Joel Uckelman on Wed, 28 Jul 2010 06:21:24 -0700 (MST)

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Re: [game-lang] Another quick tangential comment about integers and reals

Thus spake Simon McGregor:
> In regard to rationals and reals, imagine the following: a game is
> played by moving points on a board in R2, up to 1 unit's distance at a
> time (with the Euclidean metric). We can model this using the
> rationals, because rational arithmetic is adequate to determine
> whether two points with rational coordinates are within a rational
> distance of one another.
> However, suppose that the provably optimal move in some situation is
> to move your point as far as possible towards another point. Then (in
> general), no move in the rational model is optimal, although there's
> an optimal move in the continuous model. Uh-oh!

I don't follow this last bit of reasoning. It seems to imply that there's
some real r which is nearer to a target t than any rational q is. If
you have a nonstandard model of the reals where you also have
infinitessimals, then this could be the case (the move would be t-\epsilon),
but not so for the standard reals.

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