REAL CLOSED FIELDS and MODELS of FRAGMENTS of ARITHMETIC (Joint Work P

REAL CLOSED FIELDS AND MODELS OF FRAGMENTS OF ARITHMETIC (Joint work P. D’Aquino and S. Kuhlmann) PA−

PA− is the ﬁrst-order theory in the language of arithmetic L := {+, ·, <, 0, 1} of the positive parts of discretely totally ordered commutative rings with 1. Induction

If S is a set of formulas in the language L of arithmetic, then IS denotes the set of formulas

PA− ∪ {∀~a((φ(0,~a) ∧ ∀x(φ(x,~a) =⇒ φ(x + 1,~a))) =⇒ ∀xφ(x,~a)) : φ ∈ S}. Open denotes the set of quantifier-free formulas of L En denotes the set of formulas consisting of a quantifier-free formula preceded by n alternating bounded quantifiers, starting with an existential quantifier. S∞ ∆0 abbreviates i=0 Ei . Σn denotes the set of formulas consisting of a ∆0-formula, preceded by n alternating unbounded quantifiers, starting with an existential quantifier. Each of these classes corresponds to a theory of arithmetic. By S∞ PA, we denote i=1 I Σn. The theory PA is interesting not only because basically all theorems of number theory can be proved within it: its theorems are (with a suitable translation) exactly those of a version of ZFC set theory with the infinity axiom is replaced by its contrary, i.e. the statement that all sets are finite. Hence PA can be taken to express ”finite mathematics”. A bit of Algebra

A totally ordered field F is real closed (RCF ) iff −1 is not a sum of squares in F , but, in every proper field extension F ( K, it is. (Equivalently, F is elementary equivalent to the reals for the language of ordered rings. Equivalently, every positive element of F has a square root in F and every polynomial in F [X ] of odd degree√ has a root in F . Equivalently, F is not algebraically closed but F [ −1] is. Equivalently, the intermediate value theorem holds in F for all polynomials in F [X ]...) If F is a real closed field, then an integer part (IP) of F is a discretely linearly ordered subring R with 1 as its smallest element such that, for every x ∈ F , there is r ∈ R with r ≤ x < r + 1. We denote this element by bxc and write R+ for its non-negative part. Remark: If F is archimedean, then the only IP of F is (isomorphic to) Z. IOpen

By our remark above, R+ |= PA− for every integer part of a real closed field. But we can say more: Theorem: (Shepherdson) Models of IOpen are exactly the integer parts of real closed fields. Furthermore, such models are easy to obtain: Theorem: (Mourgues, Ressayre): Every RCF has an integer part. Remark: I There are ordered fields K without an integer part. (Boughattas) I However, every ordered field K has an ultrapower which has an integer part (Boughattas) I The IPs of a non-archimedean RCF K are in general not isomorphic or even elementary equivalent

I In fact, they can be very diﬀerent: Each model of IOpen is the IP of some RCF - including models of e.g. true arithmetic - but it follows from the proof√ of Mourgues-Ressayre that every RCF has an IP in which 2 is rational and the set of primes is bounded Independence from IOpen

This can be used to build recursive nonstandard models of IOpen and for ’concrete’ independence proofs over IOpen. Some results include that, in models of open induction: √ I 2 can be rational (Shepherdson)

I The set of primes can be bounded (Shepherdson)

I The set of primes can be unbounded and all but boundedly many primes are of the form 4n + 1 (Marker/Macintyre)

I The set of primes can be unbounded and all but boundedly many primes p come in twins, i.e. p − 2 or p + 2 is also prime (Marker/Macintyre)

I ... What about (stronger fragments of) PA?

Shepherdson’s result indicates that the degree of induction an integer part satisfies corresponds to algebraic properties of the field. Hence it is natural to ask: Let T be a subtheory of PA. Which real closed fields have an integer part that is a model of T ? In particular: Which RCF have an IP that is a model of PA.? Let us call such an IP and IPA. The countable case

This question has recently been answered by D’Aquino, Knight and Starchenko for countable fields. Definition: A field F is recursively saturated iff, for every recursive type t in the language of fields, if every finite subtype of t is satisfied in F , then so is t. Theorem: (Knight/D’Aquino/Starchenko) A countable RCF F has an IP which is a model of PA iff it is recursively saturated. What about the uncountable case?

