Changes

Previously someone had written that Dedekind cuts called the concepts of real numbers into question, and that Dedekind cuts relied on the use of the axiom of choice. I've never heard such an insanely wrong description of Dedekind cuts, so I removed the statement. Dedekind cuts most certainly do NOT rely on choice in their description. Moreover, the purpose behind Dedekind cuts is that they allow one to construct the real numbers out of the rationals. So far from calling the existence of the reals into question, Dedekind put the definition of the reals onto solid ground.--[[User:Mathoreilly|Mathoreilly]] 11:02, 2 July 2008 (EDT)

Previously someone had written that Dedekind cuts called the concepts of real numbers into question, and that Dedekind cuts relied on the use of the axiom of choice. I've never heard such an insanely wrong description of Dedekind cuts, so I removed the statement. Dedekind cuts most certainly do NOT rely on choice in their description. Moreover, the purpose behind Dedekind cuts is that they allow one to construct the real numbers out of the rationals. So far from calling the existence of the reals into question, Dedekind put the definition of the reals onto solid ground.--[[User:Mathoreilly|Mathoreilly]] 11:02, 2 July 2008 (EDT)

I agree.  Any mathematician who was concerned about Dedekind cuts relying on the axiom of choice was just wrong.  They have nothing to do with AC.  Nor did they "undercut the assumption of the continuity of the real line".  They put the continuity on a firm basis.  Nor did they create gaps.  They filled them. [[User:SamHB|SamHB]] 22:30, 20 August 2008 (EDT)

I agree.  Any mathematician who was concerned about Dedekind cuts relying on the axiom of choice was just wrong.  They have nothing to do with AC.  Nor did they "undercut the assumption of the continuity of the real line".  They put the continuity on a firm basis.  Nor did they create gaps.  They filled them. [[User:SamHB|SamHB]] 22:30, 20 August 2008 (EDT)

I just changed

I just changed

--[[User:MarkGall|MarkGall]] 23:07, 5 July 2009 (EDT)

--[[User:MarkGall|MarkGall]] 23:07, 5 July 2009 (EDT)

:You're correct; I was being stupid.

:You're correct; I was being stupid. In fact, looking at my initial draft, I '''did''' have parens to indicate open intervals, and must have thought, at the last minute, that it looked wrong.  In any case, the union operation is the right thing.

 

:I can't get substack to work either.  But stackrel works, as in <math>\stackrel{a}{\widehat{B}}</math> or <math>\stackrel{\textstyle{\rightarrow}}{V}</math> [[User:PatrickD|PatrickD]] 12:22, 6 July 2009 (EDT)

:Next, I have to do neighborhoods.  This will be somewhat more daunting than watching Mr. Rogers. [[User:PatrickD|PatrickD]] 12:28, 6 July 2009 (EDT)