Home Philosophy Advances in Proof-Theoretic Semantics

## Defining SetsIf we think of set definitions as abstractions • • (i.e., sets In the same manner the axiomatic approach,
but also other types of axioms such as axioms introducing measurable cardinals and other large cardinals. Although the axioms of power set and replacement, together with axioms of infinity (large cardinals starting with ℵ0), provide for strong means to build sets following the cumulative hierarchy intuition of the universe of sets, they still represent a theory marked by withdrawal from foundational disasters to more favourable positions. It is not only matters of a first order formalization of Why is 1. It contradicts the idea of sets as collections of given objects, i.e., 2. We cannot comprehend the given objects we are supposed to collect into a whole by abstraction. In both cases we say that object that behaves nicely with respect to the intended reading of logical constants and the notion of membership. But this does not really answer the question. It just says that whatever collection of given objects. The problem here is an example of what we in many cases meet as we try to define a notion where it is difficult to map out the exact borders by elementary means, the notion of a total computable function being a canonical example. From a foundational and theoretical point of view it would be nice if it were possible to make sense in some way of the initial, and very elementary, ideas of Frege and others [4]. Let us look at a very naïve and simplistic attempt to • • • tion This would answer question (A). To answer question (B) we add the following definition: • • • • • ∃ • ∀ Russell's paradox tells us of course directly that there are no such definitions satisfying the intended closure properties we have written down above. But from an intensional, i.e., definitional, point of view, we actually intend to define something by writing down these clauses. The question is just what that is, in what ways we can interpret these acts of defining? |

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