” insofar as they demonstrate that the consistency operator is not canonical in this sense. In §8.3, we will describe “positive” results that demonstrate that the consistency operator is canonical among monotone functions. Definition 8.2. A function g is monotone over a theory T if for all ϕ and ψ if T ⊢ ϕ → ψ then T ⊢ g(ϕ) → g(ψ). Note that all monotone functions are extensional but not vice-versa. Extending the analogy with Martin’s Conjecture, monotone functions are analogous to order- preserving functions. In §8.3 we will describe some of the major results demon- strating the canonicity of the consistency operator and its iterates among recursive monotone functions. We also mention some open problems. Before describing these results, let’s fix some notation. When I write that ϕ implies ψ over a theory T , I mean that T + ϕ ⊢ ψ. When I write that ϕ strictly implies ψ, I mean that ϕ implies ψ and ψ does not imply ϕ. We write [ϕ]T to denote the equivalence class of ϕ modulo T provable equivalence, i.e., [ϕ]T = {ψ : T ⊢ ϕ ↔ ψ}. In what follows by “true” I mean “true according to the standard interpretation N of first-order arithmetic.” 8.1. Slow Consistency. One can use Rosser-style self-reference to exhibit theories whose deductive strength is strictly between that of PA and PA + Con(PA). Indeed, by the second incompleteness theorem PA + ¬Con(PA) is consistent, so by Rosser’s Theorem there is some sentence ϕ independent from PA+¬Con(PA). An elementary argument then shows that PA + Con(PA) ∨ ϕ has deductive strength strictly between PA and PA + Con(PA). However, such theories are usually regarded as unnatural. In [10], S. Friedman, Rathjen, and Weiermann present what they claim is a natural theory of strictly intermediate strength between PA and PA + Con(PA). The key notion that they introduce is a variant of consistency called slow consistency, defined as follows: SlowCon(PA) := ∀x Fε0 (x) ↓→ Con(IΣx) 14 JAMES WALSH Theorem 8.3 (S. Friedman–Rathjen–Weiermann). SlowCon(PA) has strictly in- termediate deductive strength between PA and PA + Con(PA), i.e.: (1) PA + Con(PA) strictly implies SlowCon(PA); (2) PA does not prove SlowCon(PA), i.e., SlowCon(PA) strictly implies ⊤. S. Friedman, Rathjen, and Weiermann claim that this is an example of a “natural” theory with strength strictly between PA and PA + Con(PA). I am not convinced that this is so; nevertheless, there are some points that can be made in favor of their claim. In particular, note that SlowCon does not involve Rosser orderings or self-reference in any obvious way; it does not seem to be a statement about its own code or the lengths of its shortest proofs.7 This is evident in the extensionality of SlowCon. Indeed, relativizing the construction of the slow consistency state- ment yields an extensional function. Indeed, for any sentence ϕ, define the slow consistency statement for ϕ as follows: SlowCon(ϕ) := ∀x Fε0 (x) ↓→ Con(IΣx + ϕ) This function is extensional; for any ϕ and ψ, if PA ⊢ ϕ ↔ ψ then PA ⊢ SlowCon(ϕ) ↔ SlowCon(ψ). For more work on slow consistency and its variants, see [7, 8, 9, 13, 25]. 8.2. Uniform Density. Shavrukov and Visser provided another blow to the canon- ical status of the consistency operator. S. Friedman asked them, in conversation, whether “the consistency operator is, in some sense, the least way to strengthen theories.” Against this hypothesis, Shavrukov and Visser exhibited an extensional density function on the Lindenbaum algebra of PA. Let me explain. For any reasonable theory T , there is a recursive density function on the Linden- baum algebra of T that is bequeathed to us by Rosser’s version of G¨odel’s first incompleteness theorem. One can use Rosser-style self-reference to produce, for any sentences ϕ and ψ such that ϕ strictly implies ψ, a sentence θ of strictly in- termediate deductive strength. Since ψ does not imply ϕ, this means that ψ ∧ ¬ϕ is consistent. Hence, one can use Rosser-style self-reference to find a sentence θ independent from ψ ∧ ¬ϕ and then takes R(ϕ, ψ) := ϕ ∨ (ψ ∧ θ) as one’s intermediate sentence. However, even if ϕ and ψ are natural, it does not seem that R(ϕ, ψ) is natural. In particular, the definition of R(ϕ, ψ) involves self-reference in the choice of θ. One consequence of this self-reference is that R is not extensional. That is, R is sensitive to more than the mere deductive strength of its input over PA. Indeed, R is sensitive to syntactic details like the G¨odel codes of its input sentences. 