The canonical next step is to ask for criteria when an uncountable RCF can have an IPA. It turns out that such RCF ’s are rather special. In particular, the DKS theorem does not generalize. This was ﬁrst observed and proved by David Marker. Building on his strategy, we give strong algebraic conditions on IPA-RCF s. Some valuation theory I

Let (K, +, ·, 0, 1, <) be a non-Archimedean RCF . × For x, y ∈ K , let x ∼ y iﬀ ∃n ∈ N|x| < n|y| ∧ |y| < n|x|. With [x]∼ + [y]∼ = [xy]∼, the equivalence classes form a group G, the value group of K. The natural valuation v on K is the map v : K → G ∪ {∞}, given by × v(0) = ∞, v(x) = [x]∼ for x ∈ K . The valuation rank rank(K) of K is the linearly ordered set ({[x]∼ : x ∈ K}, <), where [x]∼ < [y]∼ iﬀ n|y| < |x| for all n ∈ N. Some valuation theory II

Let (K, +, ·, 0, 1, <) be an ordered field with natural valuation v. Then Rv = {a ∈ K : v(a) ≥ 0} is the valuation ring, i.e. the finite elements of K. µv = {a ∈ K : v(a) > 0} is the valuation ideal, i.e. the set of infinitesimals in K. >0 Uv = {a ∈ K : v(a) = 0} is the group of units in Rv which is a subgroup of (K >0, ·, 1, <). Exponentiation and Left Exponentiation

Definition: Let (K, +, ·, 0, 1, <) be a real closed field. An exponential map is an isomorphism between (K, +, 0, <) and (K >0, ·, 1, <). Definition: An ordered field K is said to have left exponentiation iff there is an isomorphism from a group complement A of Rv in >0 >0 (K, +, 0, <) onto a group complement B of Uv in (K , ·, 1, <). (It can be shown that such complements always exist and are unique up to isomorphism.) Main theorem (C./D’Aquino/Kuhlmann 2012)

Let K be an RCF with an IPA. Then K has left exponentiation.

Roughly, this says that an RCF with an IPA inherits a weak form of the exponentiation from the IP. Sketch of proof: Let M |= PA, Z = −M ∪ M, H := {qm : q ∈ Q ∧ m ∈ Z} the divisible hull of Z. Then H is a subgroup of (K, +) and a Q-vector space. Let B¯ be a basis of H containing 1. Wlog we may assume B¯ ⊂ Z. Now let B := B¯\{1} and deﬁne A := span(B). Then A is a group complement of Rv in the additive group of K. Proof sketch (continued)

The next step is to define an injective group homomorphism e :(A, +) → (K >0, ·). This is going to be the desired isomorphism. As M |= PA, there is an injective function exp : M → M\{0} such that exp(a + b) = exp(a)exp(b) for a, b ∈ M. 1 Via exp(−m) = exp(m) , we can extend this to the negative part of Z. Hence we have an exponentiation on Z. m Now, for q = n ∈ H, where m ∈ Z and n ∈ N, we define exp(q) to be the n-th root of exp(m). As K is real closed, it is in particular root closed, hence exp(q) exists for every q ∈ H. By an easy calculation, exp :(H, +) → (K >0, ·) is an injective group homomorphism. Restricting exp to A finally gives the desired e. Proof sketch (continued)

Now set B := e[A]. >0 >0 Then B is a group complement of Uv in (K , ·). Hence e is an isomorphism between a complement of Rv in (K, +) >0 >0 and a group complement of Uv in (K , ·), as desired. Thus, indeed, K has left exponentiation. QED. Remark: By a closer inspection of this proof, one sees that we can replace PA by I ∆0 + EXP in the assumption. An RCF with an IP satisfying I ∆0 + EXP hence has left exponentiation. An exponential analogue to Shepherdson’s theorem

