Axioms - GNU Aris Manual
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### 7.1 Axioms

Sequence Logic, often abbreviated 'seqlog', uses the following six axioms:

- VxVy(~s(x) = z(y))
- VxVy(s(x) = s(y) → x = y)
- Vx(v(S,x) = f_S(x))
- Vx(v(n,z(x)) = 0)
- Vx(~v(n,x) = \0)
- VxVy(v(n,x) = v(n,y) → x = y)

The first axiom states that no sucessor is the zero object, or, to put it differently, that the zero object is the first object. The second axiom states that no two different objects have the same sucessor. Using these two axioms, a 'Universal Sequence' can be defined, in a way similar to how the Peano Axioms define the natural numbers. The third axiom is the definition of a sequence, stating that the value under a given sequence 'S' of every object 'x' can be determined by a function. The rule 'sq' introduces such a sequence (see sq).

The fourth axiom defines zero, and introduces the natural numbers. It states that the first element in the natural numbers is zero. The fifth axiom defines the null object, and states that the natural numbers are defined on all inputs. The sixth axiom states that the natural numbers are one-to-one, or that if two numbers are equal, then the objects that they represent are equal.