Fields of surreal numbers and exponentiation
Fields of surreal numbers and exponentiation
复制标题
超现实数和幂运算领域
DOI：
--
复制
发表时间：
2001
期刊：
影响因子：
--
通讯作者：
Philip Ehrlich
中科院分区：
文献类型：
--
作者：
L. Dries;Philip Ehrlich
我们证明了具有自然指数函数的超现实数的Conway域与实数的指数域具有相同的初等性质。我们根据其输入的长度，得到了超现实数的乘积、倒数、指数和对数的长度的有序界。由此得出，长度小于给定序数的超现实数集合是所有超现实数域的子域，当且仅当该序数是ε-数。在这种情况下，该场在超幂下是偶闭的，是实指数场的初等扩展。介绍。Conway[1]引入了超现实数的有序域No，扩展了实数的域R。（见第1节对第1号的简要说明）Gonshor（[6]，第10章）根据Kruskal的建议，定义了一个指数函数exp: No→No，使得exp(x) = e，对于x∈r。在下面的第2节中，我们证明了带exp的No是实指数域的初等扩展：在实指数域为真的初等陈述在现实性数的指数域仍然为真。（参见Wilkie b[10]，以及Macintyre和Wilkie[9]，了解实指数场的基本理论。）a .麦金太尔、M. H.莫格和J. Lurie[8]也注意到了这个关于实和超现实幂的结果，并回答了[8]第8页第一作者的一个问题。下面的证明是通过将实域的限制解析函数扩展到No，并验证在[3]中对模型完备理论Tan，exp的公理被扩展后的No所满足，从而使No具有更深入的结构。本文的原始内容几乎全部在第3-5节，其中包含以下结果。设No（λ）为2000年数学学科的实数集分类：初级03C64、03C65、03H05、12J15、20F60；二级04A10, 06F。
We show that Conway’s field of surreal numbers with its natural exponential function has the same elementary properties as the exponential field of real numbers. We obtain ordinal bounds on the length of products, reciprocals, exponentials and logarithms of surreal numbers in terms of the lengths of their inputs. It follows that the set of surreal numbers of length less than a given ordinal is a subfield of the field of all surreal numbers if and only if this ordinal is an ε-number. In that case, this field is even closed under surreal exponentiation, and is an elementary extension of the real exponential field. Introduction. Conway [1] introduced the ordered field No of surreal numbers, which extends the field R of real numbers. (See Section 1 for a brief account of No.) Gonshor ([6], Ch. 10) followed suggestions by Kruskal and defined an exponential function exp : No→ No such that exp(x) = e for x ∈ R. In Section 2 below we show that No with exp is an elementary extension of the real exponential field: elementary statements true in the real exponential field remain true in the exponential field of surreal numbers. (See Wilkie [10], and Macintyre and Wilkie [9] for information on the elementary theory of the real exponential field.) This result relating real and surreal exponentiation was also noticed by A. Macintyre, by M. H. Mourgues, and by J. Lurie [8], and answers a question of the first author in [2], p. 8. The proof below consists in equipping No with even further structure, by extending the restricted analytic functions from the real field to No, and verifying that the axioms in [3] for the model-complete theory Tan,exp are satisfied by the thus expanded No. The original content of the paper lies almost entirely in Sections 3–5, which contain the following results. Let No(λ) be the set of surreals of 2000 Mathematics Subject Classification: Primary 03C64, 03C65, 03H05, 12J15, 20F60; Secondary 04A10, 06F.