
An Infinite Orchard - bladecatcher
http://www.alaricstephen.com/main-featured/2017/7/5/an-infinite-orchard
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mannykannot
The final paragraph on the topic alludes to Olber's paradox, but stars are not
infinitesimally small, so I do not think the argument applies here (though the
Big Bang makes the question moot anyway.)

[http://cmb.physics.wisc.edu/pub/tutorial/olbers.html](http://cmb.physics.wisc.edu/pub/tutorial/olbers.html)

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Retra
Stars are also not confined to an integral grid. Surely if the trees in the
orchard were not so confined, they would cover the entire field of view.

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mannykannot
Good point - though it has me wondering this: the equi-spaced grid is a case
having a straightforward proof (given that Cantor did the heavy lifting), but
is it a necessary condition?

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Retra
The proof relies on the fact that the rationals are lesser in cardinality than
the reals, but for each "rational point" from your field of view, there are
infinite trees in that line. So anything anything which doesn't force all
those trees into a rational point is going to cover something larger than the
rationals... and then you'll probably need an answer to the continuum
hypotheses?

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mcenedella
In the final problem, as the apple trees can be in more than one line, the
minimal valid solution is 9 trees, not 12.

