# Difference between revisions of "Talk:Continuum"

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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) |

## Revision as of 13:51, 8 February 2009

I would like to make a change to this article. In the second paragraph the link from continuous goes to an article on continuous functions. However the context in the article is that of a continuous set. I propose adding an article about continuous sets and then linking to that article. Any objections? (AndyJM 09:00, 12 December 2008 (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! AndyJM 10:14, 16 January 2009 (EST)

- The 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. -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 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. AndyJM 10:24, 29 January 2009 (EST)

- I have added the phrase "without endpoints", which defeats your reasoning. The complex numbers cannot be linearly ordered naturally; any ordering needs the Axiom of Choice to be constructed. -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 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. -Foxtrot 14:51, 8 February 2009 (EST)