7Though “explicit” self-reference is missing, the definition of SlowCon still makes use of the fixed point theorem, covertly, in the presentation of Fε0 . We define Fε0 using effective transfinite recursion, which we arithmetize after the fact. Effective transfinite recursion relies on a fixed point argument in the form of the fixed point theorem. ON THE HIERARCHY OF NATURAL THEORIES 15 Shavrukov and Visser introduced an alternative recursive density function on the Lindenbaum algebra of PA that is extensional. First, they introduced the following formula: 2 SV(ϕ) := ϕ ∧ ∀x Con(IΣx + ϕ) → Con (IΣx + ϕ) The function ϕ 7→ SV(ϕ) is extensional in the sense that whenever PA ⊢ ϕ ↔ ψ then also PA ⊢ SV(ϕ) ↔ SV(ψ). This formula engenders a recursive extensional density function on the Lindenbaum algebra of PA. Indeed, we may define: SV⋆(ϕ, ψ) := ϕ ∨ SV(¬ϕ ∧ ψ) ∧ ψ . This is a recursive extensional density function on the Lindenbaum algebra of PA: Theorem 8.4 (Shavrukov–Visser). For any ϕ and ψ such that ϕ strictly implies ψ: (1) ϕ strictly implies SV⋆(ϕ, ψ) (2) SV⋆(ϕ, ψ) strictly implies ψ. In particular, by Theorem 8.4, for any ϕ such that ϕ ∧ ConPA(ϕ) strictly implies ϕ, we have the following: ⋆ (1) ϕ ∧ ConPA(ϕ) strictly implies SV (ϕ ∧ ConPA(ϕ), ϕ); ⋆ (2) SV (ϕ ∧ ConPA(ϕ), ϕ) strictly implies ϕ. ⋆ That is, ϕ 7→ SV ϕ ∧ ConPA(ϕ), ϕ) is an extensional function that is strictly stronger than the identity operator but strictly weaker than the consistency oper- ator. Shavrukov and Visser wrote that this suggests a negative answer to S. Fried- man’s question whether the consistency operator is “the least way to strengthen theories.” It is worth mentioning before continuing that the Shvarukov–Visser results do not apply only to PA. Indeed, these results generalize to all recursively enumerable consistent theories that interpret R since the Lindenbaum algebras of such theories are all effectively isomorphic by a result of Kripke and Pour-El [17]. 8.3. Monotone operators. Shavrukov and Visser provided a recursive exten- sional density function on the Lindenbaum algebra of PA. They asked whether their result could be improved by providing such a function that is not only ex- tensional but also monotone. The answer to their question is that there is no such function. Indeed, the consistency operator enjoys canonical status among recursive monotone operators. In a sense to be made precise, it is the weakest recursive monotone operator that is stronger than the identity. Before stating the relevant theorems, let’s fix some notation. Let T be a sound (i.e., true in the standard model), recursively axiomatized extension of elementary arithmetic. We write [ϕ]T to denote the equivalence class of ϕ modulo T provable equivalence, i.e., [ϕ]T = {ψ : T ⊢ ϕ ↔ ψ}. Finally, by “true” we mean “true according to the standard interpretatin N of first-order arithmetic.” Theorem 8.5 (Montalb´an–W). Let g be recursive and monotone. Suppose that for all consistent ϕ, 16 JAMES WALSH (1) ϕ ∧ ConT (ϕ) implies g(ϕ) and (2) g(ϕ) strictly implies ϕ. Then for every true ϕ, there is a true ψ such that ψ implies ϕ and [g(ψ)]T = [ψ ∧ ConT (ψ)]T . This is to say that any recursive monotone function that is strictly stronger than the identity but weaker than the consistency operator must coincide “cofinally often” with the consistency operator on true sentences. More precisely, the functions induced by the consistency operator and g on the Lindenbaum algebra of T coincide