Lectures on Constructive Set Theory

(Padua, Spring 1998)

Lecture 11: 5th June (16.30 - 18.00)

See Lecture 10

The Arithmetization of Analysis

In this lecture I described the arithmetization of Analysis in Constructive Set Theory. I described (in outline only) set theoretic constructions first of the class Z of integers from the class N of integers, then the class Q of rational numbers from the class Z. Of course there are many possible ways that Z and Q could have been defined. What is important is the mathematical structures that can be constructed on these classes and axiomatic characterisations of those structures. In each case of N, Z and Q one can put a structure on these classes and give an axiom system of which the structure is a model that is unique up to isomorphism, and in fact is unique up to a unique isomorphism. We call such an axiom system a rigidly categorical axiom system.

The axiom system for N

Recall from the previous lecture that the class N was defined to be the smallest class such that 0 is in N and if a is in N then so is a^+. Here 0 is the empty set and a^+ is the set [a union {a}] for any set a.
Primitive Recursion Theorem[Dedekind]
Let A be a class and let a0 be in A and F:NxA->A. Then there is a uniqu H:N->A such that
  1. H(0)=a0,
  2. H(n^+)=F(n,H(n)) for all n in N.
Definition
The class A, with a0 and S is a Dedekind structure if
  1. a0 is in A,
  2. S:A->A,
  3. not [S(a)=a0] for a in A,
  4. [S(a1)=S(a2) implies a1=a2] for all a1,a2 in A,
  5. A is the smallest subclass X of A such that a0 is in A and S(a) is in A whenever a is in A.
These axioms for a Dedekind structure are usually called the Peano Axioms for the natural numbers. The basic facts about N are
  1. The class N exists.
  2. N, with 0 and S, where S(n)=n^+ for n in N, forms a Dedekind structure.
  3. The above Dedekind structure on N is unique up-to-isomorphism. Moreover the isomorphism is unique.
To be continued ....

See Lecture 12

Draft. Last Revised: 3rd June, 1998.