In this paper we develop general techniques for classes of computable real numbers generated by subsets of total computable (recursive functions) with special restrictions on basic operations in order to investigate the following problems: whether a generated class is a real closed field and whether there exists a computable presentation of a generated class. We prove a series of theorems that lead to the result that there are no computable presentations neither for polynomial time computable no even for $E_n$-computable real numbers, where $E_n$ is a level in Grzegorczyk hierarchy, $n \geq 2$. We also propose a criterion of computable presentability of an archimedean ordered field.
This paper is concerned with the values of Harish-Chandra characters of a class of positive-depth, toral, very supercuspidal representations of $p$-adic symplectic and special orthogonal groups, near the identity element. We declare two representations equivalent if their characters coincide on a specific neighbourhood of the identity (which is larger than the neighbourhood on which Harish-Chandra local character expansion holds). We construct a parameter space $B$ (that depends on the group and a real number $r>0$) for the set of equivalence classes of the representations of minimal depth $r$…
We give a~detailed construction of the complete ordered field of real numbers by means of infinite decimal expansions. We prove that in the canonical encoding of decimals neither addition nor multiplication is {\em computable}, but that both operations are {\em weakly computable}; we introduce both kinds of computability in greater generality. We determine which additive and multiplicative shifts (restrictions of addition and multiplication to one variable) are computable, and prove that each of these shifts becomes computable after a~permutation of encoding. We ask if it is the case for the b…
Universality has been an important concept in computable structure theory. A class $\mathcal{C}$ of structures is universal if, informally, for any structure, of any kind, there is a structure in $\mathcal{C}$ with the same computability-theoretic properties as the given structure. Many classes such as graphs, groups, and fields are known to be universal. This paper is about the class of finitely generated groups. Because finitely generated structures are relatively simple, the class of finitely generated groups has no hope of being universal. We show that finitely generated groups are as univ…
The class of surreal numbers, denoted by $\textbf{No}$, initially proposed by Conway, is a universal ordered field in the sense that any ordered field can be embedded in it. They include in particular the real numbers and the ordinal numbers. They have strong relations with other fields such as field of transseries. Following Gonshor, surreal numbers can be seen as signs sequences of ordinal length, with some exponential and logarithmic functions that extend the usual functions over the reals. $\textbf{No}$ can actually be seen as an elegant (generalized) power series field with real coefficie…
Represented spaces form the general setting for the study of computability derived from Turing machines. As such, they are the basic entities for endeavors such as computable analysis or computable measure theory. The theory of represented spaces is well-known to exhibit a strong topological flavour. We present an abstract and very succinct introduction to the field; drawing heavily on prior work by Escardó, Schröder, and others. Central aspects of the theory are function spaces and various spaces of subsets derived from other represented spaces, and -- closely linked to these -- properties of…
In this paper, we undertake a systematic model and valuation theoretic study of the class of ordered fields which are dense in their real closure. We apply this study to determine definable henselian valuations on ordered fields, in the language of ordered rings. In light of our results, we re-examine the Shelah-Hasson Conjecture (specialised to ordered fields) and provide an example limiting its valuation theoretic conclusions.
There are several equivalent characterizations of the valuation rank of an ordered or valued field. In this paper, we extend the theory to the case of an ordered or valued {\it difference} field (that is, ordered or valued field endowed with a compatible field automorphism). We introduce the notion of {\it difference rank}. To treat simultaneously the cases of ordered and valued fields, we consider quasi-ordered fields. We characterize the difference rank as the quotient modulo the equivalence relation naturally induced by the automorphism (which encodes its growth rate). In analogy to the the…
We characterize and construct linearly ordered sets, abelian groups and fields that are {\emph symmetrically complete}, meaning that the intersection over any chain of closed bounded intervals is nonempty. Such ordered abelian groups and fields are important because generalizations of Banach's Fixed Point Theorem hold in them. We prove that symmetrically complete ordered abelian groups and fields are divisible Hahn products and real closed power series fields, respectively. We show how to extend any given ordered set, abelian group or field to one that is symmetrically complete. A main part of…
We investigate what henselian valuations on ordered fields are definable in the language of ordered rings. This leads towards a systematic study of the class of ordered fields which are dense in their real closure. Some results have connections to recent conjectures on definability of henselian valuations in strongly NIP fields. Moreover, we obtain a complete characterisation of strongly NIP almost real closed fields.
This paper is concerned with the extent to which the Skolem-Bang theorems in Diophantine approximations generalise from the standard setting of $<R,Z>$ to structures of the form $<F,I>$, where $F$ is an ordered field and $I$ is an integer part of $F$. We show that some of these theorems are hold unconditionally in general case (ordered fields with an integer part). The remainder results are based on Dirichlet's and Kronecker's theorems. Finally we extend Dirichlet's theorem to ordered fields with $IE_1$ integer part.
We investigate the computability-theoretic properties of valued fields, and in particular algebraically closed valued fields and $p$-adically closed valued fields. We give an effectiveness condition, related to Hensel's lemma, on a valued field which is necessary and sufficient to extend the valuation to any algebraic extension. We show that there is a computable formally $p$-adic field which does not embed into any computable $p$-adic closure, but we give an effectiveness condition on the divisibility relation in the value group which is sufficient to find such an embedding. By checking that …
We describe the Dedekind cuts explicitly in terms of non-standard rational numbers. This leads to another construction of a Dedekind complete totally ordered field or, equivalently, to another proof of the consistency of the axioms of the real numbers. We believe that our construction is simpler and shorter than the classical Dedekind construction and Cantor construction of such fields assuming some basic familiarity with non-standard analysis.
