Undecidable “Elementary” Geometry, II

A couple of years ago, I mentioned that the elementary geometry of points, lines, and circles becomes undecidable when it includes screws or spirals. More recently, I have wondered how to express a suitable set of axioms for spirals. If we start with Tarski's axioms, this includes the necessary (but insufficient) step of replacing the two-dimensional upper- and lower-dimension axioms with their three-dimensional equivalents.

According to the Wikipedia article on Tarski's axioms, it can be easily extended to higher dimensions by changing the upper- and lower-dimension axioms. On the other hand, it didn't include examples of those and the papers that give examples appear to be hidden behind pay walls. So I had to devise my own…

In short, I have uploaded a file containing a JavaScript program to produce upper- and lower-dimension axioms to my Netcom/Earthlink site. (This included the JavaScript program mentioned here.)

Next I have to come up with a plausible way to express spirals …