Beyond Infinity: An Expedition to the Outer Limits of the Mathematical Universe by Eugenia Cheng

Author:Eugenia Cheng [Cheng, Eugenia]
Language: eng
Format: epub
Tags: Mathematics, General
ISBN: 1781252858
Google: EPQ8DgAAQBAJ
Amazon: B01KAEKITS
Publisher: Profile Books
Published: 2017-03-09T00:00:00+00:00

In the case of infinite ordinals it is going to make a difference, so we have to make a decision about what a times b means. The usual decision in mathematics is that a × b means ‘a added to itself b times’. (This is sort of consistent with ab meaning ‘a multiplied by itself b times’.)

So for 5 × 3 we have to think about 5 people arriving in the queue, followed by another 5, and then another 5. You will hand out raffle tickets 1–15. For 3 × 5 we have to think about 3 people arriving in the queue, and then another 3, and then another 3, and another 3, and another 3. You will still be able to hand out raffle tickets 1–15.

However, now let’s think about the coachloads and tandems again. If ω people turn up, followed by another ω people, you will have to use up a whole book of, say, red tickets, followed by a whole book of blue tickets.

On the other hand, if the infinite tandems turn up, you would start by handing out tickets 1 and 2 to the first tandem, then 3 and 4, then 5 and 6, and so on. You could keep going like this and never have to break open a second book of raffle tickets.

In the first case we have done ‘ω added to itself 2 times’, which we’re calling ω × 2. The answer is two whole books of raffle tickets, which is the same as ω + ω. In the second case we have done ‘2 added to itself ω times’, which in our definition of multiplication is 2 × ω. This time the answer is just one book of raffle tickets, that is, ω.

ω × 2 = ω + ω

2 × ω = ω

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