Let δ denote the primitive act of distinction. We identify the canonical arithmetic generated by iterated distinction, construct its choice-free number tower through the rationals, and determine the exact metatheoretic logical cost of the recognition quotients of its additive monoid.

The ambient metatheory provides propositions, equality, and collections of elements, which we call carriers. The ambient metatheory is intuitionistic. It provides function types, finite products, inductively generated carriers with their recursion and induction principles, quotients by equivalence relations, and the natural numbers ℕ , which also index the de Bruijn variables (Section 3). Equality and order on ℕ are decidable. We work in a fixed universe Type u . A carrier is a type K : Type u , and a predicate on K is a map P : K → Prop . Equality is the ambient equality, and quotients are formed by equivalence relations on carriers. In Theorem 2.5, induction applies to all such predicates, including the image predicate used for surjectivity. We do not use the law of excluded middle, the limited principle of omniscience, Markov’s principle, or the axiom of choice in the metatheory, except where explicitly assumed, as in Section 7. The ledger (Section 3.1) records principles used inside the δ -calculus.

The purpose of this construction is not to obtain another copy of the natural numbers, since the metatheory already contains ℕ . The generated δ -orbit is instead the canonical object obtained from the empty record by repeated one-step extension. By Theorem 1.6, it is initial among δ -algebras, and the Peano conditions characterize it up to a unique isomorphism. Thus the resulting arithmetic is unique up to isomorphism.

Each element of the orbit records a finite number of successive distinction steps, and the later stages of the number tower are constructed explicitly from the generated carrier by group completion and fraction construction.

The main question is therefore not what the natural numbers are, but what can be constructed from the generated carrier and at what logical price. For the recognition quotients of ( ℕ δ , + , 0 ) , this price is exact: a decidable congruence together with an explicit pair of distinct related elements implies the classification without additional nonconstructive principles; decidability alone requires Markov’s principle; and an arbitrary congruence requires excluded middle.

Definition 1.1.Let K be a carrier. Write

A proof h : Dist ( K ) is called a distinction witness on K .

The predicate Dist ( K ) uses no additional structure on K beyond the equality of the ambient metatheory. A carrier K is called nontrivial if Dist ( K ) holds. Dist ( K ) gives two unequal elements, but it gives no operation which can be iterated. The predicate is not used to generate the δ -orbit. The symbol δ is only the name of the one-step extension r ↦ S r in Definition 1.2. A distinction witness does not produce 0, S , or induction.

The word distinction has formal and philosophical uses [1][2]. Spencer-Brown’s Laws of Form [1] develops a calculus from a mark of distinction. Our construction is different. The primitive is not a mark in a plane, but the one-step extension of a finite record. The generated object is the δ -orbit. The condensation rules of [1] absorb repeated marking, whereas each application of S produces a further stage of the generated record. Successor nonzero, successor injectivity, and induction then give the natural-number object.

A related physical idea appears in Wheeler’s thesis it from bit, where particles, fields, and spacetime receive their physical meaning from elementary yes/no registrations [3]. Since each such registration presupposes two distinguishable alternatives, here we study the prior mathematical step: the finite iteration of distinction. We do not use the calculus of [1], and the scholastic doctrine of formal distinction [2] is not part of the construction.

In constructive algebra, a positive apartness relation is often used in place of negative inequality [4]-[6]. No apartness relation is assumed here. The ambient host is an intuitionistic type theory in the sense of Martin-Löf [7], close to the Calculus of Constructions [8]. We do not introduce a new type theory. Constructive reverse mathematics [9][10] asks which principles suffice for a theorem, the ledger of Section 3.1 instead records what a given derivation uses.

Definition 1.2.The δ -orbit Orb ( 0 , S ) is the collection of finite records generated by two formation rules:

here 0 is the empty record, and S r is the record obtained from r by one further act of distinction.

Proposition 1.3.For all r , q ∈ Orb ( 0 , S ) ,

Moreover,if a predicate P on Orb ( 0 , S ) satisfies P ( 0 ) and

P ( r ) → P ( S r ) for every r ,

then P ( r ) holds for every r ∈ Orb ( 0 , S ) .

Proof. The orbit is inductively generated, so it carries the recursion principle: for every carrier K , every k ∈ K and every f : Orb ( 0 , S ) × K → K there is a map g : Orb ( 0 , S ) → K with g ( 0 ) = k and g ( S r ) = f ( r , g ( r ) ) .

For S r ≠ 0 , apply the recursion principle to K = B = { 0 , 1 } , with k = 0 and f ( r , z ) = 1 . Then g ( 0 ) = 0 and g ( S r ) = 1 , so S r = 0 would imply 0 = 1 .

