A note on Brouwer’s weak continuity principle and the transfer principle in nonstandard analysis

A well-known model of nonstandard analysis is obtained by extending the structure of real numbers using an ultra power construction. A constructive approach due to Schmieden and Laugwitz uses inste ...


Introduction
A well-known approach to nonstandard analysis is obtained by extending the structure of real numbers using an ultrapower construction (see [4]).There is also a constructive approach, originating with Schmieden and Laugwitz [16], which uses a reduced power construction modulo a cofinite filter.It has been investigated and developed by Martin-Löf [9], the author [12,13] and Schuster [17] in the context of BISH, constructive mathematics in the sense of Bishop [1].A drawback of this approach is that the transfer principle is rather weak.We show here that this principle can be strengthened by assuming Brouwerian continuity axioms familiar from intuitionistic systems.
The Schmieden-Laugwitz extension can be described in elementary model-theoretic terms as follows.Let M be a first-order structure with respect to the many-sorted signature L. All sorts are supposed to be inhabited.Construct M , a first-order 2 Erik Palmgren structure, whose underlying sets are of the form S N , the set of sequences of elements in S for each sort S in M .Two sequences f and g are defined to be equal in the structure as follows f = S g ⇐⇒ def (∃k)(∀i ≥ k)f (i) = g(i).More generally we define for each relation R in the signature L: The below transfer theorem between the structures M and M can be proved by induction on formulas, using countable choice for the existential case.A first order formula is called existential conjunctive if it is built up from atomic formulas using only ∧ and ∃.The proof is essentially contained in Martin-Löf [9], but see also [12].
Theorem 1.1 For each L-structure M , for each existential conjunctive formula ϕ(x 1 , . . ., x n ) in the signature L, and for parameters f 1 ∈ S 1 , . . ., f n ∈ S n : Below we consider general quantified conjuctive formulas.A first order formula is called quantified conjunctive if it is built up from atomic formulas using only ∧, ∃ and ∀.Such a formula may easily be seen to be equivalent to a prenex formula of the form where ψ is a conjunction of atomic formulas and each Q i is either ∃ and ∀.
Remark It is possible to obtain a full transfer principle constructively if the model notion is relaxed, by using sheaf semantics [10] or using non-standard truth [9].However, with standard semantics, full transfer is impossible [13, p. 234].

Extended transfer principles
We shall use a Brouwerian type axiom to extend the transfer principle of Theorem 1.1.The weak continuity principle for a set S says that for any predicate Here α and β varies over S N .
Remark The special case when S = N is the familiar WC-N ( [18]).We do not know of a model for the general case but the axiom scheme is certainly possible to justify similarly to the special case.Moerdijk and van der Hoeven [3] gives model for WC-N which might be a candidate for generalization.
Under the assumption of the above axiom we can strengthen Theorem 1.1.
Theorem 2.1 Assume that weak continuity principle holds for each sort of the Lstructure M .For each quantified conjunctive formula ϕ(x 1 , . . ., x n ) in the signature L and for parameters f 1 ∈ S 1 , . . ., f n ∈ S n : (1) Proof.In view of the inductive proof of Theorem 1.1 we need only check the ∀-case.
(⇒) Suppose M |= (∀y ∈ S)ϕ(f 1 (i), . . ., f n (i), y).This implies by the inductive hypothesis that for each g : N → S there exists k ∈ N such that Hence by the weak continuity principle for S, there is ∈ N so that for each h : N → S with (∀j < )h(j) = g(j) it holds that Let g(i) = m for all i, where m is a fixed element in S. Then we get k and so that for each h : N → S with (∀j < )h(j) = g(j) it holds that We may assume that k ≥ .Let y be an arbitrary element in S. Define h : N → S by letting h(i) = g(i) for i < and h(i) = y for i ≥ .Then since k ≥ , the statement (2) gives (∀i ≥ k) M |= ϕ(f 1 (i), . . ., f n (i), y).
But y was arbitrary, so

Erik Palmgren
This proves the equivalence.
We note that the direction (⇐) did not use the continuity principle.Indeed this direction can be further strengthened to apply to so-called constructive Horn formulas along the lines of [12].Similarly, the saturation principles that depend on transfer principles can be improved.We will not pursue this is in the present paper.
A special case of the theorem is worth noting: Corollary 2.2 Assuming WC-N, the equivalence (1) holds for M = N, the structure of natural numbers, with all functions and relations.

