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→No additional numbers may be added to the continuum (real line) without losing its dense linear order.: reply 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)