Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers

2001 ◽  
Vol 66 (3) ◽  
pp. 1231-1258 ◽  
Author(s):  
Philip Ehrlich
Keyword(s):  
Set Theory ◽  
Number Fields ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Numbers ◽  
Ordered Group ◽  
Von Neumann ◽  
Ordered Field ◽  
Number Systems ◽  
Ordered Fields

Introduction. In his monograph On Numbers and Games [7], J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many other numbers including ω, ω, /2, 1/ω, and ω − π to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered “number” fields—be individually definable in terms of sets of von Neumann-Bernays-Gödel set theory with Global Choice, henceforth NBG [cf. 21, Ch. 4], it may be said to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system of real numbers bears to Archimedean ordered fields. This can be made precise by saying that whereas the ordered field of reals is (up to isomorphism) the unique homogeneous universal Archimedean ordered field, No is (up to isomorphism) the unique homogeneous universal orderedfield [14]; also see [10], [12], [13].However, in addition to its distinguished structure as an ordered field, No has a rich hierarchical structure that (implicitly) emerges from the recursive clauses in terms of which it is defined. This algebraico-tree-theoretic structure, or simplicity hierarchy, as we have called it [15], depends upon No's (implicit) structure as a lexicographically ordered binary tree and arises from the fact that the sums and products of any two members of the tree are the simplest possible elements of the tree consistent with No's structure as an ordered group and an ordered field, respectively, it being understood that x is simpler than y just in case x is a predecessor of y in the tree.

The Absolute Arithmetic Continuum and the Unification Of all Numbers Great and Small

2012 ◽  
Vol 18 (1) ◽  
pp. 1-45 ◽  
Author(s):  
Philip Ehrlich
Keyword(s):  
Nonstandard Analysis ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Numbers ◽  
Von Neumann ◽  
Ordered Field ◽  
Number Systems ◽  
Ordered Fields ◽  
Archimedean Axiom ◽  
Hierarchical Features

AbstractIn his monograph On Numbers and Games, J. H. Conway introduced a real-closed field containing the reals and the ordinals as well as a great many less familiar numbers including −ω, ω/2, 1/ω, and ω − π to name only a few. Indeed, this particular real-closed field, which Conway calls No, is so remarkably inclusive that, subject to the proviso that numbers—construed here as members of ordered fields—be individually definable in terms of sets of NBG (von Neumann–Bernays–Gödel set theory with global choice), it may be said to contain “All Numbers Great and Small.” In this respect, No bears much the same relation to ordered fields that the system ℝ of real numbers bears to Archimedean ordered fields.In Part I of the present paper, we suggest that whereas ℝ should merely be regarded as constituting an arithmetic continuum (modulo the Archimedean axiom), No may be regarded as a sort of absolute arithmetic continuum (modulo NBG), and in Part II we draw attention to the unifying framework No provides not only for the reals and the ordinals but also for an array of non-Archimedean ordered number systems that have arisen in connection with the theories of non-Archimedean ordered algebraic and geometric systems, the theory of the rate of growth of real functions and nonstandard analysis.In addition to its inclusive structure as an ordered field, the system No of surreal numbers has a rich algebraico-tree-theoretic structure—a simplicity hierarchical structure—that emerges from the recursive clauses in terms of which it is defined. In the development of No outlined in the present paper, in which the surreals emerge vis-à-vis a generalization of the von Neumann ordinal construction, the simplicity hierarchical features of No are brought to the fore and play central roles in the aforementioned unification of systems of numbers great and small and in some of the more revealing characterizations of No as an absolute continuum.

