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Section10.1Hotel Infinity
Subsection10.1.AHotel Infinity
Activity10.1.AHotel Infinity Puzzle.
In this puzzle, we’re going to a series videos on Hotel Infinty by mathematician Tova Brown. Be sure to pause after each part to answer the questions.
Watch Hotel Infinity, Part One and pause at the end. Hotel Infinity is full one new guest arrives, the President of Space. How do you find a room for her? (Watch video for rules.)
Watch Hotel Infinity, Part Two and pause at the end. Hotel Infinity is full when a bus load of infinitely many football players shows up. How do you find rooms for them?
Watch Hotel Infinity, Part Three and pause at the end. Hotel Infinity is empty when an infinite number of busses, each with infinitely many football players show up. How do you find rooms for them?
Sets \(S\) and \(T\) are the same size set (have the same cardinality) if there exists a pairing between \(S\) and \(T\text{,}\) as defined in Definition 6.3.13. Equivalently, if there exists a function \(f:S \to T\) that is a bijection (onto and one-to-one). For example, \(\{0,1,2,3\}\) is the same size as \(\{5,6,7,8\}\) because we have the pairing \(0 \leftrightarrow 5, 1 \leftrightarrow 6, 2 \leftrightarrow 7\text{,}\)\(3 \leftrightarrow 8\) or, equivalently the function \(f:\{0,1,2,3\} \to \{5,6,7,8\}\) defined by \(f(n)=n+5\) is onto and one-to-one.
A set \(T\) is countable if it is the same size as the natural numbers \(\mathbb{N}\text{.}\) That is, if there is a bijection \(f:\mathbb{N} \to T\) that is onto and one-to-one. For example, \(T = \{n \in \mathbb{Z} \mid n \ge 5\} = \{5,6,7,8, \ldots\}\) is countable because \(f:\mathbb{N} \to T\) defined by \(f(n)=n+5\) is onto and one-to-one.
A set \(T\) is uncountable if it is not countable. For example, the interval \([0,1] = \{x \in \mathbb{R} \mid 0 \le x \le 1\}\) is uncountable, as shown in the video Hotel Infinity, Part Five.
“I think I’ll stop here,” This is how, on 23rd of June 1993, Andrew Wiles ended his series of lectures at the Isaac Newton Institute in Cambridge. The applause, so witnesses report, was thunderous: Wiles had just delivered a proof of a result that had haunted mathematicians for over 350 years: Fermat’s last theorem. Source: https://wild.maths.org/andrew-wiles-and-fermats-last-theorem
