The Wonders of Infinity and Its Weirdness

  • Shlomo Vinner
Part of the Mathematics in Mind book series (MATHMIN)


The reason why this chapter follows the chapter about imagination acts is quite obvious. In order to follow it, the readers have to imagine a hotel whose rooms are located along an infinite half line. They have to imagine an infinite number of guests coming to stay in this weird hotel.

The chapter starts with questions like: Which set is bigger, the set of all natural numbers or the set of prime numbers? On one hand, it is obvious that the set of all natural numbers is bigger because it contains the set of prime numbers. I show that this is not the case.

Moreover, it is possible to show that the set of all rational numbers has also, in a way, the same size as the set of all natural numbers. Thus, one can get the impression that the set of all real numbers (the set of numbers that high-school students deal with, which includes the rational as well as the irrational numbers) has the same size as the set of all natural numbers. However, this impression is wrong. Showing that requires serious mathematical effort, both notational and conceptual. I believe that readers who studied mathematics in an accelerated track in high school are capable of following it. However, readers who do not want to undertake this mathematical challenge are advised to skip this chapter.


  1. Erlwanger, S. H. (1973). Benny’s conceptions of rules and answers in individually prescribed instruction (IPI) mathematics. Journal of Children Mathematics Behavior (CMB), 1(2), 7–26.Google Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Shlomo Vinner
    • 1
  1. 1.The Hebrew University of JerusalemJerusalemIsrael

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