Myhill's Constructive Set Theory
The subject was begun by John Myhill to provide a formal foundation for Errett Bishop's program of constructive mathematics. As he presented it, Myhill's system CST is a constructive first-order logic with three sorts: natural numbers, functions, and sets. The system is:
- Constructive first-order predicate logic with identity, and basic axioms related to the three sorts.
- The usual Peano axioms for natural numbers.
- The usual axiom of extensionality for sets, as well as one for functions, and the usual axiom of union.
- A form of the axiom of infinity asserting that the collection of natural numbers (for which he introduces a constant N) is in fact a set.
- Axioms asserting that the domain and range of a function are both sets. Additionally, an axiom of non-choice asserts the existence of a choice function in cases where the choice is already made. Together these act like the usual replacement axiom in classical set theory.
- The axiom of exponentiation, asserting that for any two sets, there is a third set which contains all (and only) the functions whose domain is the first set, and whose range is the second set. This is a greatly weakened form of the axiom of power set in classical set theory, to which Myhill, among others, objected on the grounds of its impredicativity.
- The axiom of restricted, or predicative, separation, which is a weakened form of the separation axiom in classical set theory, requiring that any quantifications be bounded to another set.
- An axiom of dependent choice, which is much weaker than the usual axiom of choice.
Read more about this topic: Constructive Set Theory
Famous quotes containing the words constructive, set and/or theory:
“Euphemisms are not, as many young people think, useless verbiage for that which can and should be said bluntly; they are like secret agents on a delicate mission, they must airily pass by a stinking mess with barely so much as a nod of the head, make their point of constructive criticism and continue on in calm forbearance. Euphemisms are unpleasant truths wearing diplomatic cologne.”
—Quentin Crisp (b. 1908)
“The splendor falls on castle walls
And snowy summits old in story;
The long light shakes across the lakes,
And the wild cataract leaps in glory.
Blow, bugle, blow, set the wild echoes flying,
Blow, bugle; answer, echoes, dying, dying, dying.”
—Alfred Tennyson (1809–1892)
“every subjective phenomenon is essentially connected with a single point of view, and it seems inevitable that an objective, physical theory will abandon that point of view.”
—Thomas Nagel (b. 1938)