Using your Head is Permitted

May 2015 riddle

SPECIAL ANNOUNCEMENT: Next month marks Using your Head is Permitted's one hundredth riddle. To celebrate this special occasion, I am inviting readers to send riddles to appear as the site's special one hundredth riddle.

Readers are welcomed to send in multiple suggestions, but please send riddles in together with their solutions.

Looking forward to hearing your ideas!

Here's a question I heard at a fun conference. I don't know its origins.

Let S be a set of subsets of the natural numbers, satisfying that in every pair of elements in S, one is a subset of the other. (The elements of S therefore have a natural ordering according to the subset relation.)

The question is this: can S have uncountably many elements (i.e., more elements than there are natural numbers)?

As always: prove your answer.

List of solvers:

Thomas Mack (1 May 16:38)
Lorenz Reichel (1 May 22:23)
Jim Boyce (1 May 22:32)
Zhengpeng Wu (2 May 13:59)
Omer Angel (2 May 20:28)
Li Li (3 May 09:29)
Gaopeng Guo and Gaoyue Guo (5 May 00:04)
Yuping Luo (5 May 13:48)
Jan Fricke (6 May 04:44)
Lousuan Xiao (11 May 09:27)
Zilin Jiang (12 May 01:27)
Joseph DeVincentis (14 May 01:21)
Radu-Alexandru Todor (16 May 19:25)
Shunyu Yao (17 May 00:58)
Reiner Martin (23 May 08:27)
Nis Jørgensen (27 May 08:27)
Liu Yi (29 May 14:28)

Elegant and original solutions can be submitted to the puzzlemaster at Names of solvers will be posted on this page. Notify if you don't want your name to be mentioned.

The solution will be published at the end of the month.


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