On o-minimal expansions of Archimedean ordered groups

1995 ◽  
Vol 60 (3) ◽  
pp. 817-831 ◽  
Author(s):  
Michael C. Laskowski ◽  
Charles Steinhorn
Keyword(s):  
Nonzero Element ◽  
Real Closed Field ◽  
Elementary Embedding ◽  
Closed Field ◽  
Real Numbers ◽  
Ordered Group ◽  
Ordered Groups ◽  
Differentiability Properties ◽  
Definable Function

AbstractWe study o-minimal expansions of Archimedean totally ordered groups. We first prove that any such expansion must be elementarily embeddable via a unique (provided some nonzero element is 0-definable) elementary embedding into a unique o-minimal expansion of the additive ordered group of real numbers . We then show that a definable function in an o-minimal expansion of enjoys good differentiability properties and use this to prove that an Archimedean real closed field is definable in any nonsemilinear expansion of . Combining these results, we obtain several restrictions on possible o-minimal expansions of arbitrary Archimedean ordered groups and in particular of the rational ordered group.

COMPARING TWO VERSIONS OF THE REALS

2016 ◽  
Vol 81 (3) ◽  
pp. 1115-1123
Author(s):  
G. IGUSA ◽  
J. F. KNIGHT
Keyword(s):  
Residue Field ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Numbers ◽  
Computing Power ◽  
Ordered Field ◽  
Image Position ◽  
Recursively Saturated

AbstractSchweber [10] defined a reducibility that allows us to compare the computing power of structures of arbitrary cardinality. Here we focus on the ordered field ${\cal R}$ of real numbers and a structure ${\cal W}$ that just codes the subsets of ω. In [10], it was observed that ${\cal W}$ is reducible to ${\cal R}$. We prove that ${\cal R}$ is not reducible to ${\cal W}$. As part of the proof, we show that for a countable recursively saturated real closed field ${\cal P}$ with residue field k, some copy of ${\cal P}$ does not compute a copy of k.

Relative randomness and real closed fields

2005 ◽  
Vol 70 (1) ◽  
pp. 319-330 ◽  
Author(s):  
Alexander Raichev
Keyword(s):  
Real Number ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Numbers ◽  
The Real ◽  
Real Closed Fields ◽  
Ordered Field ◽  
Closed Fields ◽  
Master's Thesis ◽  
National University

AbstractWe show that for any real number, the class of real numbers less random than it, in the sense of rK-reducibility, forms a countable real closed subfield of the real ordered field. This generalizes the well-known fact that the computable reals form a real closed field.With the same technique we show that the class of differences of computably enumerable reals (d.c.e. reals) and the class of computably approximable reals (c.a. reals) form real closed fields. The d.c.e. result was also proved nearly simultaneously and independently by Ng (Keng Meng Ng, Master's Thesis, National University of Singapore, in preparation).Lastly, we show that the class of d.c.e. reals is properly contained in the class or reals less random than Ω (the halting probability), which in turn is properly contained in the class of c.a. reals, and that neither the first nor last class is a randomness class (as captured by rK-reducibility).

The model theory of ordered differential fields

1978 ◽  
Vol 43 (1) ◽  
pp. 82-91 ◽  
Author(s):  
Michael F. Singer
Keyword(s):  
Meromorphic Functions ◽  
Analytic Solutions ◽  
Positive Element ◽  
Real Closed Field ◽  
Real Field ◽  
Closed Field ◽  
Ordered Field ◽  
Positive Elements ◽  
Odd Degree ◽  
Differential Fields

In this paper, we show that the theory of ordered differential fields has a model completion. We also show that any real differential field, finitely generated over the rational numbers, is isomorphic to some field of real meromorphic functions. In the last section of this paper, we combine these two results and discuss the problem of deciding if a system of differential equations has real analytic solutions. The author wishes to thank G. Stengle for some stimulating and helpful conversations and for drawing our attention to fields of real meromorphic functions.§ 1. Real and ordered fields. A real field is a field in which −1 is not a sum of squares. An ordered field is a field F together with a binary relation < which totally orders F and satisfies the two properties: (1) If 0 < x and 0 < y then 0 < xy. (2) If x < y then, for all z in F, x + z < y + z. An element x of an ordered field is positive if x > 0. One can see that the square of any element is positive and that the sum of positive elements is positive. Since −1 is not positive, an ordered field is a real field. Conversely, given a real field F, it is known that one can define an ordering (not necessarily uniquely) on F [2, p. 274]. An ordered field F is a real closed field if: (1) every positive element is a square, and (2) every polynomial of odd degree with coefficients in F has a root in F. For example, the real numbers form a real closed field. Every ordered field can be embedded in a real closed field. It is also known that, in a real closed field K, polynomials satisfy the intermediate value property, i.e. if f(x) ∈ K[x] and a, b ∈ K, a < b, and f(a)f(b) < 0 then there is a c in K such that f(c) = 0.

