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→No additional numbers may be added to the continuum (real line) without losing its dense linear order.: reply and warning to AndyJM
All that remains is to show that <* is dense. It is obvious that <* is dense when restricted to R. If a<*x we need to find b such that a<*b<*x. But choosing b=a+1 will suffice. [[User:AndyJM|AndyJM]] 10:24, 29 January 2009 (EST)
All that remains is to show that <* is dense. It is obvious that <* is dense when restricted to R. If a<*x we need to find b such that a<*b<*x. But choosing b=a+1 will suffice. [[User:AndyJM|AndyJM]] 10:24, 29 January 2009 (EST)
:I have added the phrase "without [[endpoint]]s", which defeats your reasoning. The complex numbers cannot be linearly ordered naturally; any ordering needs the [[Axiom of Choice]] to be constructed. -[[User:Foxtrot|Foxtrot]] 03:55, 5 February 2009 (EST)
:I have added the phrase "without [[endpoint]]s", which defeats your reasoning. The complex numbers cannot be linearly ordered naturally; any ordering needs the [[Axiom of Choice]] to be constructed. -[[User:Foxtrot|Foxtrot]] 03:55, 5 February 2009 (EST)
::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)