For injectivity, let us define pred : Orb ( 0 , S ) → Orb ( 0 , S ) by recursion with k = 0 and f ( r , z ) = r , so that pred ( S r ) = r for every r . Hence S r = S q gives

The last statement is the structural induction principle of the inductively generated orbit. □

The orbit is not defined as the subset { S n 0 : n ∈ ℕ } of a pre-existing carrier, so its definition does not use a prior natural-number object as an index. The notation S n 0 is used only metatheoretically, for the result of applying S n times to 0. The successor laws and the induction principle for the generated orbit are proved in Proposition 1.3.

The rigidity theorem (Theorem 1.6) reverses the usual direction of explanation. The generated δ -orbit is not merely another model of the Peano conditions. It is the object those conditions describe. Initiality sends the orbit canonically into every δ -algebra. When the successor map is injective and the base point is not a successor, this canonical map is an embedding, so every such faithful realization already contains a copy of the orbit. Induction makes the map surjective, and initiality makes the resulting isomorphism unique. The Peano conditions do not create an arithmetic structure by stipulation. They characterize, up to a unique structure-preserving relabeling, the arithmetic generated by repeated distinction. This is a representation-invariance statement within an ambient metatheory supporting inductive generation.

Initiality and Peano categoricity are classical [11]. Here we combine them with distinction-based generation, the choice-free number tower, and the exact metatheoretic logical prices of recognition quotients.

The δ -calculus is an intuitionistic first-order proof system over the signature { 0 , S , + , ⋅ } , defined in Section 3. The symbol δ is not part of the formal language. In particular, δ is not the δ -rule of the λ -calculus [12]. The calculus is called the δ -calculus because its symbols 0 and S , the distinction axioms, and the induction rule come from the generated δ -orbit. In the arithmetic projection (Definition 5.1), the empty record is written as 0, and the one-step extension of a record is written by the successor symbol S . At the first-order level, the successor theorems become the distinction axioms

They are validated in the standard model N = ( ℕ , 0 , S , + , ⋅ ) . The operations + and ∙ are governed by their recursion equations.

Let ℕ δ denote the arithmetic structure carried by the generated δ -orbit, with + and ∙ defined by recursion. Its successor laws and induction are given by Proposition 1.3.

We fix the generation grammar consisting of the generation of ℕ δ from the primitive act δ , followed by group completion and fraction construction. The stages of the construction are

The first arrow denotes the inductive generation of the δ -orbit of finite records and the natural-number object presented by it. The next stage is the group completion

The map ι ℤ δ is the canonical embedding of ℕ δ into this quotient, given by a ↦ [ ( a , 0 ) ] .

The next stage is the fraction construction ℚ δ , presented by ratio orbits with signed-orbit numerators and nonzero denominators in ℕ δ , as defined in Section 5. The canonical embedding

is induced by adjoining denominator 1.

Thus the integer and rational stages are defined by their internal quotient relations. The comparison maps into the standard systems ℤ and ℚ are constructed after these quotients are formed. The construction is choice-free. In the δ -calculus, each derivation carries a ledger recording the nonconstructive principles used. The main results are verified in Lean 4 (Section 8).

Definition 1.4.For a carrier X ,write

Enc δ ( X ) : ⇔ there exists an injection X ↪ ℕ δ .

An object satisfying this predicate is called δ -encodable.

We prove that the metatheoretic number systems ℕ , ℤ , and ℚ are δ -encodable. In this paper, we stop the tower at ℚ δ . The real-number stage is not treated in this paper. Its construction gives a further constructive problem, since the Cauchy and Dedekind constructions are not always equivalent.

We now state the universal property of the generated δ -orbit. ℕ δ denotes the δ -orbit of Definition 1.2, with constructors 0 and S .

Definition 2.1.A δ -algebra is a triple A = ( | A | , 0 A , S A ) , where | A | is a carrier, 0 A ∈ | A | , and S A : | A | → | A | is a unary operation. A homomorphism f : A → B of δ -algebras is a map f : | A | → | B | such that

for every x ∈ | A | .

The orbit ( ℕ δ , 0 , S ) is a δ -algebra. Proposition 1.5 states the freeness of the term algebra over the signature { 0 , S , + , ⋅ } with variables. Here we study the generated δ -orbit as an algebra with operations 0 and S .

Theorem 2.2.For every δ -algebra A ,there exists a unique homomorphism rec A : ℕ δ → A .It is determined by

Proof. Existence follows from the recursion principle of the inductively generated δ -orbit.