Above, we saw that, if M |= IOpen, then M is an IP of an RCF (in fact of the real closure of its fraction field). Call a discretely ordered ring exponential if there is a total exponential function (basically a function f satisfying basic laws of exponentiation like f (0) = 1, f (n + 1) = af (n) for some a) on its positive part. We then obtain the following exponential variant of Shepherdson’s theorem: Corollary M |= PA implies that M is an exponential IP of a left exponential RCF . Proof Let M |= PA, and consider K := (ff (−M ∪ M))rc , the real closure of the fraction field of −M ∪ M. Clearly, M is exponential. As M |= IOpen, M is an IP of K. By our theorem, K has left exponentiation. Non-IPA RCF ’s

Let G be an ordered abelian group. R((G)) is the ﬁeld of generalized power series over G with real coeﬃcients. g I.e., R((G)) := {Σg∈G cg t |cg ∈ R∧’{g ∈ G : cg 6= 0} is well-ordered’}. The addition is pointwise, the ordering is lexicographic and the multiplication is given by the convolution formula. Fact: If G is also divisible, then R((G)) is an RCF . Theorem (Kuhlmann/Kuhlmann/Shelah) If G is an ordered abelian group, then R((G)) has no left exponentiation. Hence, by our theorem, there is no ordered abelian group G such that R((G)) has an IPA. DKS does not generalize

Fact: If G is an ℵ0-saturated ordered abelian group (i.e. if every ﬁnite subtype of a countable type τ is satisﬁed in G, then so is τ), then R((G)) is ℵ0-saturated as well.

In particular, if G is ℵ0-saturated, then R((G)) is recursively saturated. But as we saw above, R((G)) has no IPA. This gives counterexamples for the DKS-theorem in the uncountable case. Valuation theoretical conditions for IPAs

Furthermore, it is known that, if a field K has left exponentiation, then the rank of the value group of K is a dense linear ordering without endpoints. In particular, no field with finite or discretely ranked value group can have an IPA. This gives a large class of counterexamples, containing most RCF ’s that naturally arise. Having an IPA hence turns out to be a rather special property of an RCF . A class of non-IPA-RCF s I

Definition: Let Γ be a linearly ordered set, and let {Bγ|γ ∈ Γ} be a family or Archimedean groups. The Hahn group G = ⊕γ∈ΓBγ is then defined as the group of functions f :Γ → ∪γ∈ΓBγ with fγ ∈ Bγ for all γ ∈ Γ and finite support. Addition is pointwise, the ordering lexicographic. Definition: Let G be an ordered abelian group with rank Γ and Archimedean components {Bγ|γ ∈ Γ}, and let C be an Archimedean group. G is an exponential group in C iff Γ is <0 order-isomorphic to G and each Bγ is isomorphic (as an ordered group) to C. Fact: If K is a non-archimedean, left exponential RCF , then the value group of K is exponential in the additive group (K¯ , +, 0, <) of the residue field. Hence: Theorem(C./D’Aquino/Kuhlmann): If K is a non-Archimedean RCF with an IPA, then the value group of K is exponential in (K¯ , +, 0, <). A class of non-IPA-RCF s II

Let k ⊆ R be real closed, and let G 6= ∅ be a divisible, ordered, abelian group which is not exponential in (k, +, 0, <). Denote by k(G) the subﬁeld of k((G)) generated by k and {tg |g ∈ G}. Then no RCF K with

k(G)rc ⊆ K ⊆ k((G))

