逆数学是什么?
(2019-08-02 16:08:54)逆数学是什么?
:逆数学(Reverse mathematics)。
延伸之一。
袁萌
附件:
Reverse mathematics is a program in mathematical logic that seeks to determine which axioms are required to prove theorems of mathematics. Its defining method can briefly be described as "going backwards from the theorems to the axioms", in contrast to the ordinary mathematical practice of deriving theorems from axioms. It can be conceptualized as sculpting out necessary conditions from sufficient ones.
The reverse mathematics program was foreshadowed by results in set theory such as the classical theorem that the axiom of choice and Zorn's lemma are equivalent over ZF set theory. The goal of reverse mathematics, however, is to study possible axioms of ordinary theorems of mathematics rather than possible axioms for set theory.
Reverse mathematics is usually carried out using subsystems of second-order arithmetic, where many of its definitions and methods are inspired by previous work in constructive analysis and proof theory. The use of second-order arithmetic also allows many techniques from recursion theory to be employed; many results in reverse mathematics have corresponding results in computable analysis. Recently, higher-order reverse mathematics has been introduced, in which the focus is on subsystems of higher-order arithmetic, and the associated richer language.
The program was founded by Harvey Friedman (1975, 1976) and brought forward by Steve Simpson. A standard reference for the subject is (Simpson 2009), while an introduction for non-specialists is (Stillwell 2018). An introduction to higher-order reverse mathematics, and also the founding paper, is (Kohlenbach (2005)).
Contents
1 General principles
1.1
Use of second-order arithmetic
1.2
Use of higher-order arithmetic
2
The big five subsystems of second-order arithmetic
2.1
The base system RCA0
2.2
Weak Knig's lemma WKL0
2.3
Arithmetical comprehension ACA0
2.4
Arithmetical transfinite recursion ATR0
2.5
Π11 comprehension Π11-CA0
3
Additional systems
4ω-models and β-models
5 See also
6 References
7 External links
General principles
In reverse mathematics, one starts with a framework language and a base theory—a core axiom system—that is too weak to prove most of the theorems one might be interested in, but still powerful enough to develop the definitions necessary to state these theorems. For example, to study the theorem “Every bounded sequence of real numbers has a supremum” it is necessary to use a base system which can speak of re