# ∀ x ∀ y : x * (y + 1) = x * y + y |
# ∀ x ∀ y : x * (y + 1) = x * y + x |
Like any logical theory, first-order calculus consists of
There are two types of axioms: the logical axioms which embody the general truths about proper reasoning involving quantified statements, and the axioms describing the subject matter at hand, for instance axioms describing sets in set theory or axioms describing numbers in arithmetic.
While the set of inference rules in first-order calculus is finite, the set of axioms may very well be and often is infinite. However we require that there is a general algorithm which can decide for a given well-formed formula whether it is an axiom or not. Furthermore, there should be an algorithm which can decide whether a given application of an inference rule is correct or not.
The well-formed formulas contain
The object, predicate and function constants will typically depend on the particular domain we are talking about.
Instead of giving the lengthy definition of well-formed formulas, we will simply look at some examples from arithmetic together with their ordinary interpretation. Our domain here is the set of natural numbers:
Now one would have to write down 15 logical axioms and 2 inference rules to completely specify the first-order calculus. How can one be sure that those axioms and rules are enough? This is the subject of Gödel's completeness theorem: if you start out with some subject matter axioms and you look at a certain statement, then it is possible to prove that statement using only the subject matter axioms, the 15 logical axioms and the two inference rules if and only if the statement is true in every domain in which the subject matter axioms are true.
The [Peano axioms]? for the natural numbers are sometimes formulated as second-order statements (the induction axiom talks about all "properties" or all "sets of numbers"), but this is not necessary if one is willing to allow infinitely many first-order axioms. A first-order version of the Peano axioms follows.
We are using the object constants 0 and 1, the function constants + and *, and the predicate constant =. Here are the axioms:
Axioms 1, 2 and 7 are the traditional Peano axioms while axioms 3-6 serve to define addition and multiplication. If one omits the function constant * and axioms 5 and 6 and allows in scheme 7 only formulas without *, then one gets the very interesting Presburger arithmetic.