The +1 conjecture

A self-operating system cannot complete operation in its domain.

If a system could operate on itself, for example a hand grasping itself, then the system would be able to complete the operation intrinsically.

The question then is under what circumstances would it simply complete under itself and under what circumstances would it complete under external circumstances.

If it is a truely intrinsic system then it negates its own operation because it would always complete under itself.

If there is an extrinsic factor that triggers self completion then it can operate on all factors except that extrinsic factor where the result is self-operation.

Thus there must always be one element within the domain of the input which it cannot operate upon, if it is to have self operation.

A set cannot essentially contain itself, because if it could complete under itself it would negate its containing function.

This is why a hand cannot grasp itself, for if it could then to be able to grasp anything else there would need to be an external trigger, which it could not itself grasp. Evidentally in the realm of things there are not rules enough to distinguish that special thing which would cause self-grasping. It is worth exploring this to find the rules...


On References....

The same argument then applies to references. A box can contain other boxes but it cannot contain itself. It contains instead a reference to itself.

However then it cannot then contain also that thing without breaking the reference rule. Thus to have self-membership you need a symbol which has a special meaning i.e. self. and not its obvious meaning.

Now it might seem absurd. If we can model 14 rugby players there is nothing to modelling a 15th and calling it myself. However this is not the system of self modelling. This is simply treating the self as another entity and there is no notion of selfhood.

Formally... and this is where I need to push ahead...

A universal set {A,B,C}

Now the powerset requires no new symbols. Within the rules we can construct subsets at liberty. e.g. {A,B},{A,B,C}

Now if the set {B,C} was to have self membership we would need to create a new symbol with which to reference it. T = {T,B,C} and the system increases by 1

The set of all symbols {A,B,C} and itself has a problem then because it requires a new symbol T, but that creates a new set and so on...Recursively this creates a problem...

Now let us start with a special symbol T for self reference

{A,B,C,T}

I guess the problem might be that T has no definition, and doesn't really belong to the same set as the other symbols so it can't be a member at this stage... to explore