So if last blog didn't quite get there here it is:
Let us begin with the idea of "Contradiction by Self-Reference" CBSR. This is a contradiction that is derived by use of any type of self-reference. There is debate (that I am currently reading) on whether all paradoxes involve self-reference. It seems self-reference paradoxes can be rewritten as infinite paradoxes and vice-versa. But certainly not all contradictions involve self-reference. But that type of contradiction that comes from self-reference is what interests us.
Quick note on method. Theorems taking theorems as input is usually achieved through a mapping from the ordered list of theorems to whatever domain variables have. If the logic is about numbers this is very evident. Theorem(5) can mean both the result of operating on "5" and also the theorem that is listed at position 5 in the ordered list. This is called Godel Numbering.
So Strong SRH hypothesises "all systems with self-reference can derive contradiction from that self-reference" SSRH (strong self-reference hypothesis).
Note to myself: I need give the logic here a little bit of thought to get it really accurate. Consider the a function like SR(x) that decides if the theorem x has self-reference. SR(x) is true if x involves self-self-reference and false otherwise. However I can't just do SR(SR) since SR is bound to a variable. It would be a whole family of theorems like SR(SRx) the theorem that comes from binding itself to itself with an input x. If "SRn" is the theorem at position n in the list of SR theorems then SR(n) is only self-referential if and when n = SRn. I'm unsure where we can argue self-reference is guaranteed in this construction!
So at the moment I don't know how to create a function like C(SR(x)) which would says that if x has self-reference then you have a contradiction. Which is not quite SSRH which states that the theorem itself is true i.e. SSRH(C(SR(x))) is true. Warning this could all be nonsense I'm just thinking it through.
Anyway concerns out of the way:
Suppose we construct a sequence of logic, a theorem called SSRH to illustrate this. It takes one theorem as input and outputs 1 if the theorem has CBSR and contradiction and 0 otherwise.
So SSRH(SSRH) = 0 is an obvious contradiction. SSRH says that all theorems with self-reference can derive a contradiction yet SSRH(SSRH) say that it doesn't. But that is itself a contradiction as this self-reference has provided a contradiction so the result should be 1.
But SSRH(SSRH) = 1 is highly problematic too.
It says that SSRH is a theorem with self-reference that generates a contradiction from that self-reference.
Clearly it has self-reference since it has taken itself as input. But it says that doing so derives a contradiction. If we believe it, it has said that its output is a contradiction, and if we don't believe it it means that SSRH is inconsistent. Essentially Godel's result.
If SSRH is consistent then it says its own logic is faulty, which if it was true invalidates the outcome. Not "I am lying" but "I tell contradictions." In standard logic we reject contradictions so it means "If the logic is consistent then reject an assumption that you made in deriving this contradiction." But if we do that then we are actually taking a contradiction to be true! If we accept that the logic was good it would be a contradiction to reject an assumption! This road breaks the very process of logical truth! So we reject SSRH.
So what if SSRH(SSRH) is not the self-reference that is problematic and there is another one. If SSRH contains a self-reference that is problematic within the logic not bound to the variable then the result is even worse and applies to all input. So the problem above stands.
So we must reject the consistency of SSRH all together. There is no such thing.
Now what about Weak Self-Reference Hypothesis (WSRH). This is much more interesting. It says that where there is self-reference and definable conditions then there is CBSR.
Firstly let me introduce the theorem C(x) which sees if x contains a contradiction. (Note: this is proving a useful session as forcing it into logic is really breaking up what I've been trying to say here).
Suppose we illustrate WSRH like this:
WSRH(x) := C( Conditions(x) && SR(x) )
Which says that if theorem x contains self-refence and it meets the conditions defined by Conditions then it will give a contradiction.
I'm going to leave this here for now cos I need to go back and sort out self-reference and the function SR(x). That might be very fruitful in itself.
Lots of meta-logic functions here (functions deriving results from the logic itself: Godel only had one Beq(x) which decided if a theorem was provable): probably too many.
Not getting stuck into the logic. To recap WSRH suggested that where a domain contained its range (like with fractals) you were guaranteed a "Fixed Point" and the existence of that indicated that SRH was true and problems with self-reference were guaranteed. Let me just define the meta-logic function FP(x) which for a logic decides this. FP(ZFC) is a valid statement since Zermelo-Fraenkel Choice logic enables Godel numbering and so there is guaranteed a statement with one variable in ZFC (e.g. F(x)) which calculates a number which is its own position in the list of theorems.
Note:
F(x) = x is a fixed point
F(n) where n is the position of F(x) in the list of theorems is Self-Reference i.e. n=G(x) is position in the list of theorems or a unique mapping to the domain somehow.
They are different.
OK racing through this. Need to go over the very faulty logic!!!