Peano and Osgood theorems via effective infinitesimals

We provide choiceless proofs using infinitesimals of the global versions of Peano's existence theorem and Osgood's theorem on maximal solutions. We characterize all solutions in terms of infinitesimal perturbations. Our proofs are more effective than traditional non-infinitesimal proofs found in the literature. The background logical structure is the internal set theory SPOT, conservative over ZF.


Introduction
Nonstandard analysis (NSA) is sometimes criticized for its implicit dependence on strong forms of the Axiom of Choice (AC).Indeed, if * is the mapping that assigns to each X ⊆ N its nonstandard extension * X , and if ν ∈ * N \ N is an unlimited integer, then the set U = {X ⊆ N | ν ∈ * X} is a nonprincipal ultrafilter over N. Of course strong forms of AC, such as Zorn's Lemma, are a staple of modern set-theoretic mathematics, but it is undesirable to have to rely on them for results in ordinary mathematics dealing with Calculus or differential equations (see Simpson [18] for a discussion of the distinction between set-theoretic and ordinary mathematics).The traditional proofs of most theorems in ordinary mathematics are effective: they do not use AC. 1 A few results, such as the equivalence of the ε-δ definition and the sequential definition of continuity for functions f : R → R, require weak forms of AC, notably the Axiom of Countable Choice (ACC) or the stronger Axiom of Dependent Choice (ADC).These weak forms are generally accepted in ordinary mathematics; they do not imply the strong consequences of AC such as the existence of nonprincipal ultrafilters 2 K Hrbacek and M Katz or the Banach-Tarski paradox (see Jech [12], Howard and Rubin [7]).We refer to such proofs as semi-effective.
An answer to the above criticism of NSA is offered by recent developments in the axiomatic/syntactic approach that dates back to the work of Hrbacek [8] and Nelson [15].A number of axiomatic systems for NSA have been proposed, of which Nelson's IST is the best known.We refer to Kanovei and Reeken's monograph [13] for a comprehensive discussion of such axiomatic frameworks.An accessible introduction to IST is Robert [16].
The theory IST includes the axioms of ZFC, so one could ask whether the dependence on AC could be avoided by deleting AC from the axioms constituting IST.It turns out that in the resulting theory one can still prove the existence of nonprincipal ultrafilters, by an argument similar to the one given above for the model-theoretic approach (see Hrbacek [9] and the paragraph following Lemma 2.5 below).
In Hrbacek and Katz [10] the authors have developed an axiomatic system for NSA with the acronym SPOT, a subtheory of IST.The theory SPOT is a conservative extension of ZF.This means that every statement in the ∈-language provable in SPOT is provable already in ZF.In particular, AC and the existence of nonprincipal ultrafilters are not provable is SPOT, because they are not provable in ZF.A stronger theory SCOT which is a conservative extension of ZF + ADC is also considered there.Hence proofs in SPOT are effective, and proofs in SCOT are semi-effective.Some examples of constructions in nonstandard analysis formalized in these theories are given in [10].In particular, it is shown there how the Riemann integral can be defined in SPOT using partitions into infinitesimal subintervals, and the countably additive Lebesgue measure in SCOT using counting measures.The expository article Hrbacek and Katz [11] presents in SCOT various nonstandard arguments related to compact sets and continuity.
In Section 2 we state the axioms of SPOT, list some of their consequences, and prove a stronger version of the Standard Part principle SP that is crucial in the preliminary Section 3.
In Sections 4 -6 we give nonstandard proofs in SPOT of the global versions of Peano's and Osgood's theorems concerning the existence of solutions of ordinary differential equations.While the nonstandard approach using Euler approximations with an infinitesimal step that we employ is well known for local solutions (see, eg, Albeverio, Høegh-Krohn, Fenstad, and Lindstrøm [1, page 30]), we offer three innovations: We first prove (Theorem 4.1) that every infinitesimal perturbation ε determines a unique global solution y ε (some or all of these solutions may be the same).We next prove (Lemma 5.1) that every solution that is not global is a restriction of some y ε .Hence every solution is either global or can be extended to a global one (Corollary 5.2) and every global solution is of the form y ε for some infinitesimal perturbation ε (Theorem 5.3).
Finally we state the global Osgood's theorem (Theorem 6.2).The proof shows first that there is a local maximal solution (Lemma 6.5 and the last part of the sentence that precedes it).The last paragraph of the proof obtains the global maximal solution as the union of all local ones.

