Interface:First-order field axioms

A field has two operations, addition and multiplication, which obey the familiar properties shown below. Here we present a first-order theory of fields (that is, all variables indicate an element of the field, and there is no general way to form sets of elements or variables which represent elements of some structure other than the field). This is as opposed to building the field within set theory, where the field consists of a set and addition and multiplication are relations on that set.

Our first-order theory of fields is taken from a book by Margaris.

We start by importing propositional and predicate logic.

 param (PROPOSITIONAL Interface:Classical_propositional_calculus ) param (FIRSTORDER Interface:First-order_logic_with_quantifiability (PROPOSITIONAL) ) 

Addition
There is an addition operation, which obeys the usual equality axioms.  var (object x y z x0 y0 z0 x1 y1 z1) var (variable v v0 v1 v2)

term (object (+ object object)) stmt (AdditionBuilder  (((x0 = x1) ∧ (y0 = y1)) → ((x0 + y0) = (x1 + y1)))) 

It is associative and commutative.  stmt (AdditionAssociativity  (((x + y) + z) = (x + (y + z)))) stmt (AdditionCommutativity  ((x + y) = (y + x))) 

There is an additive identity called.  term (object (0)) stmt (AdditiveIdentity  ((x + (0)) = x)) 

Every element has an additive inverse.  stmt (AdditiveInverse ((x v)) (∃ v ((x + (value v)) = (0)))) 

Terms versus variables
An astute reader comparing the axioms above with the ones in Margaris will notice that the above ones are stated in terms of terms rather than variables. For example, Margaris' states commutativity as  (where   and   are variables), whereas we state it above as , where   and   are terms. The axioms using variables would suffice; we could convert them to their term counterparts using  combined with the equality axioms. We state the axioms using terms merely to avoid a number of repetitive proofs of this sort.

A similar rationale applies to the distinct variable constraints. In a work like Margaris, the axioms are stated using variables and it is implicitly true that variables which are given different letters cannot refer to the same variable.

Multiplication
There is a multiplication operation, which obeys the usual equality axioms.  term (object (· object object)) stmt (MultiplicationBuilder  (((x0 = x1) ∧ (y0 = y1)) → ((x0 · y0) = (x1 · y1)))) 

It is associative and commutative.  stmt (MultiplicationAssociativity  (((x · y) · z) = (x · (y · z)))) stmt (MultiplicationCommutativity  ((x · y) = (y · x))) 

There is a multiplicative identity called.  term (object (1)) stmt (MultiplicativeIdentity  ((x · (1)) = x)) 

Every nonzero element has a multiplicative inverse.  stmt (MultiplicativeInverse ((x v)) ((x ≠ (0)) → (∃ v ((x · (value v)) = (1))))) </jh>

Distributivity
It is not enough to merely have addition and multiplication operations which obey the above axioms. Addition and multiplication also need to relate to each other. Specifically, multiplication must distribute over addition.  stmt (Distributivity  ((x · (y + z)) = ((x · y) + (x · z)))) </jh>

Non-degenerative
The axioms presented so far allow for a situation where there is exactly one element, but we wish to exclude that case.  stmt (ZeroOne  ((0) ≠ (1))) </jh>

Cited works

 * Margaris, Angelo (1990), First Order Mathematical Logic, ISBN 978-0-486-66269-5