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:
“The measure discriminates definitely against products which make up what has been universally considered a program of safe farming. The bill upholds as ideals of American farming the men who grow cotton, corn, rice, swine, tobacco, or wheat and nothing else. These are to be given special favors at the expense of the farmer who has toiled for years to build up a constructive farming enterprise to include a variety of crops and livestock.”
—Calvin Coolidge (1872–1933)
“The people always have some champion whom they set over them and nurse into greatness.... This and no other is the root from which a tyrant springs; when he first appears he is a protector.”
—Plato (c. 427–347 B.C.)
“There is in him, hidden deep-down, a great instinctive artist, and hence the makings of an aristocrat. In his muddled way, held back by the manacles of his race and time, and his steps made uncertain by a guiding theory which too often eludes his own comprehension, he yet manages to produce works of unquestionable beauty and authority, and to interpret life in a manner that is poignant and illuminating.”
—H.L. (Henry Lewis)