peg (02/09/83)
Here's a problem I've been working on for years. Anyone out there have a proof? Def: A tack consists of three closed line segments in the plane that share EXACTLY one point. So they look like the letters T or Y, for example. --> Question: How many tacks can you fit into the plane so that no two of them intersect? Def: A set is countably infinite if there exists a one-to-one correspondence between it and the integers. Def: A set is uncountably infinite if there exists a one-to-one correspondence between it and the real numbers. It's easy to construct a countably infinte set of tacks in the plane, so the question becomes: --> Can uncountably many tacks fit into the plane with no two of them intersecting?? My guess is that countable is the best you can do, but I've been unable to prove it. Come close now and then, but... Have fun!! Peggy **** PS - For people who want to learn about infinities and eventually prove to yourself that the axiom of choice is equivalent to the well-ordering axiom, I highly recommend the book "Manual of Axiomatic Set Theory" by Frank D. Quigley of Tulane University, published by Appelton-Century-Crofts.