Self-Reference revisited

I'm determined to get this one.

So we have the Ouroboros pictured right from wikipedia. It typifies the problem I envisage with self-reference.

What happens if the Ouroboros keeps eating? The circle will tighten and tighten, but we know for a fact that it will never be able to satisfy its consumption. Even as a singularity it will not reduce to nothing. The knot will always be left of its head rising up upon its own head. Think we have eaten ourselves and we ask, "who ate us" and then we discover ourselves again.

Another fictional figure which captures the mood is the Woozle. While sure they are following a Woozle, Pooh and Piglet find out that really they are following their own foot prints.

The Ouroboros can complete its meal if one thing happens. It breaks into two. Then the front half can eat the back half. And so it seems possible that self-operations might always create an unbreakable knot, which is resolved only by the separation of what way one into two.

At this stage without thinking it through I suggest that the Ego struggling to found itself, can only do so by causing division and separation. Is this not a creation myth to explain the plurality of a world that the wise describe as One.

Turning the enquiry around this impossibility of self-operation might indeed be a test. Whereever insluable problesm exist that threaten to break unities, maybe it is because the two halves are really the same entity - only into 2 because of a self-operation.

Dependent (DSR) and Independent Self-Reference (ISR)

Take two coins. Call one "heads" and the other "tails" and then picking up "tails" ask it which coin to toss next. Toss it to find out. If it lands on "tails" is it not saying "toss me next"?

Well not really. This is dependent self-reference because the coin means nothing by itself. It requires the player of the game to understand that "head" and "tail" are names and which coin they reference.

In Godels proof he uses this. As I understand it, it goes something like this. He creates an ordered set (in others words item 5 of the set is always the same item) of valid sentences of the logic. There is a sentence then which says that "x is not in the list" where x is the actual position of this sentence in the ordered set. This can be interpreted to say that "this sentence is not in the list"... which it must be since it has a number. "Godel numbering" which is how he coded the "names" of sentences is quite involved and i haven't looked at all this yet (as you can tell), but I think I know enough to say that the proof is a case of DSR because the reader has to consciously make the identity between the number and the sentence.

Clearly DSR happens. But it is not of great interest because it is an illusion.

"This sentence has five words" is the same in DSR terms as "the sentence 'this sentence has five words' has five words". It is in two parts. The sentence making the statement and the syntax itself which I put in "quote marks".

It is not really self-reference because 'this sentence' does not by itself refer to the whole sentence. It requires the reader to understand this (semantically) and then do a count of the sentence as though it were an arbitrary sequence of words. One cannot count the meaning of the sentence! Consider again the semantics in translation. I have no idea about Chinese Tradition but assuming that Babelfish is even part correct this is the same sentence in Chinese 這個句子有五個詞. Where I do know that 五 means five. This sentence doesn't even have any words! So the semantics and the syntax are quite different and the sentence clearly isn't referring to itself.

ISR is what I am interested in. Where reference is to the actual self. Like the snake eating itself. Or a sentence which says "I am true". Translate that into any language and you have the same self-reference, because the SR is constructed within the system - it is independent of an outside agent.

Now the hypothesis to be proven is that ISR is impossible. The problem is just what form the proof might even take?