# Absoluteness

It is one thing to use *if A is true, then B is true* as a defining condition in a definition and quite another thing to state *if A is true, then B is true* as a closure condition for a given definition. In view of an analogy between models of set theoretical axioms and definitions of set theoretical concepts we might introduce the notion of *absoluteness* (cf. [8]) also in this definitional context. Whereas in the first case we compare how a set theoretical notion (formula) behaves in a model in relation to its behavior in another model, which intuitively means outside the model if the second model is the true cumulative hierarchy *V* , in the latter case we compare how a definitional notion/condition behaves inside the definition, in the local logic of the definition, with how it behaves outside the definition in the world of *intended* interpretation of defining conditions.

A set theory *S* is a pair of definitions *SΦ* and *T SΦ*, following the ideas discussed above in Sect. 2, for a given collection of functions *Φ*. A defining condition *A* is

*(left) absolute* (with respect to *S*), if for all defining conditions *B B* follows from *A* in *T SΦ* iff *( A* is true in *T SΦ* =⇒ *B* is true in *T SΦ).*

What this means is that deriving something from *A* in *T SΦ* is the same as implication. One closure condition that is generally self-evident is the following one

*a* is true by definition *D* iff there is a defining condition *A* in *D* of a true by *D.*

This is the basic axiom of definitional theory.

Take the Russell set *S(λ(x* ∈ *x* → *F ))* (let us call it *r* ) and let *R* be a set theory

that includes this set. The set *r* is not (left) absolute in *R*. *r* → *F* is true in *R*, that is *F*

follows from *r* in *R*. But whereas *r* is true in *R*, *F* is obviously not since it is not even

defined in *R*. The argument follows from the basic definitional axiom together with an assumption that the local logic of the definition has a reasonable behavior with respect to the intended interpretation of involved logical constants. This argument demonstrates that negation is not an absolute notion, which from a proof-theoretic point of view would be a reasonable way to interpret the Russell paradox, i.e., falsity is an absolute notion, while negation is not.

This notion of absoluteness can further be specialized as follows: A defining condition *A* is

1. *(right) absolute* (with respect to *S*) if

*A* follows from *B* in *T SΦ* ⇐⇒ *(B* is true in *T SΦ* =⇒ *A* is true in *T SΦ),*

2. *upward absolute* if

*B* follows from *A* in *T SΦ* =⇒ *( A* is true in *T SΦ* =⇒ *B* is true in *T SΦ),*

3. *downward absolute* if

*A* is true in *T SΦ* =⇒ *(B* is true in *T SΦ* =⇒ *B* follows from *A* in *T SΦ),*

4. etc.

To say that a defining condition, or a set, is (left/right) absolute means that the condition, or set, with respect to local reasoning has the same meaning inside the local logic as outside it.