Theory SPOT
By an ∈-language we mean the language that contains a binary membership predicate ∈ and is enriched by defined symbols for constants, relations, functions and operations customary in traditional mathematics.For example, it contains names N and R for the sets of natural and real numbers; they are viewed as defined in the traditional way.(N is the least inductive set, R is defined in terms of Dedekind cuts or Cauchy sequences.)The symbols <, + and × denote the ordering, addition and multiplication of real numbers, and so on without further explanation.The classical theories ZF and ZFC are formulated in the ∈-language.
The language of SPOT contains an additional unary predicate st.SPOT is a subtheory of IST and its bounded version BST (see [13]).We use ∀ and ∃ as quantifiers over sets and ∀ st and ∃ st as quantifiers over standard sets.The theory SPOT has the following axioms.
ZF (Zermelo -Fraenkel Set Theory) T (Transfer) Let ϕ be an ∈-formula with standard parameters.Then: The theory SPOT proves the following results (see [10]).
Lemma 2.1 Standard natural numbers precede all nonstandard ones: Note that {0, 1, . . ., n − 1} is a finite set for every n ∈ N; it is nonstandard when n is nonstandard.
The dual form of Countable Idealization is: Countable Idealization easily implies the following more familiar form.We use ∀ st fin and ∃ st fin as quantifiers over standard finite sets.
Corollary 2.3 Let ϕ be an ∈-formula with arbitrary parameters.For every standard countable set A: The axiom SP ′ is often stated and used in the form where x is limited iff |x| ≤ n for some standard n ∈ N, and The unique standard real number r in SP is called the standard part of x or the shadow of x; notation r = sh(x).
We have the following equivalence.(SP ′′ ) Let ϕ be an ∈-formula with arbitrary parameters.Then Lemma 2.5 The statement SP ′ and the schema SP ′′ are equivalent (over the theory Standardization in full strength, as postulated in IST, BST, etc., implies the existence of nonprincipal ultrafilters over N: take a nonstandard ν ∈ N and let U be the standard subset of Nonetheless, two important special cases of Standardization can be proved in SPOT.
The scope of Countable Standardization can be expanded to a larger class of formulas.
where ψ is an ∈-formula and each Q stands for ∃ or ∀.
In other words, all occurrences of ∀ st or ∃ st in Φ appear before all occurrences of ∀ or ∃.
An st N -prenex formula is a formula of the form The theory SPOT proves the following stronger version of Countable Standardization that is used repeatedly in this paper.
Proposition 2.7 (Countable Standardization for st N -prenex formulas) Let Φ be an st N -prenex formula with arbitrary parameters.Then: Of course, N can be replaced by any standard countable set.
Proof We give the argument for a typical case Then S is standard and for all standard n: The second special case of Standardization involves st-prenex formulas with only the standard parameters.
Lemma 2.8 Let Φ(v 1 , . . ., v r ) be an st-prenex formula with standard parameters.Then ) holds for all standard v 1 . . ., v r by the Transfer principle.
The set P = {⟨v 1 , . . ., v r ⟩ ∈ S | ϕ(v 1 , . . ., v r )} exists by the Separation principle of ZF, it is standard, and has the required property.Remark 2.9 This result has twofold importance: (1) The meaning of every predicate that for standard inputs is defined by an st-prenex formula Q st 1 u 1 . . .Q st s u s ψ with standard parameters is automatically extended to all inputs, where it is given by the ∈-formula Q 1 u 1 . . .Q s u s ψ .
(2) Standardization holds for all ∈-formulas with additional predicate symbols, as long as all these additional predicates are defined by st-prenex formulas with standard parameters.

Two examples
Formulas that occur in practice are usually not in the st-prenex form, but they can often be converted to it using Countable Idealization.
Definition 3.1 (Integral of continuous functions) We fix a positive infinitesimal h and the corresponding "hyperfinite line" It is easy to show that the value of the integral does not depend on the choice of h.
(It is understood that h, n, m are not 0).This expression can be rewritten as: We swap the outmost universal quantifiers and apply the dual version of Countable Idealization (Lemma 2.2) to get which is an st N -prenex formula, clearly equivalent to One can now use Standardization for st-prenex formulas with standard parameters to conclude that, for example, for every standard f , a there exists a standard function F such that F(z) = is equivalent to an st N -prenex formula (with the parameter w).
Proof The formula Ψ(x, y) can be written as ] (let n = min {i, j}), and finally (Countable Idealization, Lemma 2.2) to the st N -prenex formula: The last formula of course simplifies to: Definition 3.4 Let w be a function, dom w = D w ⊆ I where I ⊆ R is a standard interval, and ran w ⊆ R.
• The function w is densely defined on I if for every standard x ∈ I there is α ∈ D w such that α ≈ x. • The function w is (uniformly) S-continuous if for α, β ∈ D w , α ≈ β implies w(α) ≈ w(β).Proof The usual arguments work in SPOT; see eg Hrbacek and Katz [11].
The next proposition follows immediately from the Standardization principle of IST or BST, but to prove it in SPOT we need to consider an approximation to the set W on the rationals, to which we can apply Countable Standardization for st N -prenex formulas.
Proposition 3.6 If w is S-continuous and densely defined on I , then there is a standard function W such that, for all standard x, y ∈ R, ⟨x, y⟩ ∈ W if and only if x ≈ α and y ≈ w(α) for some α ∈ D w .
The proof of Proposition 3.6 appears below, following the proof of Lemma 3.8.

