Whitehead's Problem is Undecidable
Whitehead's Problem is Undecidable
复制标题
怀特海问题是不可判定的
DOI：
10.1080/00029890.1976.11994250
复制
发表时间：
1976
影响因子：
0.5
通讯作者：
P. Eklof
中科院分区：
文献类型：
--
作者：
P. Eklof
1.导论.工作的数学家，除非他正在研究数学的基础，通常不会发现有必要明确引用集合论的公理，除非可能引用选择公理或连续统假设。只要他的论点可以在一个普遍接受的系统框架内进行，如Zermelo-Frankel集合论（ZF），他的集合论假设可以保持不表达。然而，自从哥德尔的工作以来，数学家们已经知道，在Zermelo-Frankel集合论的基础上，有一些数学陈述是不可判定的（即，既不可证明也不可反驳）。(In事实上，任何可以有效地写下来的集合论的一致公理化都会有这样的不可判定的陈述：例如，参见Monk [13; Thm 1]。）近年来，人们发现了一些不可判定陈述的具体例子。可能最著名的是连续统假设（2”o= N1）在ZFC（= ZF+选择公理）中被哥德尔和科恩证明是不可判定的。其他例子属于拓扑学和分析。(See[16]第18话：你是谁？在本文中，我们将讨论一个代数的例子：一个著名的问题，它已经抵制了最好的努力，数学家多年来，最近才证明了希拉是不可判定的ZFC。他的证明方法是证明两个公理，每一个都与ZFC一致，对这个问题产生矛盾的答案。设lA表示A上的恒等函数，Z表示整数群。阿贝尔群1 T：B~ A的满射同态称为分裂的，如果存在同态p：A~ B使得1 Tp = 1A.一个阿贝尔群A称为W-群，如果它满足以下性质：对所有满射同态1 T：B~ A，如果1 T的核同构于Z，则1 T分裂。不难看出，自由阿贝尔群是W-群。(See推论2.4）怀特海问题（Whitehead's Problem）则是要问，这个问题的匡威命题是否成立。用同调的术语来说，怀特海问题是问Ext（A，Z）= 0是否意味着A是自由的（见第3节）。这个问题也有一个等价的制定方面的拓扑群：是每一个紧凑的arcwise-connected阿贝尔群的产品的副本的循环组，R I Z？等价性由庞特里亚金对偶定理得出。
1. Introduction. The working mathematician, unless he is studying the foundations of mathematics, usually does not find it necessary to make explicit references to axioms of set theory-except perhaps to invoke the Axiom of Choice or the Continuum Hypothesis. As long as his arguments can be carried out within the framework of a commonly accepted system such as Zermelo-Frankel set theory (ZF), his set-theoretic assumptions can remain unexpressed. However, mathematicians have known since the work of Godel that there are mathematical statements that are undecidable (ie, neither provable nor refutable) on the basis of Zermelo-Frankel set theory.(In fact, any consistent axiomatization of set theory which can be effectively written down will have such undecidable statements: see, for example, Monk [13; Thm 1].) In recent years, a number of concrete examples of undecidable statements have been discovered. Probably the most famous is the Continuum Hypothesis (2" o= N1) proved undecidable in ZFC (= ZF+ Axiom of Choice) by Godel and Cohen. Other examples belong to topology and analysis.(See Rudin [16] and Shoenfield [18].) In this paper, we are going to discuss an algebraic example: a famous problem which had resisted the best efforts of mathematicians for many years before it was recently proved by Shelah to be undecidable in ZFC. His method of proof is to show that two axioms, each consistent with ZFC, yield contradictory answers to the problem.In order to state the problem, we need some definitions. Let lA denote the identity function on A and let Z denote the group of integers. A surjective homomorphism of abelian groups 1T: B~ A is said to split if there is a homomorphism p: A~ B such that 1Tp= 1A. An abelian group A is called a W-group if it satisfies the property: for all surjective homomorphisms 1T: B~ A, if the kernel of 1T is isomorphic to Z, then 1T splits. It is not hard to see that a free abelian group is a W-group.(See Corollary 2.4.) Whitehead's Problem asks whether the converse is true. In homological terms, Whitehead's Problem asks whether Ext (A, Z)= 0 implies A is free (see section 3). The problem also has an equivalent formulation in terms of topological groups: is every compact arcwise-connected abelian group a product of copies of the circle group, R I Z? The equivalence follows from the Pontryagin Duality Theorem.