Let f : ℕ δ → A be a homomorphism. We prove f ( n ) = rec A ( n ) by structural induction on n . For n = 0 , we have f ( 0 ) = 0 A = rec A ( 0 ) . If the equality holds for n , then

Hence f = rec A . □

Theorem 2.3.Let A be a δ -algebra.Suppose that S A is injective and 0 A ∉ im ( S A ) .Then the canonical homomorphism rec A : ℕ δ → A is injective.

Proof. We prove

by structural induction on a .

Let a = 0 . If b = 0 , the claim holds. If b = S b ′ , then

This contradicts the assumption 0 A ∉ i m ( S A ) .

Let a = S a ′ . The element b cannot be 0, since otherwise

Thus b = S b ′ for some b ′ . Then

Since S A is injective, rec A ( a ′ ) = rec A ( b ′ ) . The induction hypothesis gives a ′ = b ′ , and therefore a = b . □

Thus, if S A is injective and 0 A ∉ im ( S A ) , the canonical homomorphism

identifies ℕ δ with the sub- δ -algebra im ( rec A ) of A .

Definition 2.4.A δ -algebra A is a Peano realization if

1) S A is injective;

2) S A x ≠ 0 A for every x ∈ | A | ;

3) for every predicate P on | A | , if

for every x ∈ | A | , then P ( x ) holds for every x ∈ | A | .

Theorem 2.5.For every Peano realization A ,the canonical homomorphism rec A : ℕ δ → A is bijective.It is the unique δ -algebra isomorphism ℕ δ ≅ A .

Proof. Injectivity follows from Theorem 2.3. For surjectivity, let us consider the predicate

The base case holds with witness 0, since rec A ( 0 ) = 0 A . Suppose that P ( x ) holds. Let n ∈ ℕ δ satisfy

Then

Thus S n witnesses P ( S A x ) . Induction in A gives P ( x ) for every x ∈ | A | . Hence rec A is surjective.

The uniqueness of the isomorphism follows from Theorem 2.2. □

The predicate P used in the previous theorem is an ambient predicate. The third condition of Definition 2.4 applies to every such predicate, including the image predicate used for surjectivity.

Remark2.6.Theorem 2.5 is relative to an ambient metatheory supporting inductive generation. It shows that every Peano realization is uniquely isomorphic to ℕ δ (cf. [11]). For the standard Peano algebra

the canonical map rec A is the counting map D of Section 5. Hence Lemma 5.3 is the standard-model instance of Theorem 2.5.

Theorems 2.2, 2.3, and 2.5 together prove Theorem 1.6.

We now define the δ -calculus, a natural-deduction system for an intuitionistic first-order arithmetic with equality over the signature { 0 , S , + , ⋅ } . Each derivation has a ledger recording the use of the law of excluded middle (EM), the limited principle of omniscience (LPO), Markov’s principle (MP), and induction on formulas containing a quantifier. The ledger is defined in Section 3.1.

Terms are generated by de Bruijn variables, zero, successor, addition, and multiplication:

Under the arithmetic projection (Definition 5.1), the symbol 0 represents the empty record and S represents one further extension of a record.

Formulas are generated by the grammar of first-order logic with equality [16]:

The nonlogical vocabulary of the calculus is exactly 0 , S , + , ⋅ . Lifting and substitution are defined by the standard structural recursions for de Bruijn syntax. For every metatheoretic natural number n , the expression S n 0 is a closed term, called the numeral for n .

This section proves the universal property stated in Proposition 1.5. It also develops the arithmetic validated in the standard model.

Every term is generated by the constructors

and every term has a unique such constructor form. At this stage the term algebra is not quotiented by the recursion equations.

Let A be an algebra of the signature { 0 , S , + , ⋅ } , with carrier | A | : a constant 0 A ∈ | A | , a unary operation

and binary operations

Let

be an assignment of the de Bruijn variables. No equations are assumed in A .

Proof of Proposition 1.5. Define the map h on terms by structural recursion:

These equations define a homomorphism from the term algebra to A . By construction it preserves 0 , S , + , ⋅ and extends ρ , which gives existence.

For uniqueness, let h ′ be any homomorphism from the term algebra to A such that h ′ ( x n ) = ρ ( n ) for all n . A structural induction on t shows that h ′ ( t ) = h ( t ) for every term t . Hence h ′ = h . □

This section proves Theorem 1.7(a). The construction proceeds from the orbit carrier ℕ δ to its group completion ℤ δ , and then to the fraction construction ℚ δ . All constructions in this section are choice-free.

