相关论文: Recursive definitions on surreal numbers

The class $\mathbf{No}$ of surreal numbers, which John Conway discovered while studying combinatorial games, possesses a rich numerical structure and shares many arithmetic and algebraic properties with the real numbers. Some work has also…

The notion of surreal number was introduced by J.H. Conway in the mid 1970's: the surreal numbers constitute a linearly ordered (proper) class $No$ containing the class of all ordinal numbers ($On$) that, working within the background set…

Conway's real closed field $\mathbf{No}$ of surreal numbers is a sweeping generalization of the real numbers and the ordinals to which a number of elementary functions such as log and exponentiation have been shown to extend. The problems…

Conway's field No of surreal numbers comes both with a natural total order and an additional "simplicity relation" which is also a partial order. Considering No as a doubly ordered structure for these two orderings, an isomorphic copy of No…

In this note we study the convergence of recursively defined infinite series. We explore the role of the derivative of the defining function at the origin (if it exists), and develop a comparison test for such series which can be used even…

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…

A numeral system is an infinite sequence of different closed normal $\lambda$-terms intended to code the integers in $\lambda$-calculus. H. Barendregt has shown that if we can represent, for a numeral system, the functions : Successor,…

We define a sequence of positive integers recursively, where each term is determined as follows: starting with a given positive integer, if the term is odd, the next is the sum of its positive divisors; if the term is even, the subsequent…

We define a multiplication on the surreal numbers as higher inductive-inductive types.

If a semantically open language has no constraints on self-reference, one can prove an absurdity. The argument utilizes co-referring names 'a0' and 'a1', and the definition of a functional expression 'The reflection of x = y'. The…

We give a precise definition of a formal mathematical object as any symbol for an individual constant, predicate letter, or a function letter that can be introduced through definition into a formal mathematical language without inviting…

D-finite functions and P-recursive sequences are defined in terms of linear differential and recurrence equations with polynomial coefficients. In this paper, we introduce a class of numbers closely related to D-finite functions and…

We define a class of computable functions over real numbers using functional schemes similar to the class of primitive and partial recursive functions defined by G\"odel and Kleene. We show that this class of functions can also be…

This paper defines a new notion of bounded computable randomness for certain classes of sub-computable functions which lack a universal machine. In particular, we define such versions of randomness for primitive recursive functions and for…

How many odd numbers are there? How many even numbers? From Galileo to Cantor, the suggestion was that there are the same number of odd, even and natural numbers, because all three sets can be mapped in one-one fashion to each other. This…

The purpose of this paper is to clarify the relationship between various conditions implying essential undecidability: our main result is that there exists a theory $T$ in which all partially recursive functions are representable, yet $T$…

We note that if a sequence of real numbers converges to some limit, then the sequence of the corresponding strings in the surreal $+,-$ sign expansion representation converges, for a natural notion of string convergence, to the string…

Abstract convexity generalises classical convexity by considering the suprema of functions taken from an arbitrarily defined set of functions. These are called the abstract linear (abstract affine) functions. The purpose of this paper is to…

On Cuesta-Conway numbers as an extension of Cantor's ordinals: A short introduction to surreal numbers. The class of Cuesta-Conway numbers, the surreal numbers, can be defined simply, starting from their normal forms (families of…

Models of computation operating over the real numbers and computing a larger class of functions compared to the class of general recursive functions invariably introduce a non-finite element of infinite information encoded in an arbitrary…