cofinally in the ultrafilter of true sentences. As a corollary we infer the following: Corollary 8.5.1. There is no monotone function g such that for all consistent ϕ, (1) ϕ ∧ ConT (ϕ) strictly implies g(ϕ); (2) g(ϕ) strictly implies ϕ. Why is the slow consistency operator not a counter-example to this result? The reason is that if PA + ϕ ⊢ Fε0 ↓, then PA + ϕ ⊢ SlowCon(ϕ) ↔ Con(ϕ). Accordingly, the slow consistency operator and the consistency operator agree with each other on all inputs that imply the totality of Fε0 , whence they agree cofinally in the ultrafilter of true sentences. Theorem 8.5 shows that sufficiently constrained recursive monotone functions must coincide with the consistency operator cofinally in the ultrafilter of true sentences. In fact, more can be said. Indeed, in [33] it is shown that sufficiently constrained recursive monotone functions must coincide with the consistency operator in the limit within the ultrafilter of true sentences. Note that “in the limit” plays a role analogous to that which “a.e. in the sense of Martin measure” does in Martin’s Conjecture. Theorem 8.6 (W). Let g be recursive and monotone such that, for all ϕ, g(ϕ) is Π1. Then one of the following holds: (1) There is a true ψ such that, for all ϕ for which ϕ implies ψ, ϕ implies g(ϕ). (2) There is a true ψ such that, for all ϕ for which ϕ implies ψ, ϕ + g(ϕ) implies ConT (ϕ). This is roughly to say that any recursive monotone operator that must either be as weak as the identity operator in the limit or as strong as the consistency operator in the limit. Theorem 8.5 generalizes into the effective transfinite. For the rest of this section we will be interested in theories of iterated consistency that are finitely axiomatized over T (that is, equivalent to T +ϕ for some sentence ϕ). Accordingly, in this section we will work with a definition of iterated consistency sequences that guarantees this ON THE HIERARCHY OF NATURAL THEORIES 17 property. To this end, we fix an effective presentation ≺ of a recursive ordinal. Using G¨odel’s fixed point lemma, we define the predicate Con⋆ as follows: PA ⊢ Con⋆(α, T ) ↔ ∀β ≺ αCon T + pCon⋆(β,T )q α ⋆ α We use the notation Con (T ) for Con (α, T ) and we write Tα forT + Con (T ). In [22] it is shown that for any recursive monotone g, for arbitrarily strong true α β inputs, g must be either as strong as ConT or as weak as ConT for some β ≺ α. Theorem 8.7. Let g be monotone. Suppose that for all ϕ, α (1) ϕ ∧ ConT (ϕ) implies g(ϕ) β (2) If [g(ϕ)] = [⊥], then g(ϕ) strictly implies ϕ ∧ ConT (ϕ) for all β ≺ α. Then for every true ϕ, there is a true ψ such that ψ implies ϕ and [g(ψ)]T = α [ψ ∧ ConT (ψ)]T . Corollary 8.7.1. There is no monotone function g such that for all ϕ, if α [ϕ ∧ ConT (ϕ)]T 6= [⊥]T , (1) ϕ ∧ ConT (ϕ) strictly implies g(ϕ) and (2) g(ϕ) strictly implies ϕ. Ideally, one would be able to generalize Theorem 8.6 to iterates of the consistency operator in an analogous manner. However, it is not clear at present how to gen- eralize this result even beyond the first step. This leads to the following question: Question 8.8. Let g be recursive and monotone such that, for all ϕ, g(ϕ) is Π1. Must it be that one of the following holds? (1) There is a true ψ such that, for all ϕ for which ϕ implies ψ, ϕ ∧ ConT (ϕ) implies g(ϕ). (2) There is a true ψ such that, for all ϕ for which ϕ implies ψ, 2 ϕ ∧ g(ϕ) implies ConT (ϕ). Theorem 8.7 says that if the range of a monotonic function g is sufficiently con- α strained, then for some ϕ and some α,[g(ϕ)]T = [ϕ∧ConT (ϕ) 6= [⊥]T . This property still holds even when these constraints on the range of g are relaxed considerably. Indeed, the main theorem of [22] states that if the strength of any sufficiently nice function g is “bounded” by some iterate of the consistency operator, then g must somewhere coincide with an iterate of the consistency operator. Theorem 8.9 (Montalb´an–W). Let g be recursive and monotone such that, for all ϕ, g(ϕ) is Π1. Then one of the following holds: α (1) For some ϕ, ϕ ∧ ConT (ϕ) does not imply g(ϕ); β (2) For some β α and some ϕ, [ϕ ∧ g(ϕ)]T = [ϕ + ConT (ϕ)]T 6= [⊥]T . 