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

Have fun!