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INDUCTION, BOUNDING,
WEAK COMBINATORIAL PRINCIPLES,
AND THE HOMOGENEOUS MODEL THEOREM
DENIS R. HIRSCHFELDT, KAREN LANGE, AND RICHARD A. SHORE
Abstract. Goncharov and Peretyat’kin independently gave nec- essary and sufficient conditions for when a set of types of a com- plete theory T is the type spectrum of some homogeneous model of T. Their result can be stated as a principle of second order arith- metic, which we call the Homogeneous Model Theorem (HMT), and analyzed from the points of view of computability theory and reverse mathematics. Previous computability theoretic results by Lange suggested a close connection between HMT and the Atomic Model Theorem (AMT), which states that every complete atomic theory has an atomic model. We show that HMT and AMT are indeed equivalent in the sense of reverse mathematics, as well as in a strong computability theoretic sense. We do the same for an analogous result of Peretyat’kin giving necessary and sufficient conditions for when a set of types is the type spectrum of some model. Along the way, we analyze a number of related principles. Some of these turn out to fall into well-known reverse mathematical classes, such as ACA 0 , IΣ^02 , and BΣ^02. Others, however, exhibit complex interactions with first order induction and bounding prin- ciples. In particular, we isolate several principles that are prov- able from IΣ^02 , are (more than) arithmetically conservative over RCA 0 , and imply IΣ^02 over BΣ^02. In an attempt to capture the combinatorics of this class of principles, we introduce the principle Π^01 GA, as well as its generalization Π^0 nGA, which is conservative over RCA 0 and equivalent to IΣ^0 n+1 over BΣ^0 n+1.
Date: March 3, 2014. 2010 Mathematics Subject Classification. Primary 03B30; Secondary 03C07, 03C15, 03C50, 03C57, 03D45, 03F30, 03F35. Hirschfeldt was partially supported by the National Science Foundation of the United States, grants DMS-0801033 and DMS-1101458. Lange was partially sup- ported by NSF Grants DMS-0802961 and DMS-1100604. Shore was partially sup- ported by NSF Grants DMS-0554855, DMS-0852811, and DMS-1161175, John Tem- pleton Foundation Grant 13408, and a short term visiting position at the University of Chicago as part of the Mathematics Department’s visitor program. 1
Contents
- Introduction 2 1.1. The thematic level 3 1.2. The specifics 8 1.3. An outline of the paper 18
- Definitions 19 2.1. Reverse mathematics 20 2.2. Model theoretic notions 22 2.3. Other notions 25
- The Atomic Model Theorem and related principles 26
- Defining homogeneity 32
- Closure conditions and model existence 47 5.1. Type spectra of homogeneous models 48 5.2. Type spectra of general models 50 5.3. Comparing closure conditions 51 5.4. Spectrum enumeration existence theorems 53 5.5. Model existence theorems 57
- Extension functions and model existence 58 6.1. Extension functions 58 6.2. The reverse mathematics of extension functions 65
- The reverse mathematics of model existence theorems 89 7.1. Extension function approximations, AMT, and ATT 90 7.2. Comparing model existence theorems 100 7.3. Computability theoretic equivalences 101
- Open questions 103 Appendix A. Approximating generics 104 Appendix B. Atomic trees 108 Appendix C. Saturated models 110 References 111
- Introduction This paper began as an investigation into the difficulty (in terms of the axioms needed) of proving certain theorems of classical model theory about the existence of models of given theories with specific properties (in particular homogeneity). We were also motivated by, and interested in illuminating, the relations of these proof theoretic analyses with ones of the computational complexity of constructing these models. In analyzing these questions we were led into several byways: proof theoretic, computational, and combinatorial. The paper
