REAL CLOSED FIELDS AND MODELS OF FRAGMENTS OF ARITHMETIC (Joint - PowerPoint PPT Presentation

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 ))) = ⇒ ∀ x φ ( x ,� a )) : φ ∈ S } . Open denotes the set of quantifier-free formulas of L E n denotes the set of formulas consisting of a quantifier-free formula preceded by n alternating bounded quantifiers, starting with an existential quantifier. ∆ 0 abbreviates � ∞ i =0 E i . Σ n 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 PA , we denote � ∞ i =1 I Σ n .

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 ⌊ x ⌋ and write R + for its non-negative part. Remark: If F is archimedean, then the only IP of F is (isomorphic to) Z .

IOpen = PA − for every integer part of a real By our remark above, R + | 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 : ◮ There are ordered fields K without an integer part. (Boughattas) ◮ However, every ordered field K has an ultrapower which has an integer part (Boughattas) ◮ The IP s of a non-archimedean RCF K are in general not isomorphic or even elementary equivalent ◮ 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: ◮ √ 2 can be rational (Shepherdson) ◮ The set of primes can be bounded (Shepherdson) ◮ The set of primes can be unbounded and all but boundedly many primes are of the form 4 n + 1 (Marker/Macintyre) ◮ 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) ◮ ...

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 R v = { a ∈ K : v ( a ) ≥ 0 } is the valuation ring, i.e. the finite elements of K . µ v = { a ∈ K : v ( a ) > 0 } is the valuation ideal, i.e. the set of infinitesimals in K . U > 0 = { a ∈ K : v ( a ) = 0 } is the group of units in R v which is a v 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 R v in ( K , + , 0 , < ) onto a group complement B of U > 0 in ( K > 0 , · , 1 , < ). v (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 R v 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 . Now, for q = m 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 ]. Then B is a group complement of U > 0 in ( K > 0 , · ). v Hence e is an isomorphism between a complement of R v in ( K , +) and a group complement of U > 0 in ( K > 0 , · ), as desired. v 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 . = PA , and consider K := ( ff ( − M ∪ M )) rc , the real Proof Let M | 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. I.e., R (( G )) := { Σ g ∈ G c g t g | c g ∈ R ∧ ’ { g ∈ G : c g � = 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 .