Home Philosophy Advances in Proof-Theoretic Semantics
If we think of set definitions as abstractions λX , saying that a property, or functional expression, X defines a set, we may derive the following definitions of membership and equality for sets:
• A ∈ λX iff X ( A),
• A = B iff (x ∈ A ⇐⇒ x ∈ B) for all sets x
(i.e., ( A = λX & B = λY =⇒ λX = λY ) ⇐⇒ (X (x ) ⇐⇒ Y (x )) for all
sets x ).
In the same manner the axiomatic approach, ZF and other similar set theories, introduce axioms stating the existence of sets for certain specific safe defining properties, such as for example the subset property
x ∈ P( A) iff x is a subset of A
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 safe axioms, but also from a more general intensional perspective a lack of elementary foundational principles. There is a very elementary and suggestive extensional picture through the cumulative hierarchy, but this is lacking with respect to definitional issues.
Why is (x = x ), for example, not an admissible set defining condition?
1. It contradicts the idea of sets as collections of given objects, i.e., λ(x = x ) is a member of λ(x = x ).
2. We cannot comprehend the given objects we are supposed to collect into a whole by abstraction.
In both cases we say that (x = x ) does not define a set in the sense of a total
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 λ(x = x ) may define it is not a set in the extensional sense as a
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 .
Let us look at a very naïve and simplistic attempt to define sets based on the idea of sets as introduced by abstraction of defining properties. In defining sets this way it is natural to make a distinction between set expressions, i.e., sets, terms etc., and propositional expressions, i.e., propositions, formulas etc. But if we accept more open definitions this does not seem necessary, and for reasons of simplicity we will just make a distinction between sets (A) and set theoretical reasoning (B) in what follows. This would also be in line with reading Ockham's razor as saying that basic classifications and distinctions are matters of proofs and not foundational definitions. The definition of sets is:
• T and F are sets,
• A → B, A ∈ B, A = B are sets if A and B are sets,
• S( f ), ∃( f ) and ∀( f ) are sets if the world of sets is closed under the given func-
tion f .
This would answer question (A). To answer question (B) we add the following derived
• T is true,
• A → B is true if ( A is true =⇒ B is true),
• A ∈ S( f ) is true if f ( A) is true,
• A = B is true if (x ∈ A is true ⇐⇒ x ∈ B is true) for all sets x ,
• ∃( f ) is true if f (x ) is true for some set x ,
• ∀( f ) is true if f (x ) is true for all sets x .
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?