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Unit 1 Mathematics
光棍节搞笑图片Part I EST Reading
Reading 1
Section A Pre-reading Task
Warm-up Questions: Work in pairs and discuss the following questions.
1. Who is Bertrand Rusll?
实心球教案>项目风险评估Bertrand Arthur William Rusll (b.1872 – d.1970) was a British philosopher, logician, essayist and social critic best known for his work in mathematical logic and analytic philosophy. His most influential contributions include his defen of logicism (the view that mathematics is in some important n reducible to logic), his refining of the predicate calculus introduced by Gottlob Frege (which still forms the basis of most contemporary logic), his defen of neutral monism (the view that the world consists of just one type of su
bstance that is neither exclusively mental nor exclusively physical), and his theories of definite descriptions and logical atomism处女座和天秤座. Rusll is generally recognized as one of the founders of modern analytic philosophy, and is regularly credited with being one of the most important logicians of the twentieth century.
2. What is Rusll’s Paradox?
Rusll discovered the paradox that bears his name in 1901, while working on his Principles of Mathematics (1903). The paradox aris in connection with the t of all ts that are not members of themlves. Such a t, if it exists, will be a member of itlf if and only if it is not a member of itlf. The paradox is significant since, using classical logic, all ntences are entailed by a contradiction. Rusll's discovery thus prompted a large amount of work in logic, t theory, and the philosophy and foundations of mathematics.
3. What effect did Rusll’s Paradox have on Gottlob Fregg’s system?
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At first Frege obrved that the conquences of Rusll’s paradox are not immediately clear. For example, “Is it always permissible to speak of the extension of a concept, of a class? And if not, how do we recognize the exceptional cas? Can we always infer from the extension of one concept’s coinciding with that of a cond, that every object which falls under the first concept also falls under the cond? Becau of the kinds of worries, Frege eventually felt forced to abandon many of his views.
4. What is Rusll’s respon to the paradox?
Rusll's own respon to the paradox came with the development of his theory of types in 1903. It was clear to Rusll that some restrictions needed to be placed upon the original comprehension (or abstraction) axiom of naive t theory, the axiom that formalizes the intuition that any coherent condition may be ud to determine a t (or class). Rusll's basic idea was that reference to ts such as the t of all ts that are not members of themlves could be avoided by arranging all ntences into a hierarchy, beginning with ntences about individuals at the lowest level, ntences about ts of in
dividuals at the next lowest level, ntences about ts of ts of individuals at the next lowest level, and so on Using a vicious circle principle similar to that adopted by the mathematician Henri Poincaré, and his own so-called "no class" theory of class, Rusll was able to explain why the unrestricted comprehension axiom fails: propositional functions, such as the function "x is a t," may not be applied to themlves since lf-application would involve a vicious circle. On Rusll's view, all objects for which a given condition (or predicate) holds must be at the same level or of the same "type."
5. Have you ever heard of Zermelo-Fraenkel t theory.? Can you give an account of it?
Contradictions like Rusll’s paradox aro from what was later called the unrestricted comprehension principle: the assumption that, for any property 私房菜p, there is a t that contains all and only tho ts that have p. In Zermelo’s system, the comprehension principle is eliminated in favour of veral much more restrictive axioms:
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a. Axiom of extensionality. If two ts have the same members, then they are identical.
b. Axiom of elementary ts. There exists a t with no members: the null, or empty, t. For any two objects a and b, there exists a t (unit t) having as its only member a, as well as a t having as its only members a and b.
