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1、ReferenceAnswersUnit 1 Mathematics Part I EST Reading Reading 1 (http:/ Section APre-reading Task Warm-up Questions: Work in pairs and discuss the following questions. 1. Who is Bertrand Russell? Bertrand Arthur William Russell (b.1872 d.1970) was a British philosopher, logician, essayist and social
2、 critic best known for his work in mathematical logic and analytic philosophy. His most influential contributions include his defense of logicism (the view that mathematics is in some important sense reducible to logic), his refining of the predicate calculus introduced by Gottlob Frege (which still
3、 forms the basis of most contemporary logic), his defense of neutral monism (the view that the world consists of just one type of substance that is neither exclusively mental nor exclusively physical), and his theories of definite descriptions and logical atomism. Russell is generally recognized as
4、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 Russells Paradox? Russell discovered the paradox that bears his name in 1901, while working on his Principles of Mathematics (1903). The pa
5、radox arises in connection with the set of all sets that are not members of themselves. Such a set, if it exists, will be a member of itself if and only if it is not a member of itself. The paradox is significant since, using classical logic, all sentences are entailed by a contradiction. Russells d
6、iscovery thus prompted a large amount of work in logic, set theory, and the philosophy and foundations of mathematics. 3. What effect did Russells Paradox have on Gottlob Freggs system? At first Frege observed that the consequences of Russells paradox are not immediately clear. For example, “Is it a
7、lways permissible to speak of the extension of a concept, of a class? And if not, how do we recognize the exceptional cases? Can we always infer from the extension of one concepts coinciding with that of a second, that every object which falls under the first concept also falls under the second?Beca
8、use of these kinds of worries, Frege eventually felt forced to abandon many of his views. 4. What is Russells response to the paradox? Russells own response to the paradox came with the development of his theory of types in 1903. It was clear to Russell that some restrictions needed to be placed upo
9、n the original comprehension (or abstraction) axiom of naive set theory, the axiom that formalizes the intuition that any coherent condition may be used to determine a set (or class). Russells basic idea was that reference to sets such as the set of all sets that are not members of themselves could
10、be avoided by arranging all sentences into a hierarchy, beginning with sentences about individuals at the lowest level, sentences about sets of individuals at the next lowest level, sentences about sets of sets of individuals at the next lowest level, and so on Using a vicious circle principle simil
11、ar to that adopted by the mathematician Henri Poincar, and his own so-called “no class“ theory of classes, Russell was able to explain why the unrestricted comprehension axiom fails: propositional functions, such as the function “x is a set,“ may not be applied to themselves since self-application w
12、ould involve a vicious circle. On Russells view, all objects for which a given condition (orpredicate) holds must be at the same level or of the same “type.“ 5. Have you ever heard of Zermelo-Fraenkel set theory.? Can you give an account of it? Contradictions like Russell s paradox arose from what w
13、as later called the unrestricted comprehension principle: the assumption that, for any property p, there is a set that contains all and only those sets that have p. In Zermelo s system, the comprehension principle is eliminated in favour of several much more restrictive axioms: a.Axiom of extensiona
14、lity. If two sets have the same members, then they are identical. b. Axiom of elementary sets. There exists a set with no members: the null, or empty, set. For any two objects a and b, there exists a set (unit set) having as its only member a, as well as a set having as its only members a and b. c.
15、Axiom of separation. For any well-formed property p and any set S, there is a set, S1, containing all and only the members of S that have this property. That is, already existing sets can be partitioned or separated into parts by well-formed properties. d. Power-set axiom. If S is a set, then there
16、exists a set, S1, that contains all and only the subsets of S. e. Union axiom. If S is a set (of sets), then there is a set containing all and only the members of the sets contained in S. f.Axiom of choice. If S is a nonempty set containing sets no two of which have common members, then there exists a set that contains exactly one member from each member of S. g.Axiom of infinity. There exists at least one set tha