The natural-number object ℕ δ is presented by the generated δ -orbit of Definition 1.2, with constructors 0 and S . Its elements are the finite records generated by these constructors. For each metatheoretic natural number n , we write S n 0 for the element of ℕ δ obtained by applying S n times to 0.

Let Num denote the inductively generated collection of closed terms determined by the rules

Definition 5.1.The arithmetic projection is the map

defined by structural recursion:

Thus the empty record is represented by the closed term 0, and each one-step extension of a record is represented by one application of the successor symbol S .

Definition 5.2.Addition and multiplication on ℕ δ are defined by structural recursion:

and

We write

These are metatheoretic operations on ℕ δ . Lemmas 5.4 and 5.5 show that under the map D , they correspond to the interpretations of the function symbols + and ∙ in the standard model N .

Let us define

and

The map D assigns to each finite record the number of its successive extensions, while ν assigns to each n ∈ ℕ the record obtained by applying S n times to 0.

For n ∈ ℕ , let n ¯ ∈ Num denote the closed numeral S n 0 . Then

Lemma 5.3.The maps D and ν are mutually inverse:

Proof. The identity D ( ν ( n ) ) = n follows by induction on the metatheoretic natural number n . The identity ν ( D ( x ) ) = x follows by structural induction on x ∈ ℕ δ . □

By Remark 2.6, this lemma is an instance of Theorem 2.5 at the standard model.

Lemma 5.4.For all x , y ∈ ℕ δ ,

Consequently, D : ( ℕ δ , + , 0 ) → ( ℕ , + , 0 ) is a monoid isomorphism.

Proof. The identity follows by structural induction on y , using

Thus D preserves addition and zero. Since D is bijective by Lemma 5.3, it is a monoid isomorphism. □

Lemma 5.5.For all x , y ∈ ℕ δ ,it holds

Proof. We use structural induction on y . The case y = 0 is immediate. For the successor case, we have

□

Proposition 5.6.The structure ( ℕ δ , + , ⋅ , 0 , 1 ) is a commutative semiring.Moreover,

and

Proof. By Lemmas 5.3, 5.4, and 5.5, the map D : ℕ δ → ℕ is a bijection preserving 0, 1, addition, and multiplication. The proof follows from the corresponding properties of ℕ . □

Definition 5.7.A signed orbitis a pair ( p , n ) ∈ ℕ δ × ℕ δ , where p and n are its positive and negative parts. Two signed orbits are balanced, written

Lemma 5.8.The balance relation is a decidable equivalence relation.

Proof. Reflexivity and symmetry are immediate. For transitivity, suppose

Then

Adding n ″ to the first relation and using the second, we obtain

and additive cancellation (Proposition 5.6) gives p + n ″ = p ″ + n , so ( p , n ) ~ ( p ″ , n ″ ) .

The relation is decidable because addition and equality on ℕ δ are decidable. Lemma 5.3 implies that D is injective. Hence, for all x , y ∈ ℕ δ , we get

Since equality on ℕ is decidable, then equality on ℕ δ is decidable. □

Lemma5.9.The operations on signed orbits defined by

and

are compatible with the balance relation in each argument.

Proof. Suppose

and let ( r , s ) be a signed orbit.

For addition, we have

Finally, we have ( p + r , n + s ) ~ ( q + r , m + s ) .

For multiplication, we obtain

Thus multiplication is compatible with balance in the first argument. Compatibility in the second argument follows from commutativity. □

Definition 5.10.The integer object is the quotient ℤ δ : = ( ℕ δ × ℕ δ ) / ~ . Its addition and multiplication are defined by

and

These operations are well-defined by Lemma 5.9.

For a signed orbit u = ( p , n ) , define

On ℤ δ , define

If ( p , n ) ~ ( p ′ , n ′ ) , then p + n ′ = p ′ + n . By commutativity, n + p ′ = n ′ + p , that is ( n , p ) ~ ( n ′ , p ′ ) . Hence negation is well-defined on ℤ δ .

Lemma 5.11.The map

is an injective map preserving addition and multiplication.

Proof. Both identities

hold by Definition 5.10. If ι ℤ δ ( a ) = ι ℤ δ ( b ) , then ( a , 0 ) ~ ( b , 0 ) , that is a = b . □

Lemma 5.12.The map

is a bijection satisfying

and preserving addition,multiplication,and negation.

Proof. By Lemmas 5.4 and 5.5, we obtain

for all a , b ∈ ℕ δ .

