Home Courses Theory of Computation Course materials Lectures 3.A: Infinity, hotel Hilbert, and countable and uncountable sets

3.A: Infinity, hotel Hilbert, and countable and uncountable sets

Course week(s) Week 3
Course subject(s) 3: Decidabillity

Description: Hotel Hilbert is a story of infinity and how to calculate with countable, infinite sets. Cantor’s proof of the uncountability of the set of real numbers is presented using the famous technique of diagonalisation. Other examples of uncountable sets are given. The existence of languages that are not Turing recognisable is demonstrated.

Book: Introduction to the Theory of Computability, Chapter 4, pp. 176-180.

Exercises: 4.5, 4.6, 4.7 and 4.8

Key concepts:

  • Diagonalization
  • Hilbert’s Hotel
  • Two Degrees of Infinity
  • Countable and Uncountable Sets
  • Uncountability of the Reals
Module 3A
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