Math Squad

To Infinity and Beyond. Introduction
Review of Set Theory
Functions
Countability
Cardinality
Hilbert's Grand Hotel




To Infinity and Beyond

by Maliha Hossain, proud member of the Math Squad



A Brief Look at the Origins of Set Theory and its Connection to Infinity

Let us consider two infinite sets: the set of natural numbers, {1, 2, 3, ...} and the set of whole numbers {0, 1, 2, ...}. Do they have the same number of elements? They are both infinite, but is one bigger that the other? So essentially, is one infinity bigger than the other? What about the set of all integers, {..., -2, -1, 0, 1, 2, ...}? What about the set of all real numbers?

These are some of the questions we will try to answer in this tutorial. These questions were studied by German mathematician Georg Cantor (1845-1918). Cantor is known today as the creator of set theory and his work laid the foundations for 20th century mathematics. He revolutionized mathematics by showing that the origins of all mathematical ideas can be traced from a single concept: the concept of a set. Cantor was fascinated with the infinite and devised formal mathematical rules governing its nature. He claimed, and later proved, that there are different sizes of infinity (Aleph-null, Aleph-1, etc). The key lay in understanding the cardinality of a set.

Cantor's work was opposed by one of his early mentors, Leopold Kroenecker (1823-1891), a believer of finitism, which proposes that all mathematics must be based on the natural numbers. At the time, Kroenecker was a well respected and influential figure in mathematical history and prevented much of Cantor’s work from being published in his lifetime. Cantor on the other hand, suffered a series of mental breakdowns and eventually died impoverished and alone in a sanatorium.

Cantor's ideas were supported by another German mathematician David Hilbert (1862-1943) who said, No one shall expel us from the paradise that Cantor has created for us. Hilbert created the analogy of the Hotel Infinity to illustrate Cantor's ideas of the infinite. We will take a closer look at the Hilbert Hotel in a later section but first, we start with a review of set theory.



References

  • R. Kenney. MA 301. "An Introduction to Proof through Real Analysis", Lecture Notes. Faculty of the Department of Mathematics, Purdue University, Spring 2012
  • J. Lewin, "Elements of Set Theory" in An Interactive Introduction to Mathematical Analysis, Cambridge, UK: Cambridge University Press. 2003 ch. 4, pp 50-51
  • J. J. Wanko in Vol. 102, No. 7. "Mathematics Teacher" March 2009.
  • B. Clegg in Infinity: the Quest to Think the Unthinkable, New York, Carroll and Graf 2003



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