16.pdf - CMPSCI 250 Introduction to Computation Lecture#16...

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CMPSCI 250: Introduction to Computation Lecture #16: Recursive Definition David Mix Barrington 11 October 2017
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Recursive Definition The Peano Axioms for the Naturals Pseudo-Java for the Naturals Forms of the Fifth Peano Axiom Recursion and the Fifth Axiom Defining Addition and Multiplication Other Recursive Systems
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Axioms for the Naturals Our mathematical arguments should always be subject to questioning. For any step of reasoning we can ask “Why is that true?” The ultimate answers are always definitions because there is no questioning them -- if you and I disagree about how the natural numbers are defined, then we are dealing with two different number systems rather than the same one.
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Axioms for the Naturals About 100 years ago logicians sought a definition of the natural numbers that was as simple as possible, while still allowing all the familiar properties to be proved. Giuseppe Peano’s axioms define the naturals using three undefined terms: “natural”, “zero”, and “successor”. The process of axiomatization is similar to the definition of a class in Java, where need to say what the objects in the class are (their data fields) and what can be done with them (the methods they support).
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The Five Peano Axioms Zero is a natural . Every natural has exactly one successor , which is a natural. Zero is not the successor of any natural. No two naturals have the same successor. If you start with zero, and keep taking successors, you eventually reach all of the naturals.
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Clicker Question #1 In the novel Watership Down , the rabbits have a number system where the numbers are {0, 1, 2, 3, 4, 5} and all sets with more than four elements are said to have size 5. Which of these statements about the Peano axioms is true for this system? (a) There is a number without any successor. (b) There are numbers that are not reachable from zero by taking successors. (c) There is a number whose successor is zero. (d) There are two numbers that share a successor.
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Answer #1 In the novel Watership Down , the rabbits have a number system where the numbers are {0, 1, 2, 3, 4, 5} and all sets with more than four elements are said to have size 5.
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