That's not quite what's going on here. The manipulations show that, if 4=x^(x^(...)), 4=x^4. The mistake is in beginning with the assumption that the former e

That's not quite what's going on here. The manipulations show that, if 4=x^(x^(...)), 4=x^4. The mistake is in beginning with the assumption that the former equation has a solution. If it doesn't, any manipulations that assume it does are invalid to begin with. Most people aren't exposed to the idea that an equation could have no solution all by itself, basically because polynomial equations always do.