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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 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(xxF )) (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. rF 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.

 
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