Section 1: Philosophy, logic and constructivity
Philosophy, formal logic and the theory of computation all bear on problems in the
foundations of constructive mathematics. There are few places where these, often competing, disciplines converge more neatly than in the theory of realizability structures.
Uealizability applies recursion-theoretic concepts to give interpretations of constructivism
along lines suggested originally by Heyting and Kleene. The research reported in the
dissertation revives the original insights of Kleene—by which realizability structures are
viewed as models rather than proof-theoretic interpretations—to solve a major problem of
classification and to draw mathematical consequences from its solution.
Section 2: Intuitionism and recursion: the problem of classification
The internal structure of constructivism presents an interesting problem. Mathematically, it is a problem of classification; for philosophy, it is one of conceptual organization.
Within the past seventy years, constructive mathematics has grown into a jungle of fullydeveloped
"constructivities," approaches to the mathematics of the calculable which range
from strict finitism through hyperarithmetic model theory. The problem we address is taxonomic:
to sort through the jungle, set standards for classification and determine those
features which run through everything that is properly "constructive."
There are two notable approaches to constructivity; these must appear prominently in
any proposed classification. The most famous is Brouwer's intuitioniam. Intuitionism relies
on a complete constructivization of the basic mathematical objects and logical operations.
The other is classical recursive mathematics, as represented by the work of Dekker, Myhill,
and Nerode. Classical constructivists use standard logic in a mathematical universe
restricted to coded objects and recursive operations.
The theorems of the dissertation give a precise answer to the classification problem for
intuitionism and classical constructivism. Between these realms arc connected semantically
through a model of intuitionistic set theory. The intuitionistic set theory IZF encompasses
all of the intuitionistic mathematics that does not involve choice sequences. (This includes
all the work of the Bishop school.) IZF has as a model a recursion-theoretic structure,
V(A7), based on Kleene realizability. Since realizability takes set variables to range over
"effective" objects, large parts of classical constructivism appear over the model as inter¬
preted subsystems of intuitionistic set theory. For example, the entire first-order classical
theory of recursive cardinals and ordinals comes out as an intuitionistic theory of cardinals
and ordinals under realizability. In brief, we prove that a satisfactory partial solution to
the classification problem exists; theories in classical recursive constructivism are identical,
under a natural interpretation, to intuitionistic theories. The interpretation is especially
satisfactory because it is not a Godel-style translation; the interpretation can be developed
so that it leaves the classical logical forms unchanged.
Section 3: Mathematical applications of the translation:
The solution to the classification problem is a bridge capable of carrying two-way
mathematical traffic. In one direction, an identification of classical constructivism with intuitionism yields a certain elimination of recursion theory from the standard mathematical
theory of effective structures, leaving pure set theory and a bit of model theory. Not only
are the theorems of classical effective mathematics faithfully represented in intuitionistic
set theory, but also the arguments that provide proofs of those theorems. Via realizability,
one can find set-theoretic proofs of many effective results, and the set-theoretic proofs are
often more straightforward than their recursion-theoretic counterparts. The new proofs
are also more transparent, because they involve, rather than recursion theory plus set
theory, at most the set-theoretic "axioms" of effective mathematics.
Working the other way, many of the negative ("cannot be obtained recursively") results of classical constructivism carry over immediately into strong independence results
from intuitionism. The theorems of Kalantari and Retzlaff on effective topology, for instance, turn into independence proofs concerning the structure of the usual topology on
the intuitionistic reals.
The realizability methods that shed so much light over recursive set theory can be
applied to "recursive theories" generally. We devote a chapter to verifying that the realizability techniques can be used to good effect in the semantical foundations of computer
science. The classical theory of effectively given computational domains a la Scott can
be subsumed into the Kleene realizability universe as a species of countable noneffective
domains. In this way, the theory of effective domains becomes a chapter (under interpre¬
tation) in an intuitionistic study of denotational semantics. We then show how the "extra
information" captured in the logical signs under realizability can be used to give proofs of
classical theorems about effective domains.
Section 4: Solutions to metamathematical problems:
The realizability model for set theory is very tractible; in many ways, it resembles
a Boolean-valued universe. The tractibility is apparent in the solutions it offers to a
number of open problems in the metamathematics of constructivity. First, there is the
perennial problem of finding and delimiting in the wide constructive universe those features
that correspond to structures familiar from classical mathematics. In the realizability
model, it is easy to locate the collection of classical ordinals and to show that they form,
intuitionistically, a set rather than a proper class. Also, one interprets an argument of
Dekker and Myhill to prove that the classical powerset of the natural numbers contains at
least continuum-many distinct cardinals.
Second, a major tenet of Bishop's program for constructivity has been that constructive mathematics is "numerical:" all the properties of constructive objects, including
the real numbers, can be represented as properties of the natural numbers. The realizability model shows that Bishop's numericalization of mathematics can, in principle, be
accomplished. Every set over the model with decidable equality and every metric space is
enumerated by a collection of natural numbers.