On Existence in Set Theory

On Existence in Set Theory, Part II: Relative Productivity

“…Although there is general agreement between the standard view and the results obtained in [1], there is also at least one point of divergence: the extensionality axiom. According to the standard view, extensionality has no existential import.…”

“…According to the standard view, extensionality has no existential import. The results presented in [1] show a different picture: the existential power of extensionality is indeed very weak, but it is not zero. The transitivity of a domain suffices to fulfill the existential demands of extensionality, but it is also required: if the domain is not transitive then extensionality may not hold.…”

“…The following examples provide some proofs of (mathematically) interesting existential statements in which extensionality is used. (The first two were already hinted at in [1]. )…”

“…The analysis of existence assertions in Part I (see [1]) is founded on the notion of productivity: an existence assertion is either productive or admits unconditional degree 0. The productivity of a (valid) sentence is, roughly, its power of producing sets.…”

The Ontological Import of Adding Proper Classes

“…This property represents how an axiom generates ontological import. A detailed analysis of this conception leads to an understanding of existential aspects in formal theories (see [4]). But it is of no help if we want to compare different theories regarding ontology.…”