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→No additional numbers may be added to the continuum (real line) without losing its dense linear order.
::AndyJM, I do not at all appreciate your reversion without discussion on the talk page. This is not how our wiki works, and based on your past history I'm going to say that you cannot make edits to the mathematics articles without first obtaining approval of your proposed edits from [[User:Aschlafly]], myself or [[User:Ed_Poor]].
::AndyJM, I do not at all appreciate your reversion without discussion on the talk page. This is not how our wiki works, and based on your past history I'm going to say that you cannot make edits to the mathematics articles without first obtaining approval of your proposed edits from [[User:Aschlafly]], myself or [[User:Ed_Poor]].
::Your second example (mentioned only in the edit summary) is also not a counterexample: the long line needs the uncountable ordinal <math>\omega_1</math> to be constructed. The elements of the long line are not numbers, but cartesian pairs of numbers with infinite ordinals. So the long line cannot possibly be a counterargument to the statement "No additional ''numbers'' may be added to the continuum (real line) without losing its dense linear order." (emph. added) which you removed now for the third time. The fact of the matter is, if you try to include another number into the reals, you will lose the dense linear ordering without endpoints of the reals. So including an imaginary number, for example, destroys the properties of the ordering. This is the issue at the crux of why the complex numbers cannot be linearly ordered without the Axiom of Choice: the additional elements all violate the essential properties of the original, natural ordering of the reals, and you need AC to rectify the situation artificially. Andy, you may be trying hard to insert your Wikipedia bias here, but the beauty of mathematics is that it follows logical reasoning and flawed arguments are often easy to pierce. -[[User:Foxtrot|Foxtrot]] 14:51, 8 February 2009 (EST)
::Your second example (mentioned only in the edit summary) is also not a counterexample: the long line needs the uncountable ordinal <math>\omega_1</math> to be constructed. The elements of the long line are not numbers, but cartesian pairs of numbers with infinite ordinals. So the long line cannot possibly be a counterargument to the statement "No additional ''numbers'' may be added to the continuum (real line) without losing its dense linear order." (emph. added) which you removed now for the third time. The fact of the matter is, if you try to include another number into the reals, you will lose the dense linear ordering without endpoints of the reals. So including an imaginary number, for example, destroys the properties of the ordering. This is the issue at the crux of why the complex numbers cannot be linearly ordered without the Axiom of Choice: the additional elements all violate the essential properties of the original, natural ordering of the reals, and you need AC to rectify the situation artificially. Andy, you may be trying hard to insert your Wikipedia bias here, but the beauty of mathematics is that it follows logical reasoning and flawed arguments are often easy to pierce. -[[User:Foxtrot|Foxtrot]] 14:51, 8 February 2009 (EST)
::: Ok, I'll leave the maths articles alone. I have this last point to make though.
:::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)
*'''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''']]<sub><small><small>/Admin</small></small></sub><sup>[[User_Talk:TK|/Talk]]</sup> 20:20, 8 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''']]<sub><small><small>/Admin</small></small></sub><sup>[[User_Talk:TK|/Talk]]</sup> 20:20, 8 February 2009 (EST)