So you're saying, since there are \aleph_1 infinite patterns, and each can have at most \aleph_0 descendants, no one of them can include all the patterns in its future history. I agree... I couldn't find "cemetery" on the LifeWiki (http://www.conwaylife.com/), but maybe you mean empty space or still-life patterns. But considering just those maps Z^2 -> {0,1} in which all of the NxN squares contain a small fixed finite periodic configuration, and since we took N to be finite, then the infinite grid has a finite period (the LCM of the periods of all the NxN squares). Such a map would be ineligible for the Adam and Eve question, because we only care about maps Z^2 -> {0,1} that have an arbitrarily long line of ancestors, where every ancestor is distinct. That breaks the argument, but it can be fixed easily. Just leave an empty strip somewhere (say of width N and length infinity) and place a spaceship inside it. The presence of the spaceship makes every ancestor distinct. Another similar problem comes up (spaceships are just another oscillator), which can be fixed by again restricting the set of maps Z^2 -> {0,1}. I suppose if we were to continue adding conditions to which infinite patterns are eligible, and eventually run into limits on our ability to completely define what should and should not be eligible. On Wed, Dec 7, 2011 at 11:19, Dan Asimov <dasimov@earthlink.net> wrote:
Both of these options are impossible.
First tile the plane by NxN squares for big enough N. By using either a small fixed finite periodic configuration, or a cemetery, in each NxN square, this shows the existence of uncountably many maps Z^2 -> {0,1} each having at least one infinite line of ancestors, which covers both questions.
But any one map Z^2 -> {0,1} can only have countably many descendants. QED
--Dan
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