Wednesday, July 23, 2008

Formalizing predicate boundedness

Informally, we defined a predicate F to be bounded if the entities for which F is true can be enclosed in a reasonably "small" class. Of course, for this definition to be of any use we have to clearly specify what we mean by "small", or at least use our intuition to propose some axioms for this concept:

  • If A and B are small, A U B is small.
  • If A and B are small, AB is small.
  • If A is small and B is included in A, B is small.
  • If A is small then −A, the class of all entities outside A, is not small.

In the definition of −A given above we have assumed that we are working inside some universal class comprising any conceivable entity.

The axioms for "small" are far from categorical. We give a few models for this notion:

  • If we decide to work within a mathematical class theory such as NBG, there is an easy definition for "small": A class is small if it is a set (not all classes are sets).
  • We can take "small" as meaning "finite", provided the universal class is infinite, or as having cardinality less than or equal to some fixed infinite cardinal κ, provided the universal class is larger than κ.
  • Suppose we have a fixed set of "primitive" predicates Fi with domains Di such that UDi is smaller than the universal class. We can define the class of small sets as the least class S comprising all Di that is closed under finite union, finite intersection and comprehension: if X belongs to S and Y is a subset of X, Y belongs to S. This definition of "small" can be seen as a formal model for Quine's notion of natural kinds.

This is probably as far as we can get with respect to defining what "small" means in the context of predicate boundedness.

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