Mathematical Complexity
نویسنده
چکیده
Mathematicians often speak informally about how one definition of an object is preferable to other equivalent choices, about how one proof of a mathematical theorem is more transparent or constructive than alternatives, or about the “complexity” of an object guaranteed to exist by a theorem (i.e. can we actually “find it” or somehow get our hands on it). Turning these somewhat philosophical and aesthetic questions into precise mathematical questions is nontrivial but can be accomplished within the framework of mathematical logic. For a concrete example of an object with various definitions, consider the Jacobson radical of a (commutative) ring. One standard definition is that Jac(R) is the intersection of all maximal ideals of R. However, a basic result in commutative algebra says that a ∈ Jac(R) if and only if ab − 1 is a unit for all b ∈ R. The first definition, while perhaps more elegant, is typically more cumbersome if we wanted to “calculate” Jac(R). In the second definition, we have replaced quantifying over potentially complicated subsets of R with quantifying over elements of R. Since mathematical logic provides a framework to formally define certain aspects of mathematical language, it allows us to precisely compare these definitions and to ask whether there is an even “simpler” definition of Jac(R). In terms of constructive proofs and the complexity of objects, consider a result of combinatorics known as König’s Lemma. We work with the set of binary sequences of 0’s and 1’s, which we denote by {0, 1}∗. In this context, a tree is a subset of {0, 1}∗ which is closed under initial segments and an infinite branch of a tree T is an infinite sequence of 0’s and 1’s such that every initial segment is in T .
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تاریخ انتشار 2006