Field Axioms in Constructive Mathematics

• Field Axioms are defined as algebraic laws derived from order completeness and a dense rational embedding in the real numbers.
• Addition and multiplication are constructed via suprema and infima of rational-bounded sets, ensuring properties like continuity and associativity.
• Constructive proofs avoid classical principles such as the law of excluded middle, establishing a Heyting field rather than a classical field.

A field is a mathematical structure consisting of a set equipped with two binary operations, addition and multiplication, satisfying a specific collection of algebraic laws known as the field axioms. In constructive mathematics, the derivation and status of these axioms can diverge significantly from classical treatments. The precise constructive development emphasizes order, completeness, and density properties as primitive, with field properties subsequently derived. Recent constructive axiomatizations, such as that of Jean S. Joseph, reconstruct the operations and algebraic properties of $\mathbb{R}$ systematically from minimal order-theoretic data and the completeness of certain subsets, eschewing any use of the law of excluded middle or trichotomy (Joseph, 2018).

1. Order-Theoretic Base Axioms

The foundation for constructing a field structure on $\mathbb{R}$ relies first on an axiomatic ordered set $(R,<)$ equipped with the following properties for all $x, y, z \in R$:

• Asymmetry: $x < y \implies \neg(y < x)$
• Cotransitivity: $x < y \implies (x < z \lor z < y)$
• Negative Antisymmetry: $(\neg(x < y) \wedge \neg(y < x)) \implies x = y$

The symbol $x \leq y$ abbreviates $\neg(y < x)$.

The definition of upper order located and supable subsets underpins constructive notions of completeness. A set $S \subseteq R$ is upper order located if, for all $x < y$, either $x < s$ for some $s \in S$ or some upper bound $u$ of $S$ satisfies $u < y$. $S$ is supable if non-empty, bounded above, and upper order located.

The Completeness Axiom further posits that every supable subset $S$ has a least upper bound in $R$.

Embedding of the rationals $\mathbb{Q}$ is included as an order-embedding $i: Q \to R$ such that $i(Q)$ is almost dense ($$x < y \implies \exists q, q' \in Q: x \leq q < q' \leq y$$) and bicofinal ($$\forall x \in R, \exists q, q' \in Q: q \leq x \leq q'$$). The rational field $(Q, +, 0, -, \cdot, 1, <)$ is assumed to be Archimedean and endowed with the usual analytical structure; multiplication is continuous in the order topology.

2. Constructive Operations: Addition and Multiplication

Addition on $R$ is constructed by extending the abelian group structure of $Q$:

• For $x \in R$, define $L_x := \{ a \in Q : a \leq x \}$.
• The sum $x + y$ is defined by $x + y := \sup(L_x + L_y)$, where $L_x + L_y := \{ a + b : a \in L_x, b \in L_y \}$.
• For $x \in R$, the additive inverse is defined by $-x := \inf \{ -a : a \in L_x \}$.

Multiplication leverages rational bounds:

• For $x, y \in R$, set $$P_{x, y} := \{ \min(aa', ab', ba', bb') : a, b, a', b' \in Q; a \leq x \leq b, a' \leq y \leq b' \}$$.
• The product $x \cdot y$ is $x \cdot y := \sup P_{x, y}$.

Constructive proofs for these operations use the supability of relevant subsets, the density of $Q$ in $R$, and order properties, circumventing any appeal to excluded-middle arguments.

3. Derivation of the Field Axioms

The field axioms, including associativity, commutativity, distributivity, and the existence of neutral elements and inverses, are not assumed but derived as theorems using the operations defined above:

• $(R, +)$ forms an abelian group: associativity, commutativity, the neutral element $0_R$, and additive inverses $-x$ exist and are constructed.
• $(R \setminus \{0\}, \cdot)$ is an abelian group: associativity, commutativity, neutral element $1_R$, and multiplicative inverses for nonzero elements are all established.
• Distributive law is verified constructively via continuity: the maps $x \cdot (y + z)$ and $x \cdot y + x \cdot z$ agree on dense subsets and, by continuity, on all of $R^3$.
• The positivity law ($0 < x$ and $0 < y$ imply $0 < xy$) is also derived.
• $0 \neq 1$ follows from the properties of the rational embedding.

The resulting structure on $\mathbb{R}$ is a Heyting field (a field with tight apartness), not just an ordinary field, due to the constructive setting (Joseph, 2018).

4. Constructive Proof Techniques

Constructive approaches avoid classical principles such as the law of excluded middle and trichotomy. Instead, the following techniques supplant classical reasoning:

• Supremum and infimum of supable or infable subsets define operations and inverses.
• Cotransitivity replaces trichotomy: order comparisons are managed without assuming strict total comparability.
• Negative antisymmetry provides equality criteria in the absence of classical dichotomies.
• Density and bicofinality of $\mathbb{Q}$ in $\mathbb{R}$ ensure that arbitrary real numbers can be approximated by rationals, facilitating constructive arguments.
• Continuity of rational addition and multiplication allows extension of field laws to all of $R$ via density arguments.

5. Comparison with Classical Field Axiomatizations

Classical ZF-based constructions, such as those via Dedekind cuts or Cauchy sequences, typically take field axioms as primitive for $\mathbb{R}$, and completeness as an additional axiom or property. In the constructive framework detailed above, only order-theoretic completeness and dense rational embedding are taken as primitive. All algebraic field axioms are derived subsequently.

This constructive approach eliminates the need for the law of excluded middle, ensuring validity in settings such as any topos or constructive set theory, whereas classical proofs often rely essentially on case distinctions and trichotomy. Furthermore, operations on $R$ are defined via universal properties (suprema of rational-bounded sets) rather than as equivalence classes of sequences or cuts (Joseph, 2018).

A consequence is a fully constructive, sup-based axiomatization of the real field, reconstructing all standard field properties from a minimal order-completion framework.

6. Summary Table: Constructive Field Laws for $\mathbb{R}$

Property	Constructive Basis	Method of Derivation
Abelian group under $+$	Order-completeness, $\mathbb{Q}$-embedding	Suprema/infima of rational sums
Abelian group under $\cdot$ on $R{\setminus}\{0\}$	Local boundedness, continuity	Suprema of $P_{x,y}$-sets
Distributivity	Continuity, density	Density plus continuity argument
Positivity	Constructive order	Properties of $Q$ and their transfer
$0 \neq 1$	Rational structure	Inherited from $Q$

These derived laws ensure that $\mathbb{R}$ forms a Heyting field on a constructive foundation, with each property following from completeness, embedding, and rational approximation, rather than being postulated as axioms (Joseph, 2018).