The Peano Axioms: An IBL Unit Constructing the Natural Numbers

Abstract

A problem sequence is presented developing the basic properties of the set of natural numbers (including associativity and commutativity of addition and multiplication, among others) from the Peano axioms, with the last portion using von Neumann’s construction to provide a model satisfying these axioms. This sequence is appropriate for mathematically inclined students in their first or second years, and has regularly required about a week of class time. The sequence could be included as one unit within any of several appropriate courses, and could be an ideal “trial size” first attempt at guiding inquiry-based learning in the tradition of R.L. Moore.

Keywords:

• Inquiry-based learning
• Peano axioms
• natural numbers

ACKNOWLEDGEMENTS

I would like to thank my students and colleagues at Coe College for their feedback and perseverance over the years. Kent Herron and Cal Van Niewaal deserve particular credit for helping to shape this material, and for going above and beyond reasonable expectations in so many different ways. Thanks also go to Mariah Birgen and her students in the spring of 2015 at Wartburg College for the first external testing of this sequence. Obviously all remaining defects are my own responsibility.

Additional information

Notes on contributors

Jonathan J. White

Jonathan J. White completed his B.A. at Coe College, his M.S. at the University of Iowa, and his Ph.D. from the University of Oklahoma (with the Research in Undergraduate Mathematics Education option). His interests include everything related to undergraduate pedagogy and everything related to games. This, along with his two kids, ensures that he is always very tired.