Murder on the Einstein Express and Other Stories by Harun Šiljak

Author:Harun Šiljak
Language: eng
Format: epub, pdf, mobi
Publisher: Springer International Publishing, Cham

Lecture One: Infinity

Half a dozen students showed up for the first lecture. I had no intention of explaining the idea of the course – I just waved the papers, same as I did with Volodya and said: “This is the mandatory reading material for the course… and this is the additional” pointing at two thick volumes of Encyclopedia Poldevica.

“During this course, we will read and analyse a cheap detective story called Murder on the Einstein Express. Criticism of the author’s literary style is strictly forbidden.” The last sentence was followed by a smile on my behalf, not to be taken seriously (criticism is probably frowned upon elsewhere, but at our little institution, it’s welcome).

I started reading the story out loud.

“Two new guests were seen that morning at the reception desk of Hilbert Hotel Grand. All rooms were occupied, as usual, so the receptionist had to repeat his tiresome procedure once again: move the guest from room no. 1 to room no. 2, the guest from room no. 2 to room no. 3 and so on to free two rooms for the new guests. “You know, we’ve got a special crowd in these days”, he apologised. The Yearly Congress of Monkey Typists was in town – after typing all Shakespeare’s works the last time, this time they got the task to write the whole Borges’ Library of Babel. And of course, all monkeys were staying in Hilbert Hotel Grand.”

I couldn’t stand the puzzled looks, so I started explaining right away:

“I know I promised a physics course, but we have to start with mathematics. How do you imagine an infinite set?”

“Set with a number of elements larger than any imaginable number?” one of the students said, and I nodded. “How big is the set of positive integers, 1, 2, 3, …?”

“Infinite!” they all said.

“And how about the set of all positive even numbers? 2, 4, 6,…?”

“Also infinite!”

“Which one’s bigger?”

“All positive integers.”

“At first, it seems your logic makes sense. Quick mental calculation says that there’s the same number of even and odd numbers, and the positive integers are comprised of those two. So, even numbers are a twice smaller set than the set of integers. However, let’s take a different perspective: what if we take a function from the set of positive integers to the set of even integers defined by f(n) = 2n. Is that a bijection?”

One of the students, Leonid Yuryevich, confidently replied: “A map is bijective if every element from one set is related with exactly one element in the other set and vice versa. It is obvious in this case that it holds – as the “originals” in the mapping we use the integers, and as “images” the even integers, where no original has two images, nor any image has two originals.”

“Correct, Lyonya! A map is bijective if it is injective, i.e. there are no two originals sharing the same image and if it is surjective, i.e. there is no image without an original. Do the two sets between which we

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.