can have an IPA. Remark: Note that such G are easily obtained. Consider e.g. the Hahn group ⊕γ∈ΓBγ with Γ finite or the Bγ not pairwise isomorphic. We have seen that our theorem provides us with strong necessary conditions for the existence of an IPA. However, in contrast to the DKS-theorem, they are not sufficient. In fact, no statements on the rank can characterize RCF ’s with an IPA: Theorem(Carl/D’Aquino/Kuhlmann): Let ∆ be a dense linear ordering without endpoints. Then, there is a real closed field K with valuation rank ∆ such that K does not have an IPA. In fact, we can choose all archimedean components of the value group to be divisible and pairwise isomorphic, say to C, and the rank Γ to be a dense linear ordering without endpoints, but choose the residue field so that K¯ is no isomorphic to C. It is now natural to further investigate the relation between models of arithmetic theories and the existence of exponential functions for RCF s of which they are integer parts. Regarding this question, we have two partial results. Theorem(C.) Let M |= Th(N). Then there exist an RCF K and a function f : K → K such that M is an IP of K and (K, +, ·, f , <) is elementary equivalent to (R, +, ·, exp, <). Th(N) cannot be replaced by I ∆0. Theorem(C.) Let M |= Th(N), and let f : M → M be an exponentiation on M in the sense of arithmetic. Let K := [ff (−M ∪ M)]rc . Then there is no f ⊆ fˆ : K → K such that (K, +, ·, fˆ, <) is elementary equivalent to (R, +, ·, exp, <). We conjecture that Th(N) can be replaced by PA here. Beyond Exponentiation I

PA proves the totality of exponentiation. This is what allows to deduce the existence of left exponentiation and the corresponding consequences for the value group. But PA proves much more: For some exponential function exp, deﬁne sexpa(b) by sexp (a) sexp1(a) = a and sexpb+1(a) = a b . sexp is called a superexponential function. It is easy to see that PA proves that sexp is total, which is not true for I ∆0 + EXP. This suggests that RCF s with IPAs are in a strong sense ’larger’ than mere I ∆0 + EXP-RCF s. Beyond Exponentiation II

In fact, PA proves the totality of the ﬁrst ε0 functions of the fast-growing hierarchy. Hence even superexponentiation is an extremely weak criterion. Idea: Try to make valuation-theoretical sense of the fast-growing hierarchy.

However: In the countable case, the existence of an I ∆0 + EXP-IP suﬃces to imply the existence of an IPA (and more). It is hence not clear that the fast-growing hierarchy is relevant to our question. In fact, Jerabek and Kolodziejczyk have shown how to considerably weaken the base theory further. Weakening the base theory

One might ask how strong a theory T of arithmetic is needed to ensure left exponentiation of an RCF having a T -IP. We have seen that I ∆0 + EXP suffices. On the other hand, we get: Theorem: There is an RCF K with an I ∆0-IP without left exponentiation. On the other hand, already quite weak fragments are sufficient to enforce strong conditions on the valuation rank: Theorem: If K is an RCF with an I ∆0-IP, then the valuation rank of K contains a subset which is a dense linear ordering with out endpoints. (In particular, there is an RCF K without an I ∆0-IP.) In fact, there is n ∈ ω such that the valuation rank of every RCF with an IEn-IP has a DLOWEP as a subset. Remark: A closer inspection of the proof of the last statement suggest that n = 7 is sufficient. An interesting question is whether we can get n = 1. Remark: These results are obtained by translating results on the structure of models of certain subtheories of arithmetic into conditions on real closures using valuation theory. This method appears to be rather fruitful. Further Work

Despite the works of D’Aquino/Knight/Starchenko, our work and recent results of Marker and Steinhorn about real closures of ω1-like models of PA, the guiding problem is hence still open: Give stronger necessary or suﬃcient criteria, or even better, an algebraic characterization of IPA-RCF ’s.

I More generally, we want to understand how algebraic properties of a real closed ﬁeld correspond to arithmetical properties of its integer parts.

I All explicit constructions of IPs for RCF s known so far lead to very little arithmetic (not even IE2 can hold in these IPs). This suggest the general question how concrete such an IP can be given. We plan to study this in terms of models of inﬁnitary computations. Further further Work

For IOpen ⊆ T1, T2 ⊆ T (N), the complete theory of N, let T1 ≤RCF T2 iﬀ every RCF K with an T2-IP also has a T1-IP. ≤RCF is obviously transitive and reﬂexive. Writing T1 ≡RCF T2 for T1 ≤RCF T2 ∧ T2 ≤RCF T1, we get an equivalence relation on arithmetical theories. Clearly, if T1 is provable in T2, then T1 ≤RCF T2. We are currently studying the question whether the reverse statement (under appropriate side conditions) also holds and, more generally, properties of the ordering ≤RCF : Is PA maximal? Is there recursively axiomatizable maximal element at all? Thank you!