Transfer and boundedness
A subset A ⊆ N is bounded if there is n ∈ N so that A ⊆ {0, 1, 2, . . ., n}.A useful characterization [6,15] of a pseudo-bounded subset A ⊆ N is that for each f : N → A there is some k such that f (n) ≤ n for all n ≥ k.The boundedness principle BD of Ishihara [5] is Every inhabited pseudo-bounded set of natural numbers, is bounded.
A weaker principle BD-N is: Every inhabited countable pseudo-bounded set of natural numbers, is bounded.
The principle BD-N has the intriguing property of being a theorem of classical mathematics CLASS, intuitionistic mathematics INT and Russian recursive mathematics RUSS (using the nomenclature of [2]), and yet it is not provable in BISH as shown by Lietz [8].BD-N is equivalent to the completeness of the uniform space D(R) of test functions in distribution theory [6] and to several important properties involving sequential continuity [5].
For each S and g : S → N let (Tr g ) be the proposition where η : N → N is the identity η(i) = i.Note that (Tr g ) is an instance of Theorem 2.1.Parts (⇐) of the following characterization are due to Nordvall-Forsberg [11].It can be regarded as a nonstandard formulation of Richman's criterion [15], and suggests that the boundedness principles may have further interesting strengthenings.
Theorem 3.1 (a) BD holds if, and only if, for every inhabited set S and each g : S → N, (Tr g ) holds.
(b) BD-N holds if, and only if, for each g : N → N, (Tr g ) holds.
Proof.(a, ⇐) Suppose that A ⊆ N is an inhabited pseudo-bounded set.According to Richman's characterization this means that for each sequence x : N → A, there is k ∈ N so that x(n) ≤ n for all n ≥ k.Letting g : A → N be the inclusion, we thus have (∀x ∈ A )g (x) ≤ η where η(n) = n.Hence by (Tr g ) (∃k)(∀n ≥ k)(∀x ∈ A)g(x) ≤ η(n).Letting n = k we see that A is bounded by k.
(a, ⇒) Assume BD.Let g : S → N be arbitrary with S inhabited.Note that the direction (⇐) of (Tr g ) is trivial and does not require BD.We prove the opposite direction.Suppose (∀x ∈ S )g (x) ≤ η.
Thus for every x : N → S, there is k such that g(x(n)) ≤ n for n ≥ k.Let A = Im(g).For any f : N → A, there is by countable choice x : N → S so that g(x(n)) = f (n) for all n ∈ N. Hence f (n) ≤ n for all n ≥ k.Thus A is inhabited and pseudobounded, so by BD the set A is bounded by some N .Hence (∀x ∈ S)g(x) ≤ N .Thus (∀n ≥ N)(∀x ∈ A)g(x) ≤ η(n) as required.
(b, ⇐) Observe that if A is inhabited and countable, then A = Im(g) for some g : N → N. Suppose A is pseudo-bounded.Let x : N → N be arbitrary then f (n) = g(x(n)) ∈ A for all n ∈ N. Thus there is k with f (n) ≤ n for all n ≥ k.Hence (∀x ∈ N )g (x) ≤ η and thus (Tr g ) implies (∃k)(∀n ≥ k)(∀x ∈ N)g(x) ≤ η(n).Hence A is bounded by k, thus proving BD-N.
(b, ⇒) We note that for the case S = N in (a, ⇒) BD-N suffices. .