Addition
The theory of the natural numbers with a successor function has signature consisting of a constant 0 and a unary function S ("successor": S(x) is interpreted as x+1), and has axioms:
- ∀x ¬ Sx = 0
- ∀x∀y Sx = Sy → x = y
- Let P(x) be a first-order formula with a single free variable x. Then the following formula is an axiom:
- (P(0) ∧ ∀x(P(x)→P(Sx))) → ∀y P(y).
The last axiom (induction) can be replaced by the axioms
- For each integer n>0, the axiom ∀x SSS...Sx ≠ x (with n copies of S)
- ∀x ¬ x = 0 → ∃y Sy = x
The theory of the natural numbers with a successor function is complete and decidable, and is κ-categorical for uncountable κ but not for countable κ.
Presburger arithmetic is the theory of the natural numbers under addition, with signature consisting of a constant 0, a unary function S, and a binary function +. It is complete and decidable. The axioms are
- ∀x ¬ Sx = 0
- ∀x∀y Sx = Sy → x = y
- ∀x x + 0 = x
- ∀x∀y x + Sy = S(x + y)
- Let P(x) be a first-order formula with a single free variable x. Then the following formula is an axiom:
- (P(0) ∧ ∀x(P(x)→P(Sx))) → ∀y P(y).
Read more about this topic: List Of First-order Theories
Famous quotes containing the word addition:
“But the best read naturalist who lends an entire and devout attention to truth, will see that there remains much to learn of his relation to the world, and that it is not to be learned by any addition or subtraction or other comparison of known quantities, but is arrived at by untaught sallies of the spirit, by a continual self-recovery, and by entire humility.”
—Ralph Waldo Emerson (18031882)
“The most important American addition to the World Experience was the simple surprising fact of America. We have helped prepare mankind for all its later surprises.”
—Daniel J. Boorstin (b. 1914)
“Depend upon it there comes a time when for every addition of knowledge you forget something that you knew before. It is of the highest importance, therefore, not to have useless facts elbowing out the useful ones.”
—Sir Arthur Conan Doyle (18591930)