Definition 3.7
The existence of the standard set For q ∈ I ∩ Q let Z q = {r ∈ Q | ⟨q, r⟩ ∈ Z} and W 0 (q) = inf Z q , if it exists (it can happen that Z q = ∅ or Z q = Q, in which cases W 0 (q) is undefined).Finally, let W be the closure of (the graph of) W 0 .We show below that the standard set W has the property from Proposition 3.6.
Lemma 3.8 If q ∈ I ∩ Q is standard, then q ∈ dom W 0 if and only if there exists α ∈ D w such that α ≈ q and w(α) is limited.If this is the case, then W 0 (q) = sh(w(α)).
Conversely, if ⟨x, y⟩ ∈ W , then for every standard ϵ > 0 there is q  Here and elsewhere, if c ∈ R is an endpoint of an interval I = dom y, y ′ (c) is the appropriate one-sided derivative of y at c.We call a solution of the initial value problem ( * ) that cannot be continued to any interval [0, a ′ ) with a ′ > a a global solution.
We generalize the familiar construction of Euler approximations with an infinitesimal step by allowing infinitesimal perturbations.This is a variation on an idea in Birkeland and Normann [2] (the main difference being that we perturb the construction of the solution, while Birkeland and Normann perturb the function F ).
We will prove the theorem for standard F ; the stated result follows by Transfer.The construction proceeds as follows.
Let N be a positive unlimited integer and h = 1/N .We fix x 0 ≥ 0, x 0 ≈ 0, y 0 ≈ 0, and let The concept is not needed for the proof of Theorem 4.1, where the simplest choice ε k = 0 for all k suffices, but it is used for its generalization in Section 5.
We define y k recursively: Observe that: We next define: The existence of Y in SPOT follows from Proposition 3.6 (let I = [0, ∞) and w(x k ) = y k for 0 ≤ k < N 2 ).The strategy for the rest of the proof is to show thatY is a (graph of) a continuous function defined on an open subset of [0, ∞), and the restriction y of Y to the connected component of its domain containing 0 has the required properties.
x + e) can be replaced by (x − e, x + e).
Proof By continuity of F at ⟨x, y⟩ there exist standard c, d, M > 0 such that Fix a standard e such that 0 < e < min{c, d/(M + 1)}.
We prove by induction on k that  It remains to prove that Y ′ (x) = F(x, Y(x)) holds for x ∈ dom Y .Let I be as above, x, z ∈ I be standard and without loss of generality x ≤ z.In the notation of the previous paragraph, we have ( 1)  Proof We prove that for every standard r > 0 there is a standard ϵ > 0 such that for all standard x, a − ϵ < x < a implies |y(x)| ≥ r.
Assume that the statement is false and fix a standard r > 0 such that for every standard n ∈ N there is a standard x ∈ (a − 1 n , a) such that Y(x) ∈ (−r, r).Hence for every standard n ∈ N there is k < N 2 such that x k ∈ (a − 1 n , a) and y k ∈ (−r, r) (take ⟨x k , y k ⟩ ≈ ⟨x, Y(x)⟩).By Countable Idealization (Lemma 2.2), there exists p < N 2 such that y p ∈ (−r, r) and x p ∈ (a − Remark 4.6 Note that the solution y is determined by the choice of the starting point x 0 , y 0 and the infinitesimal perturbation ε.Thus we can single out a particular global solution of ( * ) by fixing N and letting x 0 = 0, y 0 = 0 and ε k = 0 for all k < N 2 .Remark 4.7 There are obvious generalizations that do not require any additional nonstandard ideas.For example, the two-sided version: Let F : R 2 → R be a continuous function.For every ⟨a, b⟩ ∈ R 2 there is an interval (a − , a + ) with −∞ ≤ a − < a < a + ≤ ∞ and a function y : (a − , a + ) → R such that y(a) = b, y ′ (x) = F(x, y(x)) holds for all x ∈ (a − , a + ), and if a − and/or a + is in R, then lim x→(a − ) + y(x) = ±∞ and/or lim x→(a + ) − y(x) = ±∞.
The domain R 2 of F can be replaced by an open set D ⊆ R 2 .One obtains a solution that tends to the boundary of D, in the sense that for every compact K ⊆ D there is c < a + such that y(x) / ∈ K holds for all c < x < a + , and analogously for a − .
The method generalizes to systems of equations.has a noncontinuable solution.
Proof For u = ⟨u 0 , . . ., u n−1 ⟩ and v = ⟨v 0 , . . ., v n−1 ⟩ in R n we let u ≈ v if u i ≈ v i for all i < n, and u ≥ v if u i ≥ v i for all i < n.With this understanding, the material in Section 3, and in particular Proposition 3.6, generalizes straightforwardly to functions w with ran w ⊆ R n .One can then follow the proof of Theorem 4.1.
Recall (see the conclusion of the proof of Theorem 4.1) that y ε is a standard function defined via ( * * ).
Lemma 5.1 Let F be standard.For every standard solution y of ( * ) defined on a standard interval [0, a) and every standard c < a, c > 0, there is an infinitesimal perturbation ε such that y(x) = y ε (x) holds for 0 ≤ x ≤ c.
Proof By the mean value theorem, for each k such that x k+1 ≤ c there is t ∈ [x k , x k+1 ] such that y(x k+1 ) − y(x k ) = y ′ (t) • h.Let t k be the least such t (as y ′ is continuous, the set of t with this property is closed).Then let ε k = F(t k , y(t k )) − F(x k , y(x k )) = y ′ (t k ) − y ′ (x k ) ≈ 0. For x k+1 > c let ε k = 0. Let y 0 = y(0); it follows that y k = y(x k ) for all k such that x k+1 ≤ c: assuming the claim is true for k, we have If x ∈ [0, c] is standard, take x ≈ x k for x k+1 ≤ c; then y ε (x) ≈ y k = y(x k ) ≈ y(x), so y ε (x) = y(x).
Corollary 5.2 Every solution of ( * ) extends to a global solution.
Proof Let y defined on [0, c) be a standard solution of ( * ) with F standard.If y is not global, then it has a standard continuation ỹ to an interval [0, a) with c < a.By Lemma 5.1 y has a continuation y ε which is global by Theorem 4.1.By Transfer, the claim holds for all solutions y and all functions F .Theorem 5.3 For every standard global solution y of ( * ) there is an infinitesimal perturbation ε such that y = y ε .
Proof Assume the domain of y is a standard interval [0, a) (possibly a = +∞).
We fix a standard strictly increasing sequence ⟨c n | n ∈ N⟩ such that c 0 > 0 and lim n→∞ c n = a.The proof of Lemma 5.1 (with c = c n ) justifies the following statement.For every standard n ∈ N there is ε = ⟨ε k | 0 ≤ k < N 2 ⟩ such that for all m ≤ n and for all k < N 2 : . By Countable Idealization (Lemma 2.2) there is ε such that for all standard n ∈ N and for all k < N 2 :

