How Abundant are the Sets of Numbers? Students Comparing Infinite Sets of Numbers

Authors: Virginia Montoro, Nora Scheuer, Ma. del Puy Pérez Echeverría

Abstract: We studied how students with different mathematical background
conceive infinite cardinality of number sets. We analyzed a task in which high
school and college students were requested to compare infinite number sets.
Students were classified according to their ideas on infinity. Using this classification
together with the students’ level of math education, we performed a
correspondence factorial analysis. A gradient was found in the depth of students’
ideas. At one end we found what we called horror infiniti, based students
propensity avoid infinity and instead construe it as something undefined. These
views were associated with students with less mathematical education. In an
intermediate zone, the finitist conception was placed. It was the most frequent
way of thinking among the participants of the study, with three versions: tacitly
infinitist, explicitly finitist, or taking the integers as model of inclusion. At the
other end, the infinitist conception was placed. It was present among students
with college mathematical education, according to two types: thinking of the
cardinality of number sets as a unique infinite quantity; or conceiving different
infinite cardinals. The latter was found only in advanced mathematics students.

Keywords: comparison, number sets, cardinal infinity, mathematics education.




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