Comments on: When are the Real Numbers Necessary? http://xorshammer.com/2008/10/13/when-are-the-real-numbers-necessary/ Some things in mathematical logic that I find interesting Wed, 18 Jan 2012 18:13:09 +0000 hourly 1 http://wordpress.com/ By: none http://xorshammer.com/2008/10/13/when-are-the-real-numbers-necessary/#comment-196 Wed, 17 Jun 2009 07:39:08 +0000 http://xorshammer.wordpress.com/?p=184#comment-196 Eh? The whole point of the reals is that if two curves in the plane cross over each other when you draw them on paper, then they actually intersect at a point. That requires completeness, i.e. some form of the least upper bound property. If you’re just on the rationals, or just on the algbraic points, or something like that, the curves can pass “through” each other without intersecting.

]]> By: mkoconnor http://xorshammer.com/2008/10/13/when-are-the-real-numbers-necessary/#comment-138 Mon, 10 Nov 2008 12:34:09 +0000 http://xorshammer.wordpress.com/?p=184#comment-138 Sorry it took me so long to respond. What you said is right, except the notion of polytope equivalence also extends to higher dimensions: i.e., the bijection must preserve whether or not their is a face bounded by edges between vertices, and a solid bounded by faces bounded by edges between vertices, and so on.

]]> By: anon http://xorshammer.com/2008/10/13/when-are-the-real-numbers-necessary/#comment-134 Sun, 19 Oct 2008 17:00:31 +0000 http://xorshammer.wordpress.com/?p=184#comment-134 I’m a little confused by what is covered by the “etc” in the definition of polytope equivalence. I think that two polytopes are equivalent if there is a bijection between their vertex sets that preserves adjacency, collinearity and noncollinearity. Is that all?

]]> By: mkoconnor http://xorshammer.com/2008/10/13/when-are-the-real-numbers-necessary/#comment-130 Fri, 17 Oct 2008 09:30:27 +0000 http://xorshammer.wordpress.com/?p=184#comment-130 Ah, good point. The convex hull is over the reals.

To clean up the statement of the result a bit: A polytope is the convex hull (in the reals) of a finite set of points in \mathbb{R}^n. A polytope is called rational if it’s equivalent to one which is the convex hull (in the reals) of a finite set of points in \mathbb{Q}^n. The result is that there are polytopes which are not rational in dimensions greater than 3.

]]> By: Mark Reid http://xorshammer.com/2008/10/13/when-are-the-real-numbers-necessary/#comment-128 Fri, 17 Oct 2008 03:40:07 +0000 http://xorshammer.wordpress.com/?p=184#comment-128 That is a striking result!

I was wondering if you could just clarify one small detail: is the convex hull in the phrase “convex hull of a set of points with only rational coordinates” a convex hull over the reals or rationals? That is, if x and y are two rational coordinates then ax + (1-a)y is in the convex hull. Are the coefficients a and b real or rational?

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