In this post I want to discuss the solution to Puzzle #4. Let’s recall the puzzle: it was to find an uncountable chain in .
Here is my next puzzle! Sorry that it’s been greatly delayed.
I mentioned previously the problem of constructing the midpoint of a given circle, using only ruler and compass. One student solved it nearly immediately (congratulations!), so here I offer a more general problem.
Sylvy’s puzzle #5
Given a parabola in the plane, construct its vertex using straightedge-and-compass construction (sometimes called `ruler-and-compass’ construction).
It seems that my first-year students (who form a large part of the target audience for my puzzles) are unfamiliar with the `countable’ and `uncountable’ terminology that I used in puzzle #4. …oops!
Instead of introducing it now, I’ll give a completely different puzzle which was originally shown to me by a friend (more on this later).
Sylvy’s weekly puzzle #4a
Show by means of a `geometrical drawing’ that
deadline: 11am Monday 9th of November
how to hand-in: send typed solution by email or put hand-written solution into the folder on my office door
prize: I’m very pleased to announce that the winner of this puzzle will get a book of mathematical puzzles!!
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
I can’t get enough of Terry Tao’s blog What’s New, he writes so much and so brilliantly. Anyway, I found two really nice old-ish posts about model theory:
- this which talks about completeness, compactness, zeroth-order logic, and Skolemisation;
- and this which talks about nonstandard analysis, notions of `elementary convergence’ and `elementary completion’, countable saturation, compactness and saturation re-written from an analytical point-of-view, and the Szemerédi regularity lemma (something I always want to know more about).
I don’t have time to write anything more now, but later I will get back to it.