0, and 1 and an induction principle saying that every set containing 0 and closed under successor contains all the numbers (or, equivalently, that every nonempty set has a least element). It may then be aug- mented by additional comprehension axioms asserting that sets with some properties (e.g. definable by formulas in some given class) exist. One may also add axioms asserting induction principles of the form that if 0 has some property (e.g. as specified by a formula of some class) and the numbers with this property are closed under successor then every number has the specified property (or the corresponding least number principle as above). We give more precise definitions and examples in Section 2.1. There is a close connection between reverse and effective mathemat- ics and between the standard axiomatic systems of reverse mathematics and the calibration schemes from computability theory for the complex- ity of sets and functions. To make this correspondence clear (and for many other purposes), we need to specify the semantics for axiom sys- tems for second order arithmetic. A structure for this language is one of the form M = 〈M, S, +, ×, 6 , 0 , 1 , ∈〉 where M is a set (the set of the “numbers” of M) over which the first order quantifiers and vari- ables of our language range; S ⊆ 2 M^ is the collection of subsets of the “numbers” in M over which the second order quantifiers and variables of our language range; as usual, + and × are binary functions on M ; and 6 is a binary relation on M while 0 and 1 are members of M. We always interpret ∈ as the usual membership relation between elements of M and elements of S. The standard weak base system for reverse mathematics, RCA 0 , contains, in addition to the usual basic axioms for arithmetic: IΣ^01 , induction for Σ^01 formulas (i.e., ones with one (or a block of) existen- tial quantifier(s) followed by a quantifier free matrix); and ∆^01 -CA 0 , comprehension for sets determined by equivalent Σ^01 and Π^01 formulas (the latter being ones with one universal quantifier or block of univer- sal quantifiers). This system corresponds to computable (or recursive) mathematics in the sense that the models of this theory whose numbers M are just the usual natural numbers N are precisely the ones whose sets are closed under join (i.e., effective union) and Turing reducibility; i.e., if A, B ∈ M then any set computable from the pair A ⊕ B is also in M. (The structures M with M = N are called ω-models.) Thus the theorems of RCA 0 are essentially theorems of computable mathematics and the converse usually holds as well except at times when the classi- cal computability theory style proofs rely on more than Σ^01 induction. In this paper we also can often follow proofs from computable model
theory to derive their analogs in RCA 0. More frequently, though, RCA 0 does not suffice. Reverse mathematics had its beginnings in the work of Harvey Fried- man in the late ’60s and early ’70s ([11, 12, 13]). Its early development is well chronicled in Simpson’s classic text [46], which is also the source for many important ideas and results. It has four basic systems in addition to RCA 0 , and the ω-models of each correspond in a similar way to closure conditions on their sets familiar from computability the- ory. The primary story of the first few decades of reverse mathematics was that almost every standard theorem from the literature of classical mathematics that was analyzed was equivalent over RCA 0 (in the sense of reverse mathematics described above) to one of these five systems. They were thus dubbed the “big five”. Moreover, each of these systems corresponds to a well known philosophical system for mathematics as well as to a standard level of complexity in computability theory. (For details, here and elsewhere, as well as a thorough general introduction to reverse mathematics, we refer the reader to the second edition of the standard text by Simpson [47]. For a shorter introduction focused on combinatorial and model theoretic principles, see Hirschfeldt [20]. A very brief introduction can be found in Shore [44].) In addition to RCA 0 , the two next stronger of the big five are the ones most relevant to this paper: ACA 0 and WKL 0. The first extends RCA 0 by adding on Σ^0 n-CA 0 for every n ∈ N, i.e., arithmetic compre- hension: all