18 JAMES WALSH In [33] some limitative results on the scope of this approach are established. In particular, it is shown that the assumption that g is recursive is necessary in the statement of Theorem 8.6. We do this by exhibiting a function g that is limit- recursive but not recursive which meets the other hypotheses of Theorem 8.6 but does not satisfy the conclusion. In particular, g vacillates between behaving like the identity operator and the consistency operator, without converging on either. Theorem 8.10 (W). There is a limit-recursive monotone function g such that, for every ϕ, g(ϕ) is Π1, yet for arbitrarily strong true sentences [ϕ ∧ g(ϕ)]T = [ϕ + ConT (ϕ)]T and for arbitrarily strong true sentences [ϕ ∧ g(ϕ)]T = [ϕ]T . 9. Reflection ranks and ordinal analysis To understand the apparent well-foundedness of natural theories by proof-theoretic strength, we now turn our attention to descending sequences of theories in proof- theoretic hierarchies. In particular, we turn to results, inspired by Steel’s Theorem 6.6, to the effect that such descending sequence cannot be simple. Indeed, these re- sults show that descending sequences in proof-theoretic hierarchies are descriptively complex or contain elements that are unsound. 9.1. Descending sequences. A few theorems concerning descending sequences in proof-theoretic hierarchies appear in [23]. The first such theorem concerns, not the ordering ≺Con on axiomatic theories, but a closely related structure. We say that a theory is Σ2-sound if all its Σ2 consequences are true. It is proved that there are no “simple” descending sequences in the Σ2-soundness hierarchy in the following sense: Theorem 9.1 (Pakhomov–W). There is no recursively enumerable sequence (Tn)n∈N of Σ2-sound extensions of BΣ1 such that, for each n, Tn proves the Σ2-soundness 8 of Tn+1. Theorem 9.1 was inspired by an alternative proof, discovered by H. Friedman, of Steel’s Theorem 6.6. The adaptability of this proof arguably strengthens the analogy between the proof-theoretic and recursion-theoretic well-foundedness phe- nomena; we revisit this idea in §10. Theorem 9.1 shows that descending sequences in a certain proof-theoretic hierarchy cannot be simple in the sense that they cannot be recursively enumerable. However, the following question remains: Question 9.2. Is there a limit-recursive sequence (Tn)n∈N of Σ2-sound extensions of BΣ1 such that, for each n, Tn proves the Σ2-soundness of Tn+1? 8In [23] this is stated as a result about EA rather than BΣ1. This is because we formalize our results in terms of “smooth provability” instead of using the ordinary provability predicate. Our proof works for smooth provability, but not ordinary provability, over EA, whereas it works for both over BΣ1. Rather than explicate the notion of smooth provability here we state our result about extensions of BΣ1. ON THE HIERARCHY OF NATURAL THEORIES 19 In fact, Theorem 9.1 is very similar to an earlier theorem of H. Friedman, Smorynski, and Solovay (see [18, 28]). Gaifman asked whether there is a recursive sequence hT0,T1,... i of consistent r.e. extensions of PA such that for all n, Tn ⊢ Con(Tn1 ). The answer is positive; however, there are none that are PA-provably descending. Theorem 9.3 (H. Friedman, Smorynski, Solovay). There is no recursively enumer- able sequence (Tn)n<ω of consistent r.e. extensions of PA such PA ⊢ ∀xPrTx Con(Tx+1) . An alternative proof of Theorem 9.3, similar in style to the proof of Theorem 9.1, is presented in [23]. Theorem 9.1 is the best possible theorem with respect to the syntactic complexity classes it concerns. To state this precisely, we recall that a theory is Σ1-sound if all its Σ1 consequences are true. The following theorem demonstrates that Theorem 9.1 is the best possible. Theorem 9.4 (Pakhomov–W). There exists a recursive