in the layer logic defined in this thread there is an inconsequence: The axioms 5 and 6 for meta statements, especially that the truth values of meta statements are the same for all layers t>0 are in objection to the principle, that no information about higher (or same) layers is available at a lower level. I.e. W(W(A,t)=w,1)=w woul allow to conclude in layer 1, that W(A,t)=w even for t>1.
Without axioms 5 and 6 for meta statements it is more complicated to define a value for statements over all layers, nevertheless I think we should go this more consequent way.
Now we have two basic layer logic axioms:
A1): statements have truth values only in combination with a layer t = 0,1,2,3,?
A2) Layers are hierarchically ordered, i.e. truth values can be defined using truth values of lower layers, conversely in lower layers nothing is known about higher (or same) layers.
As a consequence: W(W(A,t)=w,t)=u; W(W(A,t)=w,d)=u für d<=t
We therefore need the value ?u? (undefined/unknown) not only in layer 0 but in all layers. (Vice versa W(A,0)=u now is no longer a isolated formula but a kind of spezial case of W(W(A,t)=w,d)=u für d<=t für d=0)
Annother consequence: The equity of layer statements becomes difficult to be proofed: 1st attempt: W(A=B,d+1) := W ( For all t: W(A,t)=W(B,t) , d+1) after A2: W( For all t: W(A,t)=W(B,t) , d) = W ( For all t<=d: W(A,t)=W(B,t) , d+1) and W(For all t>d: W(A,t)=W(B,t) , d+1) = W ( For all t<=d: W(A,t)=W(B,t) , d+1) and u. Therefore W (A=B, d+1) = u if A=B and W(A=B,d+1)=-w if W(A,t0)-=W(B,t0) for t0<d+1. Equality could so be (sometimes) disproved and never be proofed positively.
2nd attempt: Layer statements have to be defined finitely, as they could not be used otherwise. Often they are recursively defined for layer t+1 using values of statements in layer t.
So we can restrain the statements with a finite periodic value pattern over the layers. Now positive statements over all layers are possible, if the period k (and the advance v) is fully known.
Two statements A and B are equal (in layer k+v+1), if they have the same advance and the same period in the layers. The value for the of equality of two statements is constant for layers t >= v+k+1.
Axiom A2) shows that from layer t+1 we only have a perspective to look ?downwards? to layer t or smaller, the same layer or higher is beyond the information horizon and undefined.
The definition of arithmetics now also becomes more complicated, for example the succesor function is usually defined using equality of sets.
But why do things easy if they can be done complicated?