sets defined by formulas of first order arithmetic (with set parameters) exist. It corresponds to closure under the Turing jump (i.e., the halting problem for machines with oracles in the structure). (The Turing jump and its iterates provide the most common measure of complexity in computability theory.) The second (and weaker) system corresponds to structures in which the collection of sets forms what is called a Scott set; i.e., every infinite binary tree in the collection has an infinite path in it as well. Each of these three systems has many equivalents in most areas of mathematics. Following the classical story of reverse mathematics we provide more equivalences in this paper. On the other hand, a very interesting and, in our view, important trend in reverse mathematics has emerged in the past decade or so. Many theorems of standard mathematics have been found to lie outside the scope of the big five systems. Some are stronger than all of them, some are weaker even than WKL 0 , and some are incomparable with WKL 0 but still below ACA 0. Starting with papers such as [5], [22], and [23], combinatorics and model theory have been primary sources of such examples. Earlier examples can be found in papers like [42], but these more recent papers began a trend of exploring the reverse mathematics
than Σ^01 formulas). Here, three routes are followed for different the- orems. One applies the method of Shore blocking from generalized recursion theory ([43]) to carry out, in RCA 0 , arguments that seem to need IΣ^02. (Examples include Theorem 6.10 as well as Theorem 4. of [23].) The second provides reversals to show that the theorems of interest are reverse mathematically equivalent to some induction-like principle. (Examples are Theorem 4.3 for IΣ^02 and Theorem 4.4 for BΣ^02 , the bounding principle for Σ^02 formulas, which will be defined in Section 2.1 and has been shown by Slaman [48] to be equivalent to ∆^02 -induction.) The third, and most interesting, route is one that leads to new axiomatic systems, theorems, and combinatorial principles of strength intermediate among, or incomparable with, the standard hi- erarchy of induction type axioms (which includes the usual bounding axioms). The driving examples include entries labeled 3 or 4 in Fig- ure 1 on page 67 that lead to the combinatorial principle Π^01 GA used extensively in Section 6. This principle is generalized in Appendix A to principles Π^0 nGA that fit into all levels of the induction hierarchy in a most unusual way. Each follows from the next level of induction: RCA 0 +IΣ^0 n+1 Π^0 nGA. However, Π^0 nGA is strictly weaker in two ways. First, even BΣ^02 does not follow from any Π^0 nGA. Second, for every n, we have RCA 0 Π^0 nGA + BΣ^0 n+1 ↔ IΣ^0 n+1. A foundationally challenging phenomenon arising in this paper is provided by theorems or constructions that have essentially two prov- ably different proofs. In particular, we have theorems of model theory that are provable from, for example, either WKL 0 or IΣ^02 and others from WKL 0 ∨ BΣ^02 (as described, e.g., following Proposition 4.5) or the disjunction of other pairs of comprehension and induction type ax- ioms (see, e.g., the remarks preceding Question 8.5 and items labeled 3 or 4 in Figure 1). When such a principle is not provable in RCA 0 , there may not be any canonical “best” proof or axiomatic system for it. The only earlier example of this phenomenon of which we are aware is the fact that the principle that iterations of continuous functions are continuous is equivalent to WKL 0 ∨ IΣ^02 (Friedman, Simpson, and Yu [14]). Many additional examples of this phenomenon and related ones can be found in recent work of Belanger [3]. (Some of his equivalents for WKL 0 ∨ IΣ^02 in terms of amalgamation of types are mentioned in Theorem 5.12 below.) An interesting question then is what is the im- pact of such results on the philosophical or foundational program of reverse mathematics? (A different foundational question is posed by other results by Belanger ([2, 3]) where model theoretic facts are shown to be equivalent to ACA 0 ∨ ¬WKL 0 .)
