From: HS Brandsma Subject: Re: old (set-)geometric enigma Date: Fri, 21 Jul 2000 15:58:31 GMT Newsgroups: sci.math Summary: [missing] NimbUs (nimbus@oreka.com) wrote: : Hi ! : As I'm discovering the news fora, may I ask if a question which I heard : of years ago has a known solution ? Here is the enigma : : : " Is there a set E of points of the euclidean plane, such that : every straight line ''cuts'' E in exactly 2 ( distinct ) points ? " : : True ? False ? Maybe undecidable ( or needs clarification of axioms ? ) It is true. But it seems to require the axiom of choice.. The construction is due to Kuratowksi I believe, and is found in his book on set theory (with Mostowski). THe idea is to enumerate the lines of the plane in a well-order and inductively choose points. At each stage there are still ponts left to chose from given the points already chosen and the lines already met in two points. I think the recent book on set theory by Cieselski also gives the construction as a prototypical example of a set that can be made with the axiom of choice. It is still open whether such a set can be Borel, so "nice" in a certain way. Here on the VU a PhD student is working on this an similar problems.. Eg it is known that such a set cannot be a countable union of closed sets.. Henno Brandsma : : Silly as it be, I couldn't make my mind... But I'm just an unskilled amateur? : hoping to hear the Truth. : : < nimbus@oreka.com > (France) : : : ------ : User of http://www.foorum.com/. The best tools for usenet searching.