ADAPTING COMPUTATIONAL DATA STRUCTURES TECHNOLOGY TO REASON ABOUT INFINITY.

The NCTM [6] curriculum states that students should be able to "compare and contrast the real number system and its various subsystems with regard to their structural characteristics." In evaluating overall conformity to the 1989 standard, NCTM [11] requires that "teachers must value and encourage the use of a variety of tools rather than placing excessive emphasis on conventional mathematical symbols." Finally, the Educational Testing Service in PRAXIS II [2] "assess(es) the subject matter knowledge necessary for a beginning teacher of secondary school mathematics" to demonstrate competence in "the concept of countability as related to infinite sets" and this test "conforms to the NCTM standards" of 1989 and 1991. In compliance with the above standards and goals, this paper presents to educators a deeper understanding of the concept of countability and as to which sets are countable and which are not. Cantor [3] introduced the diagonalization method to determine whether an infinite set of real numbers is uncountable. This paper presents a template for standardizing diagonalization type proofs by borrowing notions of applied software engineering technology to "store" infinite sets. This template is applied to a number of specific examples to illustrate the proper application of the diagonalization argument. With the abstract use of computational data structures for describing infinite sets, we present a formal analysis of the template proof to show that countable sets do not provide adequate information for the contradiction required by diagonalization proofs. This further elucidates the associated computer and mathematical theory concepts. [ABSTRACT FROM AUTHOR]

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