1.2. The specifics. In order to describe the actual problems analyzed in our results, we require specific formal definitions of the standard axiomatic systems being used (and some variations on them) as well as the model theoretic notions being analyzed in both the computability theoretic and reverse mathematical settings. These are provided in Section 2. Here we give a description for the reader who already has a basic familiarity with reverse mathematics and elementary model theory or who will refer to Section 2 as needed. For now we just note that all languages (and so theories) and structures are assumed to be countable. Additionally, we assume all theories to be complete and consistent. So in the setting of computable model theory, for us, the notions of computable and decidable coincide for theories. In the setting of reverse mathematics, we include the full elementary diagram in the presentation of a structure. (From the viewpoint of effective model theory then, we study only decidable structures for which the full elementary diagram is computable rather than computable structures for which only the atomic diagram is assumed to be computable.) The initial motivation for our study was the investigation by Hirsch- feldt, Shore, and Slaman of the reverse mathematical complexity of several classical model theoretic theorems in [23] and related work from the viewpoint of computability theory that both preceded and followed it, in particular the analysis of what in [23] is called AMT, the Atomic Model Theorem: every atomic theory has an atomic model. Harrington [19] and independently Goncharov and Nurtazin [16] proved an important early result on atomic models: An atomic, (com- plete) decidable theory T has a decidable atomic model if and only if there exists a uniformly computable listing of all of the principal types of T , i.e., all the types realized in the atomic model. From the relativized version of the above characterization, it easily follows that every atomic, decidable theory has a 0 ′-decidable atomic model, i.e., one whose full elementary diagram is computable in 0 ′. Csima [7] greatly improved this result by showing that such a theory always has an atomic model decidable in some low degree. Csima, Hirschfeldt, Knight, and Soare [9] studied atomic bounding degrees, where a de- gree d is atomic bounding if every atomic, decidable theory has a d- decidable atomic model. They showed that the ∆^02 atomic bounding degrees x (i.e., x (^6) T 0 ′) are exactly the nonlow 2 ones (i.e., x′′^ >T 0 ′′). (These computability theoretic papers used the word “prime” in place of “atomic”. The two notions are classically equivalent (for countable models) but differ in our context, as discussed extensively in [23] and a bit in this paper.) The question of which degrees not below 0 ′^ are atomic bounding is more complicated (see Conidis [6]) but is connected
is determined up to isomorphism by its type spectrum; see Proposi- tion 4.7 for a reverse mathematical analysis of this fact.) The first question, even classically, is then which sets of types are type spectra of homogeneous models. Consider the situation in computable model theory. Clearly, for any decidable model, there is a computable enu- meration of the types it realizes. Goncharov [15] and Peretyat’kin [41] independently showed that there are additional conditions on a com- putable listing of types X that are necessary and sufficient for there to be a decidable homogeneous model A such that X is an enumer- ation of the type spectrum of A. Classically, it is easy to see that the set of types realized in a homogeneous model must satisfy certain amalgamation properties for formulas and types. To get a decidable homogeneous model, an additional effectiveness condition is needed on the listing of types. It must have what we call a computable extension function approximation that indicates (in terms of their indices) how types and formulas can be amalgamated. (See Section 6.1 for a formal definition.) The results of Goncharov and Peretyat’kin also answer the question classically if one ignores the computability theoretic restric- tions and simply requires that the set of types be closed under the appropriate amalgamation procedures (see Theorem 5.1). In [27] and [28], Lange studied the degree theoretic analog of the second view of AMT for homogeneous models. She showed that for any computable list of types X with the appropriate amalgamation properties, there is a model A decidable in some low degree such that X is an enumeration of the type spectrum of A. She also investigated the question of which degrees d have the property that, given any X as above, there is a d-decidable homogeneous model A such that X is an enumeration of the type spectrum of A. These we might call the homogeneous bounding degrees. (Lange and Soare [29] called them the 0 -bounding degrees. Note that the papers [8, 9, 29] use the term “homogeneous bounding degrees” in the other sense mentioned above, where the class of such degrees coincides with that of the PA degrees.) Lange showed that, as for atomic models, the homogeneous bounding degrees d 6 0 ′^ are precisely the d with d′′^ > 0 ′′. Comparing these degree theoretic results (and others) with the aforementioned ones of Csima [7] and Csima, Hirschfeldt, Knight, and Soare [9] for atomic models suggested a deeper connection between model existence princi- ples for atomic and homogeneous models. However, the proofs of these results for homogeneous models and for atomic models seemed quite different. In this kind of situation, where analogies between disparate principles suggest but do not immediately provide a precise connec- tion, reverse mathematical and computability theoretic analysis can
often clarify the situation. In this paper, we show that the model ex- istence principle studied by Lange and AMT are indeed connected in a strong way, as seen from the viewpoints of reverse mathematics and computability theory. In particular, our results explain the similar- ity between Lange’s results for homogeneous models and the ones for atomic models by showing that, in a precise sense, both sets of results are dealing with the same principle, in two different guises. First, we must provide an analog to the principle AMT. Recall that the view of AMT that we are trying to emulate states that every atomic theory T has a model whose type spectrum consists of the principal types of T. In the homogeneous case, we want a principle, the Homo- geneous Model Theorem (HMT), asserting that if a list of types could be an enumeration of the type spectrum of a homogeneous model, then there in fact exists a homogeneous model whose type spectrum is enu- merated by this list. The characterization of such lists of types in the computability theoretic setting in [15] and [41] mentioned above pro- vides us with the obvious starting point. In Section 5.1, we present several variations on the conditions given in these characterizations that are classically equivalent but differ when viewed reverse mathe- matically. In Section 4, we give several definitions of homogeneity that are also classically equivalent, but differ in the setting of reverse math- ematics. We obtain three versions of the HMT principle depending on how we describe the closure conditions on the list of types and which definition of homogeneity we use. In Section 7.2, we show that these versions of HMT imply AMT and that AMT implies one of them over RCA 0 (and the other ones with sufficient additional induction assump- tions). Thus the “right” versions supply us with the theorem on the existence of homogeneous models with specified type spectra that is reverse mathematically equivalent to the atomic model theorem. Our proofs of these reverse mathematical results actually provide de- gree theoretic information that greatly strengthens the computational analogy between atomic and homogeneous models (from the second viewpoint above). Indeed, we show (Corollary 7.12) that the atomic bounding degrees are precisely what we have called the homogeneous bounding degrees. At this point, we also note that while we were originally motivated by the idea of the analogy between existence principles for atomic and homogeneous models, and the analogy proved precise for the second view of AMT, there is also a close analogy between these principles and existence principles for arbitrary models. As we mentioned above, the pure model existence problem for sets of consistent sentences is reverse mathematically (and degree theoretically) equivalent to WKL 0
simple induction arguments. These proofs are not quite simple enough, however, to work in RCA 0. There is one additional standard version (Definition 4.1(iv)), which we call strong 1 -homogeneity, that requires the existence of an automorphism taking one of the two given sequences with the same type to the other. This is “obviously” a much stronger condition. The usual proof is not computable and seems to require the jump of the model to carry it out. Not surprisingly, we can show that this level of complexity is computationally and reverse mathematically necessary. The equivalence of this characterization of homogeneity to any of the others is itself reverse mathematically equivalent to ACA 0 (Proposition 4.2). On the other hand, given this result, it is surprising to note that the existence theorem that every theory has a homoge- neous model in this strong sense remains at the level of WKL 0 (Lange [26] and Belanger [3]). The analysis of the relations among the other variations on homo- geneity (Section 4) are more interesting and unusual, as are the ones (Section 5) of the variations on the amalgamation closure conditions in the definition of the “possible” sets of types that could be type spectra of homogeneous models. These two sets of variations are corresponding ones, in that the closure conditions are the ones that obviously hold for models satisfying the corresponding notion of homogeneity. (Thus there is a match up for possible candidates for the “right theorem”.) The analyses of each set of variations and the relations between them provide reverse mathematical equivalents to induction type axioms: IΣ^02 (Theorems 4.3, 5.11, and 5.14 and the entries labeled 2 in Figure