Generalization of Hölder's Theorem to Ordered Modules

1969 ◽  
Vol 21 ◽  
pp. 149-157 ◽  
Author(s):  
T. M. Viswanathan
Keyword(s):  
Sufficient Conditions ◽  
Open Interval ◽  
Order Structure ◽  
Real Numbers ◽  
Ordered Group ◽  
Quotient Field ◽  
Interval Topology ◽  
Ordered Field ◽  
Topological Completion ◽  
Necessary And Sufficient

Hölder's theorem on archimedean groups states:An ordered (abelian) group G is order isomorphic to an ordered subgroup of the ordered group R of real numbers if and only if it is archimedean.We comprehend this theorem in the following setting: G is a Z-module and Ris the completion with respect to the open interval topology of the ordered field Q; Qitself is the ordered quotient field of the ordered domain Z.Rephrasing the situation, we raise the following question: We start with a fully ordered domain A,let Kbe its ordered quotient field. We endow Kwith the open interval topology and consider , the topological completion of K. Is it possible to impose a compatible order structure on and if this can be done, when can we say that an ordered A-module Mis order isomorphic to an ordered A-submodule of ? In Theorem 3.1, we obtain a set of necessary and sufficient conditions for this isomorphism to hold.

scholarly journals Transfer principle in quantum set theory

2007 ◽  
Vol 72 (2) ◽  
pp. 625-648 ◽  
Author(s):  
Masanao Ozawa
Keyword(s):  
Quantum Mechanics ◽  
Quantum Logic ◽  
Set Theory ◽  
Von Neumann Algebra ◽  
Transfer Principle ◽  
Real Numbers ◽  
Von Neumann ◽  
Equality Relation ◽  
Adjoint Operators ◽  
Transfer Theorems

AbstractIn 1981, Takeuti introduced quantum set theory as the quantum counterpart of Boolean valued models of set theory by constructing a model of set theory based on quantum logic represented by the lattice of closed subspaces in a Hilbert space and showed that appropriate quantum counterparts of ZFC axioms hold in the model. Here, Takeuti's formulation is extended to construct a model of set theory based on the logic represented by the lattice of projections in an arbitrary von Neumann algebra. A transfer principle is established that enables us to transfer theorems of ZFC to their quantum counterparts holding in the model. The set of real numbers in the model is shown to be in one-to-one correspondence with the set of self-adjoint operators affiliated with the von Neumann algebra generated by the logic. Despite the difficulty pointed out by Takeuti that equality axioms do not generally hold in quantum set theory, it is shown that equality axioms hold for any real numbers in the model. It is also shown that any observational proposition in quantum mechanics can be represented by a corresponding statement for real numbers in the model with the truth value consistent with the standard formulation of quantum mechanics, and that the equality relation between two real numbers in the model is equivalent with the notion of perfect correlation between corresponding observables (self-adjoint operators) in quantum mechanics. The paper is concluded with some remarks on the relevance to quantum set theory of the choice of the implication connective in quantum logic.

scholarly journals Models of true arithmetic are integer parts of models of real exponentation

2021 ◽  
Vol 13 ◽  
Author(s):  
Merlin Carl ◽  
Lothar Sebastian Krapp
Keyword(s):  
Multiplicative Group ◽  
Peano Arithmetic ◽  
Real Closed Field ◽  
Closed Field ◽  
Integer Part ◽  
Real Numbers ◽  
The Real ◽  
Positive Elements ◽  
Closed Fields ◽  
Schanuel's Conjecture

Exploring further the connection between exponentiation on real closed fields and the existence of an integer part modelling strong fragments of arithmetic, we demonstrate that each model of true arithmetic is an integer part of an exponential real closed field that is elementarily equivalent to the real numbers with exponentiation and that each model of Peano arithmetic is an integer part of a real closed field that admits an isomorphism between its ordered additive and its ordered multiplicative group of positive elements. Under the assumption of Schanuel’s Conjecture, we obtain further strengthenings for the last statement.

