Self - containing systems again

Have made some progress...

the general argument from Godel to Logical Positivist school goes like this...

Some statement about the world... what if that statement apply to itself? and it always leads to "paradoxes".

So with the LP principle of verification, one asks what empirical evidence leads to the VP?

Especially this is true of truth itself. If a system can encode truth then you can get the absurdities of true statements that say they are false.

Godel showed that in numerical systems a wff can be formed which is unprovable. Provability is thus not the same a truth, and truth lies outside the system of provability. Systems are either incomplete - where unprovable statement exist or are inconsistent.

This is more obvious when one considers the "truth" of axioms. Clearly they are true "outside" a system since the system relies upon their truth. Such Judgement one must assume as a condition for the system.

Other fundamental conditions also exist like the education of users of a system, reason, existence, thought etc a pro pos Kant. There is a hermeneutic circle of given conditions which must be present before any system can be constructed. To construct that hermeneutic circle would require a position outside itself, and presumably another hermeneutic circle. The question formally that these speculations are true.

The implications are that there is no "fundamental" system with which to construct the world. That indeed the world is at root indefinable, and that is a construction of a notion of absolute Liberty.

So can this process be generally shown?

From Wittgenstein let us assume that the "meaning" of a statement is its "use".

To know the use of something we can simply create a set of all the possible uses of that statement.

A system of axioms and logic is thus a "compression" of the set of all possible statements but extracting the information to simply show that it can be reconstructed.

Thus in a logical system we could have a number refering to the sequence of rules applied to generate a statement, which would be identical with all the statements, and that set of numbers would constitute the meaning of the system.

The difference between that set and the normal expression of axioms however would be that the axioms no longer have a meaning outside the system from which the set of possible statements can be constructed, instead the set of possible statements becomes its meaning.

consider further....