- and BΣ^02 (Theorems 4.4, 5.15, and 5.16). Now there are many logical, computability theoretic, and combinato- rial equivalents of the standard versions of these induction type axioms at all levels of their hierarchies (IΣ^0 n and BΣ^0 n). Some can be found, for example, in the standard (and quite comprehensive) text on first order arithmetic by H´ajek and Pudl´ak [17, Section A.I.2]. Others in the setting of second order arithmetic are scattered throughout the reverse mathematical literature. Our results here, however, are about model theoretic properties. Their proofs are often given by delicate and at times complicated constructions of theories and specific models. The models constructed are designed to satisfy one variant of homogene- ity but such that, if they satisfied another, we could prove one of the standard equivalents of IΣ^02 or BΣ^02. Much more unusual and surprising are the roles that induction plays in the analysis of the construction of some of the variations of homo- geneous models. Here, we want to construct a model with a suitable given type spectrum. Of course, we are assuming that the potential
type spectrum satisfies some version of the conditions required for there to be such a model in the computability theoretic setting. (As men- tioned above, these conditions were given by both Goncharov [15] and Peretyat’kin [41] independently.) Examples of such constructions in- clude Theorems 5.1 and 6.4. In particular, we assume that there is an “extension function approximation” (Definition 6.2) that approximates the action (in terms of indices) on the given list of types that the amal- gamation relations require. Here we refer the reader especially to the entries marked 3 of Figure 1 on page 67. The ones in the column labeled “pairwise full amalgamation” are all construction principles for homogeneous or arbitrary models with given type spectra of the appropriate kinds. (The pairwise full amalga- mation closure conditions form one of our versions of the Goncharov- Peretyat’kin conditions mentioned above.) Our original proofs for them (and all the entries marked 3) showed that the constructions can be carried out assuming either IΣ^02 or the comprehension principle Π^01 G from [23] which essentially asserts the existence of generics for (i.e., meeting each one of) any given uniformly Π^01 collection of dense sets. Moreover, each of them is equivalent to IΣ^02 over BΣ^02. This presented us with a very unusual situation. Let us consider one of these examples. P: For every type sequence X satisfying the (pairwise full) amalga- mation closure conditions and having an extension function approxi- mation, there is a (1-)homogeneous model realizing exactly the types in X.
(1) P is essentially the formulation of the computable construction principle for homogeneous models that appears in the literature (see [15] and [41]). (2) P makes no mention of recursiveness, Turing reducibility, arith- metic, or formulas of any specified quantifier rank. (3) The proof of P is nontrivial. (It requires a priority argument.) (4) P is a consequence of IΣ^02. (5) P is Π^11 -conservative over IΣ^01 and so does not imply even BΣ^02 , let alone IΣ^02. (6) P + BΣ^02 is equivalent to IΣ^02. Thus P is quite unusual reverse mathematically. It is a result in the standard literature with a proof not primarily by induction. Indeed, it makes no explicit mention of induction, recursiveness, or formulas in the arithmetic hierarchy. Nonetheless, it occupies a place that is within the usual hierarchy of induction axioms but is different from all the standard ones. We then thought that perhaps we could capture the combinatorial essence of these arguments in a way that would isolate
