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→No additional numbers may be added to the continuum (real line) without losing its dense linear order.: new section
Please, before anyone bans me for making the changes that I have made, I ask that you read http://mathworld.wolfram.com/Continuum.html. Thanks! [[User:AndyJM|AndyJM]] 10:14, 16 January 2009 (EST)
Please, before anyone bans me for making the changes that I have made, I ask that you read http://mathworld.wolfram.com/Continuum.html. Thanks! [[User:AndyJM|AndyJM]] 10:14, 16 January 2009 (EST)
:The <math>\mathfrak{c}</math> is mainly used by topologists--the remaining mathematicians that think about such large cardinalities tend to use the aleph system. But since I doubt we'll get that far into the details on CP, I can agree to that symbol. -[[User:Foxtrot|Foxtrot]] 22:43, 18 January 2009 (EST)
:The <math>\mathfrak{c}</math> is mainly used by topologists--the remaining mathematicians that think about such large cardinalities tend to use the aleph system. But since I doubt we'll get that far into the details on CP, I can agree to that symbol. -[[User:Foxtrot|Foxtrot]] 22:43, 18 January 2009 (EST)
== No additional numbers may be added to the continuum (real line) without losing its dense linear order. ==
The article asserts that "No additional numbers may be added to the continuum (real line) without losing its dense linear order." This is not true. I don't know what the author is trying to get at here. Unless anyone has any objections could we please remove this line? If anyone has any doubts that about this then here is a proof:
Claim: An additional point can be added to the real line without the line losing its dense linear order.
Proof: Let R denote the real line and < denote its natural linear ordering. Fix x not in R. Let X=R union {x}. Extend the order < on R to a new order <* on all of X as follows:
If a,b are in R then a<*b iff a<b.
If a is in R then a<*x.
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)