Changes

:::The statement "Any linear order of the complex numbers needs the Axiom of Choice to be constructed." is not true. I will demonstrate a linear order < on the complex numbers that doesn't need AC.

:::The statement "Any linear order of the complex numbers needs the Axiom of Choice to be constructed." is not true. I will demonstrate a linear order < on the complex numbers that doesn't need AC.

:::We define < a linear order on C. Fix a+bi,c+di in the complex plane. Then we define a+bi<c+di iff (b<* d) (where <* is the natural order on the reals) OR (b=d and a<*b). It is easy to verify that this is a linear order on the complex numbers. In fact it is even a dense linear order. [[User:AndyJM|AndyJM]] 15:35, 9 February 2009 (EST)

:::We define < a linear order on C. Fix a+bi,c+di in the complex plane. Then we define a+bi<c+di iff (b<* d) (where <* is the natural order on the reals) OR (b=d and a<*b). It is easy to verify that this is a linear order on the complex numbers. In fact it is even a dense linear order. [[User:AndyJM|AndyJM]] 15:35, 9 February 2009 (EST)

I would suggest deleting the second paragraph until you get something that is mathematically correct. The axiom of choice is needed for well-orderings, not ordinary linear orderings. A theorem that characterizes the real numbers might be more appropriate on the Real Numbers article. [[User:RSchlafly|RSchlafly]] 14:15, 16 February 2009 (EST)

I would suggest deleting the second paragraph until you get something that is mathematically correct. The axiom of choice is needed for well-orderings, not ordinary linear orderings. A theorem that characterizes the real numbers might be more appropriate on the Real Numbers article. [[User:RSchlafly|RSchlafly]] 14:15, 16 February 2009 (EST)

 

:Deleted it, at least until I can sort the concepts out better. I know there is some theorem that you can't order the complex numbers (I guess as an ordered ring) because ''i'' causes violations of the ordering. I had thought there was something more basic than that, but I guess that unless you can talk about the algebra of the complex numbers (and thus, talk about "numbers" rather than topological "points"), the complex numbers won't look any different than the long line or other topological spaces that look like continua. -[[User:Foxtrot|Foxtrot]] 14:39, 16 February 2009 (EST)

*'''No one given the right to block is going to be issuing instructions about who is allowed to edit, or imposing conditions like the above. That is the exclusive province of Administrators, and we only impose such restrictions after talking about the possibilities with each other.  ''You guys work it out amongst yourselves.'' That said, AndyJM, Foxtrot is correct about giving notice, on his user talk page, or here, about reverting major portions. That is simple courtesy. If you have further disputes on math topics, politely ask [[user:RSchlafly|Roger Schlafly]] and see if he will mediate.''' --[[User:TK|'''₮K''']]_{<small><small>/Admin</small></small>}^{[[User_Talk:TK|/Talk]]} 20:20, 8 February 2009 (EST)