than simply technical point of view are the ones asking for a precise determination of the reverse mathematical strength of groups of prin- ciples. In particular Question 8.6 asks whether all the entries marked 3 in Figure 1 are equivalent and, if so, whether they are all equivalent to Π^01 GA. If, indeed, all of these principles are equivalent, this would give some evidence for the robustness of Π^01 GA as discussed above. Returning to the relationship between reverse and computable model theory, we would like to point to a general theme brought out by our investigations here as well as those in [23]. The motivating issue is the phenomenon in computable model theory of what one might call partial effectivization. One begins with a classical type of model such as atomic (prime) or (strongly 1-) homogeneous and the question of how hard it is to construct such a model. Typically, the effective version assumes that the underlying theory is decidable and asks about the the possible degrees of models of the desired kind and, in particular, whether (or under what conditions) there is a decidable one. Our investigations of these questions from the viewpoint of reverse mathematics point out that the standard computability theoretic work ignores the issue of the complexity of verifying that the model con- structed is of the desired kind and just accepts the classical proof. Often, there is no difference. When, however, the definition of the kind of model is not first order but calls for the existence of morphisms, for example, then additional effectiveness considerations can be raised about the complexity of the morphism required to verify the construc- tion. A reverse mathematical analysis automatically levels the playing field for both the construction and verification. In the reverse math- ematical setting, one can show that a model constructed is prime or strongly 1-homogeneous only by showing that the morphisms required can also be constructed in the same system. It makes sense to carry over these questions to computable model theory and ask how com- plicated (in terms of Turing degree) it is to show that the morphisms exist. As an illustration, we take the two aforementioned topics investi- gated reverse mathematically in [23], [3], and this paper: the existence of prime models for every atomic theory (PMT) and the existence of strongly 1-homogeneous models for every theory. As we have mentioned, [9] shows that given a decidable atomic the- ory T , every d 6 0 ′^ with d′′^ > 0 ′′^ computes a prime model M of T. However, if we look at the verification that the M computable from d is prime (rather than atomic), we are required to produce, for every N T , an elementary embedding f : N → M. Now even if N is decidable there may be no such f computable even in d. From the
viewpoint of reverse mathematics [23] shows that PMT is equivalent to ACA 0. Thus in terms of a closure operation one needs the jump operator and the morphisms are arithmetic in the data. However, the computable model theorist should look more carefully at the question. What is then seen is that one can always construct the required f com- putably in T ′^ ⊕M⊕N (for any prime model M of T and any model N of T ). The proof of the reverse mathematical equivalence then shows that this level of complexity is necessary. Turning to strongly 1-homogeneous models, [8] shows that given a de- cidable T one can construct a 1-homogeneous model M computably in any PA degree. To verify that the model M is strongly 1-homogeneous one must construct, for every pair of tuples ¯a and ¯b from M with the same type, an automorphism f of M taking ¯a to ¯b. Now for an ar- bitrary 1-homogeneous model, it takes the jump of M to construct such an automorphism. This can be seen from the proof that reverse mathematically this general implication is equivalent to ACA 0 (Propo- sition 4.2). The computable model theorist can easily verify that M′ is also sufficient. On the other hand, as we have noted, Lange [26] and Belanger [3] have shown that the existence of strongly 1-homogeneous models for every theory is equivalent to WKL 0. Belanger’s proof ex- plicitly shows that if one constructs M still computably in a PA degree d but carefully, then one can also simultaneously construct all the re- quired automorphisms computably in d. Thus our reverse mathematical investigations suggest a general type of problem for computable model theory. When one proves that there is a decidable or d-decidable model M with some property that requires (in its definition) the existence of more than the elementary diagram of M, then one should ask how hard it is to compute the other objects (e.g. morphisms) required to verify that the definition holds of M. Other natural examples for investigation include issues dealing with categoricity for theories (as is common in model theory, rather than for structures as is common in computable model theory), universality, and saturation. Some of these questions have recently been investigated reverse mathematically by Belanger in [2] and [3]. Another general connection between reverse and computable mathe- matics (as well as degree classes more broadly) is suggested by the var- ious notions of bounding degrees and their analyses as discussed above. Given any principle expressed by a sentence of the form ∀X∃Y Φ(X, Y ) we can form a bounding operator BΦ : D → P(D) (where D is the set of Turing degrees) defined by BΦ(x) = {y | (∀Z (^6) T x)(∃W (^6) T y)Φ(Z, W )}. Examples considered in this paper include the ones gen- erated by relativizing the atomic, homogeneous, and model bounding
