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# CRYPTOGRAPHY MATH 0209A

UCLA

GPA 3.55

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This 1 page Study Guide was uploaded by Kaylin Wehner on Friday September 4, 2015. The Study Guide belongs to MATH 0209A at University of California - Los Angeles taught by Staff in Fall. Since its upload, it has received 132 views. For similar materials see /class/177832/math-0209a-university-of-california-los-angeles in Mathematics (M) at University of California - Los Angeles.

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Date Created: 09/04/15

Qualifying Examination LOGIC Spring 2006 All questions have equal value so try to answer all of them You may and you will need to use some of the big theorems of logic the Code Completeness and Incompleteness Theorems Tarski s Theorem Kleene s Normal Form Theorem the Condensation Lemma etc and when you do make sure you quote them correctly You may also assume that Peano arithmetic PA is sound ie its theo rems are all true in its standard interpretation and that ZermeloFraenkel Set Theory with Choice ZFC is consistent Question 1 Consider the theory T of an in nite discrete linear ordering Without end points Show that there is a countable model of T into which every countable model of T can be elementarily embedded Conclude that T is complete Question 2 Give an example of a countable complete theory with ex actly three non isomorphic countable in nite models Question 3 Suppose A is an in nite re but not recursive set of natural numbers You must prove your answers to the following T or F questions lBl is the cardinality of the set B 3a True or False there is a total recursive function f m such that for every 33 Ht 6 A l t S fw12 3 3b True or False there is an unbounded total recursive function f x such that for every 2 llt 6 A t S f139 Question 4 Assume that ZFC has a well founded model ie there exists a set A and a well founded relation E on A such that the structure A E is a model of ZFC prove that ZFC has a least transitive model ie there exists a transitive set M such that M e I M is a model of ZFC and M is a subset of every other such transitive model of ZFC Question 5 For each set of ordinals A let LA Ug LglA Where L0A 0 Item X LnlAl X is de nable with parameters in LnlAl A 139 Lid116 n Lan X LnlAll and for limit A L UEltA L Al Prove that if A Q cal and X Q to is a set of nite ordinals then X e 13W gt am lt Mix 6 Law on

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