User blog:LittlePeng9/Infinite time game of life

In this blog post I will try to prove some facts about extension of Conway's game of life to transfinite number of steps.

Definitions
For a given cell C we denote by \(C_\alpha\) the value of C at step \(\alpha\) (where 1 means the cell is alive, 0 means cell is dead). If \(C_\alpha=1\) and exactly 2 or exactly 3 of its neighbours are alive at step \(\alpha\), set \(C_{\alpha+1}=1\). If \(C_\alpha=0\) and exactly 3 of its neighbours are alive at step \(\alpha\), set \(C_{\alpha+1}=1\). Otherwise, set \(C_{\alpha+1}=0\). If \(\alpha\) is a limit ordinal, set \(C_\alpha=\text{limsup}_{\beta<\alpha}C_\beta\).

For the pattern P which is a function \(C\rightarrow C_0\) we say that P is stable if \(\forall\alpha\forall C: C_\alpha=C_0\). Equivalently, \(\forall C:C_0=C_1\).

We say that P stabilizes if \(\exists\alpha\forall C: C_\alpha=C_{\alpha+1}\). We say that P dies0 if \(\exists\alpha\forall C: C_\alpha=0\).

We say that P is an oscillator if \(\forall\alpha\exists\beta>\alpha:C_\beta=C_0\).

For the patterns I'm going to use either RLE or macrocell file format, which can be copy-pasted directly into Golly.