Suppose ( p , n ) ~ ( p ′ , n ′ ) . Then p + n ′ = p ′ + n , hence D ( p ) + D ( n ′ ) = D ( p ′ ) + D ( n ) , and therefore

Thus D ℤ is well-defined. Since D ( 1 ) = D ( S 0 ) = D ( 0 ) + 1 = 1 , we have

and

For addition, we have

For multiplication, we have

so D ( p + n ′ ) = D ( p ′ + n ) . Since D is injective by Lemma 5.3, we get p + n ′ = p ′ + n , hence ( p , n ) ~ ( p ′ , n ′ ) and [ ( p , n ) ] = [ ( p ′ , n ′ ) ] . Thus D ℤ is injective.

If z ≥ 0 , then D ℤ ( [ ( ν ( z ) , 0 ) ] ) = z , and if z < 0 , then D ℤ ( [ ( 0 , ν ( − z ) ) ] ) = z . Thus D ℤ is surjective. □

Corollary 5.13.The structure ( ℤ δ , + , ⋅ , − , 0 ℤ δ , 1 ℤ δ ) is a commutative ring,and D ℤ is a ring isomorphism.

Proof. By Lemma 5.12, D ℤ is a bijection preserving 0, 1, addition, multiplication, and negation. Hence the commutative ring structure of ℤ transfers to ℤ δ , and D ℤ is a ring isomorphism. □

We now construct the fraction object over ℤ δ , using signed-orbit representatives as numerators and nonzero elements of ℕ δ as denominators.

Definition 5.14.A ratio orbit is a pair ( u , d ) ,where u is a signed orbit and d ∈ ℕ δ is nonzero.We identify d ∈ ℕ δ with the signed orbit ( d , 0 ) .

Two ratio orbits ( u , d ) and ( u ′ , d ′ ) are cross-equal, written ( u , d ) ≈ ( u ′ , d ′ ) , if the signed orbits u ⋅ d ′ and u ′ ⋅ d are balanced.

Lemma 5.15.Let u , v be signed orbits and let d ∈ ℕ δ be nonzero.If

Proof. Let u = ( p , n ) and v = ( q , m ) . We have

and therefore

Since d ≠ 0 , using cancellation from Proposition 5.6, we have

Hence u ~ v . □

Lemma 5.16.Cross-equality is a decidable equivalence relation on ratio orbits.

Proof. Reflexivity and symmetry follow from reflexivity and symmetry of the balance relation. For transitivity, suppose

Then

By Lemma 5.9, multiplying the first relation by d ″ and the second by d , and using associativity and commutativity of multiplication, we have

We use cancellation with respect to balance. Since d ′ ≠ 0 , cancellation gives

Therefore ( u , d ) ≈ ( u ″ , d ″ ) , so cross-equality is transitive.

Finally, cross-equality is decidable because multiplication of signed orbits is recursively defined and balance is decidable. □

Lemma 5.17.Addition and multiplication are well-defined on cross-equality classes.

Proof. By Proposition 5.6, the semiring ℕ δ has no zero divisors, so the product of two nonzero denominators is nonzero. Suppose

For any ratio orbit ( w , f ) , the products w ⋅ d ⋅ e ⋅ f and w ⋅ e ⋅ d ⋅ f coincide by Proposition 5.6, and Lemma 5.9 applied to u ⋅ e ~ v ⋅ d gives

so addition is compatible with cross-equality. Also, we have

so multiplication is compatible with cross-equality. The same argument applies to the second variable. □

Definition 5.18.The rational object is the quotient

Addition and multiplication are induced by

and

The zero and unit of ℚ δ are represented by

By Lemma 5.17, these formulas induce well-defined operations on ℚ δ . For a signed orbit u = ( p , n ) , let − u = ( n , p ) . The formula

defines a negation on ℚ δ . If ( u , d ) ≈ ( v , e ) , then

implies

Hence negation is well-defined.

Lemma 5.19.The map

is injective and preserves addition and multiplication.

Proof. If ( p , n ) ~ ( p ′ , n ′ ) , then ( ( p , n ) , 1 ) ≈ ( ( p ′ , n ′ ) , 1 ) , so the map is well-defined. The two operation formulas of Definition 5.18. reduce to the operations of Definition 5.10. when both denominators equal 1, so ι ℚ δ preserves addition and multiplication. Finally, [ ( ( p , n ) , 1 ) ] = [ ( ( p ′ , n ′ ) , 1 ) ] gives ( p , n ) ⋅ 1 ~ ( p ′ , n ′ ) ⋅ 1 , hence ( p , n ) ~ ( p ′ , n ′ ) . □