Self

The Sierpinsky triangle contains “itself” in that each level (zooming in) is composed from exactly 3 of the previous larger level. The triangle set runs to infinity so each smaller copy is in bijection to its parent (every point is countable and in 1:1 correspondence). The only difference is size… and the fact that the set tops out … i.e. there is a largest final set (the starting triangle) in which all the others are bounded. In this sense each copy is exactly that: a copy, and so in strict sense does not have the “same” identity. Badly phrased but all I am saying that the triangle contains an infinite number of triangles which are only similar: the set does not “contain itself”.

It is this sense of self that I am trying to find a concrete example of. A mathematical expression of identity: because I realise that the SRH is actually saying it is this type of Self that is impossible. Isomorphic copies of the self are fine because they are only possible because they are different things, and different selves. Godel numbering or any other type of countability is only finding names for things by which to reference them – it is not actually dealing with “self”. In a sense self-reference is a misnoma: one can only reference the “self” via something which is not self and which bears a resemblance or unique connection with the parent thing… and that connection implies a meta language which I’m arguing in one place here cannot exist in either the self or the thing which is the reference. That is well documented already in the literature W.V.Quine’s “mention” and “use”. But there is the other problem that “self” because it can only expressed by the “self” itself is beyond analysis and expression… or something like that.

Interesting after some thought that the Mandelbrot set is unique. There is no other fractal like it. I’ve looked for something and found nothing and all the other sets like Phoenix etc are bad imitations. The Buddhabrot is interesting because it expressed the Mandelbrot set better than the usual rendition. What is not obvious from the usual Mandelbrot rendition is down densely the set extends into the usual coloured region. The set is created from all the points that are visited by each iteration that do not escape in magnitude (if the final iteration does not escape then any of the points in the path do not escape either). The size of the set is actually a lot bigger than is shown in the usual set rendition and it is easier here to see the way that rotations of the set produce the final shape. Never-the-less amazing that so simple a condition as Z=Z^2+C should have such a complex solution. But the existence of self-similar symmetry is not a common feature as this is unique set.