sequence (Tn)n∈N of Σ1- sound extensions of elementary arithmetic such that, for each n, Tn proves the Σ1-soundness of Tn+1. The method introduced to prove Theorem 9.1 easily adapts to rule out all descend- ing sequences in another hierarchy of proof-theoretic strength, namely, the one 1 1 1 given by Π1-soundness. Recall that a theory is Π1-sound if all its Π1 consequences are true. 1 Theorem 9.5 (Pakhomov–W). There is no sequence (Tn)n∈N of Π1-sound exten- 1 sions of ACA0 such that, for each n, Tn proves the Π1-soundness of Tn+1. Theorem 9.5 rules out descending sequences of theories according to a certain metric of logical strength. This makes it possible to rank theories according to this metric of strength. We call the rank of a theory in this hierarchy its reflection rank. It is worth pausing to note that when one proves that U proves the consistency of 1 T , one often also establishes the stronger fact that U proves the Π1-soundness of T . 1 Moreover, of course, many natural theories are Π1-sound extensions of ACA0. Ac- cordingly, Theorem 9.5 delivers some insight into the phenomenon that the natural axiomatic theories are pre-well-ordered by consistency strength. 9.2. Ordinal analysis. Recall that in ordinal analysis, values called proof-theoretic ordinals are systematically assigned to theories and that, according to conventional wisdom, proof-theoretic ordinals measure the proof-theoretic strength of theories. We vindicate this conventional wisdom by proving that for most theories T , the reflection rank of T equals the proof-theoretic ordinal of T . We make the notion of + “most theories” precise in the following way (ACA0 is an axiomatic theory extending ACA0 with the axiom “every set is contained in an ω-model of ACA0”). 1 + Theorem 9.6 (Pakhomov–W). For any Π1-sound extension T of ACA0 , the re- 1 flection rank of T equals the Π1 proof-theoretic ordinal of T . 20 JAMES WALSH Theorem 9.6 is derived from a variant of Schmerl’s conservation theorem (see The- orem 5.1) in the context of second-order arithmetic. To precisely state this variant 1 0 of Schmerl’s theorem, we introduce some terminology. Let Π1(Π3) be the syntactic 0 complexity class consisting of formulas of the form ∀Xϕ(X) where ϕ(X) is Π3. 1 Though all Π1 formulas are provably equivalent to such formulas in ACA0, this is α not the case in RCA0. We define RΓ(T ) as the result of iterating Γ reflection α many times over T . Roughly, the iterates of reflection are defined as the solutions of the following equation: α β RΓ(T ) := RFNΓ T + {RΓ(T ): β ≺ α} We are now ready to state the variant of Schmerl’s formula for second-order arith- metic: Theorem 9.7 (Pakhomov–W). For any ordinal notation α, α 1 0 εα R 1 (ACA0) is Π1(Π3) conservative over R 1 0 (RCA0). Π1 Π1(Π3) 9.3. Measuring ω-model reflection via dilators. The results described in §9.1 1 demonstrate that sufficiently sound theories can be ranked according to their Π1 consequences. More robust claims are made about theories’ consequences from 1 higher complexity classes, e.g., Π2, in [24]. The main tool for explicating these 1 claims is a conservation theorem that reduces ω-model reflection to iterated Π1 reflection. Theorem 9.6 yields thorough proof-theoretic understanding of the latter in terms of ordinal analysis. Accordingly, these reductions yield proof-theoretic analyses of ω-model reflection principles. The results in this subsection are formalized in ACA0 and concern the language L2 that extends the standard language of second-order arithmetic with set-constants CX for all sets X. L2 formulas can be encoded as sets and reasoned about in ACA0. Boldface notation is used to define the standard syntactic complexity classes for L2. 