main narrative, we leave their discussion to Appendix A. The principle FATT mentioned above, which is also related to AMT, is discussed in Appendix B. In Section 4, we compare several classically equivalent definitions of homogeneity from the reverse mathematical point of view. We also an- alyze the strength of theorems relating homogeneous models to atomic, prime, and saturated models. One tangential result on saturated mod- els is left to Appendix C. In Section 5, we introduce the characterizations by Goncharov and Peretyat’kin of possible type spectra of (homogeneous) models. We give several classically equivalent versions of the conditions in these re- sults, and compare them reverse mathematically. We also analyze the strength of the easier direction of these results (that the type spectrum of a (homogeneous) model must satisfy these conditions). Finally, in Section 5.5, we introduce HMT and its variants, including ones such as WMT concerning the existence of general models (rather than homo- geneous ones). In Section 6.1, we discuss the effective versions of the above char- acterizations (also due to Goncharov and Peretyat’kin). In particular, we introduce the notions of extension function and extension function approximation central to these effective versions, and compare them re- verse mathematically. In Section 6.2, we study the reverse mathematics of several versions of the computability theoretic results of Goncharov and Peretyat’kin, summarizing our results in Figures 1 and 2 (see also Figures 3 and 4). As mentioned above, along with versions provable in RCA 0 and ones equivalent to IΣ^02 , we obtain several that exhibit behavior similar to that of Π^01 GA. In Section 7, we study principles asserting the existence of exten- sion function approximations under various versions of the conditions of Goncharov and Peretyat’kin, and use our results to compare ver- sions of AMT and HMT reverse mathematically and computability theoretically. We show in particular that AMT, HMT, and WMT are equivalent, both over RCA 0 and in the sense of uniform computability theoretic reducibility. Finally, in Section 8, we gather several open questions arising from our work.
- Definitions A few definitions given in this section have already been mentioned in the introduction, but we include them here as well for ease of reference.
2.1. Reverse mathematics. We assume familiarity with the basics of reverse mathematics, but briefly describe some of its commonly studied axiom systems. For a complete introduction to the field, see Simpson [47]. For a shorter introduction focused on combinatorial and model theoretic principles, see Hirschfeldt [20]. We work in the language of second order arithmetic, with lower case letters representing number variables and uppercase letters representing set variables. In particular, every set we consider is assumed to be countable. We think of first order objects, such as finite strings, as encoded by natural numbers, and of second order objects, such as trees or models, as encoded by sets of natural numbers (see [47] for more details). Let P consist of axioms stating that the natural numbers form a discrete ordered commutative semiring, together with set induction:
(0 ∈ X ∧ (∀n)[n ∈ X → n + 1 ∈ X]) → (∀n)n ∈ X.
Let IΣ^0 n be the following induction principle, expressed as an axiom scheme.
(ϕ(0) ∧ (∀n)[ϕ(n) → ϕ(n + 1)]) → (∀n)ϕ(n)
for all Σ^0 n formulas ϕ. Our base axiom system is RCA 0 , which consists of P together with IΣ^01 and the following set existence axiom scheme, which is just strong enough to prove the existence of computable sets.
(∀n)[ϕ(n) ↔ ψ(n)] → (∃X)(∀n)[n ∈ X ↔ ϕ(n)]
for all Σ^01 formulas ϕ and Π^01 formulas ψ in which X does not occur free. A useful principle provable in RCA 0 is bounded Σ^01 -comprehension (see [47]), which for our purposes we state as follows. For any finite sequence s of natural numbers and any Σ^01 property P , there is a sub- sequence of s consisting of those elements of s that satisfy P. Of course, by taking complements, bounded Π^01 -comprehension also holds in RCA 0. Another useful fact is that IΣ^02 is equivalent over RCA 0 to the finite Π^01 -recursion principle (see [22]), which states that if ϕ is a Π^01 formula defining a total function, then for each z and n, there is a sequence x 0 ,... , xn such that x 0 = z and ϕ(xi, xi+1) holds for all i < n. Another collection of axioms schemes, related to IΣ^0 n, is that of re- stricted bounding principles. The bounding principle BΣ^0 n is given by the axiom scheme
(∀m)[(∀i < m)(∃u)ϕ(i, u) → (∃v)(∀i < m)(∃u < v)ϕ(i, u)]