Ohio University（俄亥俄大学）；University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
In [26], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field $\mathbf{No}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$-vector space) to be isomorphic to an initial subfield ($K$-subspace) of $\mathbf{No}$, i.e. a subfield ($K$-subspace) of $\mathbf{No}$ that is an initial subtree of $\mathbf{No}$. In this sequel to [15], piggybacking on the just-said results, analogous results are established for ordered exponential fields, making use of a slight generalizati…
An expansion of a definably complete field either defines a discrete subring, or the image of a definable discrete set under a definable map is nowhere dense. As an application we show a definable version of Lebesgue's differentiation theorem.
It is known that the theory of any class of normed spaces over the reals that includes all spaces of a given dimension d > 1 is undecidable, and indeed, admits a relative interpretation of second-order arithmetic. The notion of a normed space makes sense over any ordered field of scalars, but such a strong undecidability result cannot hold in the more general case. Nonetheless, we find that the theory of any class of normed spaces in the more general sense that includes all spaces of a given dimension d > 1 over some ordered field admits a relative interpretation of Robinson's theory Q and hen…
The main goal of this project is to prove the equivalency of several characterizations of completeness of Archimedean ordered fields; some of which appear in most modern literature as theorems following from the Dedekind completeness of the real numbers, while a couple are not as well known and have to do with other areas of mathematics, such as nonstandard analysis. Continuing, we study the completeness of non-Archimedean fields, and provide several examples of such fields with varying degrees of properties, using nonstandard analysis to produce some relatively "nice" (in particular, they are…
We introduce a notion of compatible quasi-ordered groups which unifies valued and ordered abelian groups. It was proved in a paper by Fakhruddin that a compatible quasi-order on a field is always either an order or a valuation. We show here that the group case is more complicated than the field case and describe the general structure of a compatible quasi-ordered abelian group. We also develop a notion of quasi-order-minimality and establish a connection with C-minimality, thus answering a question of F.Delon.
Suppose that $Ω$ is a complex lattice that is closed under complex conjugation and that $I$ is a small real interval, and that $D$ is a disc in $ \mathbb{C}$. Then the restriction $\wp|_D$ is definable in the structure $(\bar{\mathbb{R}},\wp|_I)$ if and only if the lattice $Ω$ has complex multiplication. This characterises lattices with complex multiplication in terms of definability.
In [15], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field No of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field to be isomorphic to an initial subfield of No, i.e. a subfield of No that is an initial subtree of No. In this sequel to [15], analogous results for ordered abelian groups and ordered domains are established which in turn are employed to characterize the convex subgroups and convex subdomains of initial subfields of No that are themselves initial. It is further show…
In this paper, we show that VC-minimal ordered fields are real closed. We introduce a notion, strictly between convexly orderable and dp-minimal, that we call dp-small, and show that this is enough to characterize many algebraic theories. For example, dp-small ordered groups are abelian divisible and dp-small ordered fields are real closed.
Let $\mathcal K=\langle\mathcal R, δ\rangle$ be a closed ordered differential field, in the sense of M. Singer, and $C$ its field of constants. In this note, we prove that, for sets definable in the pair $\mathcal M=\langle \mathcal R, C\rangle$, the $δ$-dimension and the large dimension coincide. As an application, we characterize the definable sets in $\mathcal K$ that are internal to $C$ as those sets that are definable in $\mathcal M$ and have $δ$-dimension $0$. We further show that, for sets definable in $\mathcal K$, having $δ$-dimension $0$ does not generally imply co-analyzability in $…
The following strong form of density of definable types is introduced for theories T admitting a fibered dimension function d: given a model M of T and a definable subset X of M^n, there is a definable type p in X, definable over a code for X and of the same d-dimension as X. Both o-minimal theories and the theory of closed ordered differential fields (CODF) are shown to have this property. As an application, we derive a new proof of elimination of imaginaries for CODF.
We prove that for no nontrivial ordered abelian group G, the ordered power series field R((G)) admits an exponential, i.e. an isomorphism between its ordered additive group and its ordered multiplicative group of positive elements, but that there is a non-surjective logarithm. For an arbitrary ordered field k, no exponential on k((G)) is compatible, that is, induces an exponential on k through the residue map. This is proved by showing that certain functional equations for lexicographic powers of ordered sets are not solvable.
Let $\mathcal{R}$ be an expansion of the ordered real additive group. When $\mathcal{R}$ is o-minimal, it is known that either $\mathcal{R}$ defines an ordered field isomorphic to $(\mathbb{R},<,+,\cdot)$ on some open subinterval $I\subseteq \mathbb{R}$, or $\mathcal{R}$ is a reduct of an ordered vector space. We say $\mathcal{R}$ is field-type if it satisfies the former condition. In this paper, we prove a more general result for arbitrary expansions of $(\mathbb{R},<,+)$. In particular, we show that for expansions that do not define dense $ω$-orders (we call these type A expansions), an appr…
