Re: Questions about Axiom of Choice


Daniel Grubb wrote:
1.- It is a "pure existence" statement, that is it tells something
exists but not what it is. Is math not full of these kinds of
statements?

It is now. Until relatively recently, an existence proof was required
to give a method of construction.

2.- Some of its implications are weird or counterintuitive. Can
someone please explain some of these?

My favorite is the Banach-Tarski paradox: it is possible to take
the unit ball (i.e. radius 1) in three dimensional euclidean space,
divide it into a finite number of pieces, move those pieces around
via rigid motions (i.e. translations and rotations), and re-assemble
them into a ball of radius 2.

--Dan Grubb

Thank you and everyone who replied. This is a really dumb question but
is it not possible to prove or disprove this physically? That is we
can construct a solid unit radius ball, cut it up into a finite number
of pieces (according to some procedure) and produce the new (solid?)
bigger ball? Thanks again.

.

Relevant Pages

  • Re: Screw Mastercam x
    ... > Double ouch... ... >> I determined for sure the smallest space will fit a .249 ball. ... > measure the bottom radius (curve radius measure is easier than surface ... >> little while due to the crappy graphics card. ...
    (alt.machines.cnc)
  • Re: Worst case performance for kd-tree queries?
    ... inside a ball of radius considerablz smaller than r. ... terminate traversal for that branch. ... I traverse the children of the current node. ...
    (comp.graphics.algorithms)
  • Re: Questions about Axiom of Choice
    ... to give a method of construction. ... them into a ball of radius 2. ... Herman Rubin, Department of Statistics, Purdue University ...
    (sci.math)
  • Re: Radius of largest n+1 balls in n-dimensional unit cube?
    ... sqrtis maximal distance between two corners of the cube, ... Then the distance from the center of the ball in that corner to the ... the same radius R), within the unit hypercube such that: ...
    (sci.math)
  • Re: Questions about Axiom of Choice
    ... the unit ball in three dimensional euclidean space, ... them into a ball of radius 2. ... to the group of rotations being amenable. ...
    (sci.math)