April 9th, 2017, 10:19 PM  #21 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,217 Thanks: 93 
Changed my mind about expanding points. Back to Euclid and Deascartes. The line is the fundamental geometric figure of Euclidean geometry. Points are not geometric figures, they are locations. (2) is a line from an assumed location of length 2. (2,3) are two "different" lines from an assumed location. (1,3,..an) are n different "lines" from an assumed location. The lines above also identify locations. You can't create geometry from points, but you can discuss and analyze sets of points. That's why Point Set Geometry is misleading, it is ambiguous it implies you start with points and create geometry, making point sets foundational they aren't. You can abstract infinite sets of "things" and relate them with "rules" using geometric names, which is fine until you derive things using your abstract things and rules and give the result geometric names BanachTarski paradox for example; or claim you have established the foundation of geometry. EDIT This should be generalized to: You can't create a geometric dimension from a zero dimension. Example, you can't create a square by laying lines (0 width) next to each other. Example, Banach Tarski Paradox. Last edited by zylo; April 9th, 2017 at 11:14 PM. 
April 10th, 2017, 07:32 AM  #22  
Senior Member Joined: May 2008 Posts: 259 Thanks: 50  Beer soaked query follows. Quote:
 
April 10th, 2017, 09:03 AM  #23 
Math Team Joined: Oct 2011 From: Ottawa Ontario, Canada Posts: 11,640 Thanks: 740  

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geometry, point, set 
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