1 The main syntactic reflection principle we consider, Π1-RFN(T ), informally says 1 1 “all Π1 theorems of T are true.” We will also be interested in the theories Π1- Rα(T ) that result from iterating this principle along well-orderings. Informally, one can think of them as defined inductively, according to the following equation: 1 α 1 1 β Π1-R (T ) := T + {Π1-RFN T + Π1-R (T ) : β < α} The main semantic principle we consider is an ω -model reflection principle. An ω-model is an L2 structure whose first-order part is N and whose second-order part is some subset of P(N). The semantic reflection principle we work with is “every set is contained in an ω-model of T .” The reduction principle is the following: 1 Theorem 9.8 (Pakhomov–W). ACA0 proves that for any Π2-axiomatizable T , the following are equivalent: (1) Every set is contained in an ω-model of T . 1 1 α (2) ∀α WO(α) → Π1-RFN(Π1-R (T )) . ON THE HIERARCHY OF NATURAL THEORIES 21 To systematically measure proof-theoretic strength as a function of iterated ω-model reflection, we must turn to a notion more robust than that of a proof-theoretic ordinal. The key notion here is that of the dilator of a theory, which is roughly a function encapsulating the closure conditions that a theory imposes on the ordinals. More formally, we use the following definition: Definition 9.9. The proof-theoretic dilator of a theory T is the following function ω1 ∪ {∞} → ω1 ∪ {∞}: |α| 7−→ |T + WO(α ˙ )|Π1 1 where α ranges over countable linear orders. We write |T |Π1 to denote the proof- 2 theoretic dilator of T . One can pin down how theories’ dilators climb the Veblen hierarchy as a function of the amount of ω-model reflection postulated. First we recall the basics of the Veblen hierarchy. A fixed point of a function f on the ordinals is any ordinal α such that f(α)= α. We then define the Veblen functions as follows: γ • φ0(γ)= ω . • φα+1 enumerates the fixed points of φα. • φλ enumerates the simultaneous fixed points for δ < λ, i.e., the ordinals that are fixed points of φδ for each δ < λ, for a limit λ. + Now we introduce some notation. For linear orders α,β,γ we write φα (β) to denote the standard notation system for the least ordinal strictly above β that is a value +γ of φα function. And we write φα (β) to denote the standard notation system for th α the γ ordinal above β that is a value of φα-function. We write ωR (T ) for the result of iterating the principle “every set is contained in an ω-model of...” along α starting with T . The strength of iterated ω-model reflection is characterized via dilators as follows: Theorem 9.10 (Pakhomov–W.). ACA0 proves that for any linear order α α + |ωR (ACA )|Π1 = |φ |. 0 2 1+α 10. Hyperdegrees 1 1 Theorem 9.5 demonstrates that the restriction of the Π1-reflection ordering to Π1- sound extensions of ACA0 is actually well-founded, not just well-founded for natural theories. In fact, a similar phenomenon happens on the recursion-theoretic side when we shift our attention from the Turing degrees to the hyperdegrees. For reals A and B, we say that A ≤H B if A is hyperarithmetical in B. For any real A, there A A is a canonical real O , known as the hyperjump of A, such that A Theorem 10.1 (Spector). There is no sequence (An)n∈N of reals such that, for An+1 each n, O ≤H An. 22 JAMES WALSH Whereas Spector’s proof relies on the theory of admissible ordinals, the [19] proof uses G¨odel’s second incompleteness theorem. The analogy between Theorem 9.5 and Theorem 10.1 raises the question: What is the analogue of ordinal analysis in the hyperdegrees? Indeed, Theorem 10.1 states A that the relation A ≺ B defined by O ≤H B is a well founded partial order. We call the ≺ rank of a real its Spector rank. There is a recursion-theoretically natural characterization of the Spector ranks of reals: Theorem 10.2 (Lutz–W.). For any real A, the Spector rank of A is α just in case A th ω1 is the (1 + α) admissible ordinal. Corollary 10.2.1. Assuming suitable large cardinal hypotheses, for all X on a X ♯ cone, the Spector rank of X is ω1 . In particular, for all X ≥H 0 , the Spector rank X of X is ω1 . These new proofs and results strengthen the analogy between the proof-theoretic and recursion-theoretic well-ordering phenomena. According to this analogy, cal- culating the least ordinal not recursive in a real is analogous to calculating the proof-theoretic ordinal of a theory. 