This is an old one, but good fun.
Sylvy’s weekly puzzle #4
Let denote the powerset of the natural numbers. For clarity, we will consider 0 to be a natural number. Then the pair
is a partial order: the (weak) ordering is reflexive, antisymmetric, and transitive. But it is very definitely not a total order: given we may have neither nor . A chain is a subset of on which is a total order. For example:
is a chain, but note that is countable.
This week’s problem is to find an uncountable chain, that is a subset of which is totally ordered by and uncountable.
Deadline: 10am Thursday 5th of November
(you can either send your solution by email or put a hand-written solution into the folder on my office door)
Prize: I’m very please to anounce that the winner will get a book of mathematical puzzles!!
Solution class: there will be a class at 10am on Thursday 5th of November in my office to go over the solution to this puzzle.
The webpage for these puzzles is http://anscombe.sdf.org/puzzle.html