x=x+1
x=x+1; x=x+1 [[ this extends the line at the end ]]
x = f[x]
x = {x, x} ; x = {x, x} (more ramified?)
If x = {x, x} then x is an infinite binary tree
Then what is {x, x}?
Then what is {x, x}?
In[]:=
{KaryTree[15],KaryTree[15]}
Out[]=
,
In[]:=
Graph[{u->x,x->x,x->x}]
Out[]=
Nest[f,x,Infinity]==Nest[f,x,Infinity+1]
This infinitary application is like a CTC ...
LeafCount[x={x,x}]->2^ω
LeafCount[x=f[x]]->ω
x={f[x],f[x]}
LeafCount[x=(x=Append[x,x])]
LeafCount[x=Table[x,Infinity]]->ω^ω
Table[Table[x,Infinity],Infinity]
Table[x,Infinity,Infinity]->ω^2
Nest[Table[#,Infinity]&,x,Infinity]ω^ω
Nest[Nest[Table[#,Infinity]&,x,Infinity]
Nest[Nest[Table[#,Infinity]&,#,Infinity]&,x,Infinity]ω^ω^...==
ϵ
0
Function[x,]
x|->(x=f[x])
Pure infinite nesting:
#0[]&[]
#0[]&[]&[]&[]&[]
Do all infinite rewritings have a stem cell that repeats? [Or one could have a generator of stem cells ... ]
But there has to be a way to have “surprises forever”
But there has to be a way to have “surprises forever”