NUMBER SYSTEMS WITH SIMPLICITY HIERARCHIES: A GENERALIZATION OF CONWAY’S THEORY OF SURREAL NUMBERS II

2018 ◽  
Vol 83 (2) ◽  
pp. 617-633
Author(s):  
PHILIP EHRLICH ◽  
ELLIOT KAPLAN
Keyword(s):  
Abelian Groups ◽  
Sufficient Conditions ◽  
Integer Part ◽  
Ordered Field ◽  
Number Systems ◽  
Ordered Fields ◽  
Image Position ◽  
Necessary And Sufficient ◽  
Class Models ◽  
Surreal Numbers

AbstractIn [16], the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway’s ordered field ${\bf{No}}$ 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 ${\bf{No}}$, i.e., a subfield of ${\bf{No}}$ that is an initial subtree of ${\bf{No}}$. In this sequel to [16], 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 ${\bf{No}}$ that are themselves initial. It is further shown that an initial subdomain of ${\bf{No}}$ is discrete if and only if it is a subdomain of ${\bf{No}}$’s canonical integer part ${\bf{Oz}}$ of omnific integers. Finally, making use of class models the results of [16] are extended by showing that the theories of nontrivial divisible ordered abelian groups and real-closed ordered fields are the sole theories of nontrivial densely ordered abelian groups and ordered fields all of whose models are isomorphic to initial subgroups and initial subfields of ${\bf{No}}$.

A note on valuation definable expansions of fields

1998 ◽  
Vol 63 (2) ◽  
pp. 739-743 ◽  
Author(s):  
Deirdre Haskell ◽  
Dugald Macpherson
Keyword(s):  
Valuation Ring ◽  
Quantifier Elimination ◽  
Real Closed Field ◽  
Closed Field ◽  
Real Closed Fields ◽  
Ordered Field ◽  
Valued Field ◽  
Closed Fields ◽  
Binary Predicate ◽  
Algebraically Closed Fields

In this note, we consider models of the theories of valued algebraically closed fields and convexly valued real closed fields, their reducts to the pure field or ordered field language respectively, and expansions of these by predicates which are definable in the valued field. We show that, in terms of definability, there is no structure properly between the pure (ordered) field and the valued field. Our results are analogous to several other definability results for reducts of algebraically closed and real closed fields; see [9], [10], [11] and [12]. Throughout this paper, definable will mean definable with parameters.Theorem A. Let ℱ = (F, +, ×, V) be a valued, algebraically closed field, where V denotes the valuation ring. Let A be a subset ofFndefinable in ℱv. Then either A is definable in ℱ = (F, +, ×) or V is definable in.Theorem B. Let ℛv = (R, <, +, ×, V) be a convexly valued real closed field, where V denotes the valuation ring. Let Abe a subset ofRndefinable in ℛv. Then either A is definable in ℛ = (R, <, +, ×) or V is definable in.The proofs of Theorems A and B are quite similar. Both ℱv and ℛv admit quantifier elimination if we adjoin a definable binary predicate Div (interpreted by Div(x, y) if and only if v(x) ≤ v(y)). This is proved in [14] (extending [13]) in the algebraically closed case, and in [4] in the real closed case. We show by direct combinatorial arguments that if the valuation is not definable then the expanded structure is strongly minimal or o-minimal respectively. Then we call on known results about strongly minimal and o-minimal fields to show that the expansion is not proper.

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