11. Conclusions Why are the natural axiomatic theories pre-well-ordered by proof-theoretic strength? We have described a few results that point towards an answer, interweaving the three themes discussed in §7. The emerging picture is that natural theories are proof-theoretically equivalent to iterated reflection principles. Moreover, ordinal analysis is a means of calculating the ranks of natural theories in these hierarchies. I write “hierarchies” since there will be different hierarchies attending different no- tions of proof-theoretic strength, depending, e.g., on the reflection principle they are based on. One may have to adapt the notion of ordinal analysis to investigate these different notions of proof-theoretic strength. The claim that ordinal analysis is a means of calculating the ranks of natural theories should be refined by further de- veloping the systematic connections between iterated reflection and proof-theoretic ordinals. Nevertheless, many questions remain. I have already discussed two of these in §8.3 and §9.1. I hope that both questions will be answered in such a way that leads to broad classification theorem. Let me say a bit more about these questions. First, Question 8.8: Must any recursive monotone operator that is stronger than the consistency operator coincide with an iterate of the consistency operator in the limit? I hope that a positive answer to Question 8.8 leads to the following classification: Conjecture 11.1. Let T be a sound r.e. extension of EA. Let g be a non-constant, 9 recursive, monotone function such that, for all ϕ, g(ϕ) is Π1. Let ≺ be a nice α presentation of a well-ordering. Suppose that for every ϕ, T + ϕ + ConT (ϕ) ⊢ g(ϕ). Then for some β α and some true ϕ, for all ψ such that T + ψ ⊢ ϕ, β [ψ + g(ψ)]T = [ψ + ConT (ψ)]T . 9For a definition see [1] §2.3 Definition 1. ON THE HIERARCHY OF NATURAL THEORIES 23 Second, Question 9.2: How much must we assume to rule out descending sequences in certain reflection hierarchies? In particular, is there a limit-recursive sequence of Σ2-sound theories each of which proves the Σ2-soundness of the next? I hope that a positive answer to Question 9.2 leads to the following classification: Conjecture 11.2. For every n ∈ N, both of the following hold: (n) (1) There exists a 0 -recursive sequence (Tn)n<ω of Σn+1-sound r.e. exten- sions of BΣ1 such that, for each n, Tn proves the Σn+1-soundness of Tn+1. (n) (2) There is no 0 -recursive sequence T (Tn)n<ω of Σn+1-sound r.e. exten- sions of BΣ1 such that, for each n, Tn proves the Σn+1-soundness of Tn+1. Questions 8.8 and 9.2 are precise, but broad, thematic questions also remain. The canonicity problem—that is, the problem of precisely formulating the distinction between natural and unnatural ordinal notation systems—remains. For the reasons indicated in §3–5, progress on this canonicity problem would almost certainly yield insights into the pre-well-ordering problem. However, at present, it is not even clear what sorts of progress on the canonicity problem are possible. Finally, there are many open problems—such as Martin’s Conjecture and its kin— on the recursion-theoretic side of the analogy between the two well-ordering phe- nomena. I suspect that answers to these recursion-theoretic questions will shed light on the proof-theoretic questions as well. References [1] Lev Beklemishev. Iterated local reflection versus iterated consistency. Annals of Pure and Applied Logic, 75(1-2):25–48, 1995. [2] Lev Beklemishev. Proof-theoretic analysis by iterated reflection. Archive for Mathematical Logic, 42(6):515–552, 2003. [3] Lev Beklemishev. Provability algebras and proof-theoretic ordinals, I. Annals of Pure and Applied Logic, 128(1-3):103–123, 2004. [4] Lev Beklemishev. 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PhD thesis, UC Berkeley, 2020.