Lemma 2 . 4 5 SP
The statements SP ′ and SP are equivalent (over the theory ZF + O + T).Journal of Logic & Analysis 15:6 (2023) Peano and Osgood theorems via effective infinitesimals ′ can also be reformulated as an axiom schema (Countable Standardization for ∈-formulas):

z a f
(x) dx for all standard z ∈ [a, b].By Remark 2.9 (1), the last equation holds for all z ∈ [a, b].Of course, the usual arguments show that the above definition of the integral agrees with the traditional ϵ-δ one for all standard f , a, b, r.The following observation is crucial for the proof of Proposition 3.6.Journal of Logic & Analysis 15:6 (2023) Lemma 3.3 Let w be a function, dom w = D w ⊆ R and ran w ⊆ R. Then the formula
The case k = p is clear.If the claim is true for k and x k+1 < x + e, we have |y k − y p | ≤ (M+ε)•|x k −x p | < (M+1)•e ≤ d and hence the point ⟨x k , y k ⟩ ∈ [x, x+c]×[y−d, y+d].Now |F(x k , y k )| ≤ M , so |y k+1 − y k | ≤ (|F(x k , y k )| + |ε k |) • h ≤ (M + ε) • h and: a symmetric "backward" argument shows that the statement holds also on the interval (x − e, x].It follows easily that Y as in ( * * ) is the graph of a real function.If ⟨x, y 1 ⟩, ⟨x, y 2 ⟩ ∈ Y are standard, then there are k and ℓ such that ⟨x, y 1 ⟩ ≈ ⟨x k , y k ⟩ and ⟨x, y 2 ⟩ ≈ ⟨x ℓ , y ℓ ⟩.Then x k ≈ x ℓ ≈ x and y 1 ≈ y k ≈ y ℓ ≈ y 2 and we conclude that y 1 = y 2 .Hence Y is the graph of a function, by Transfer.From now on we write Y(x) for the value of Y at x ∈ dom Y .Lemma 4.4 The domain of the function Y is an open subset of [0, ∞) containing 0, Y is continuous on dom Y , and Y ′ (x) = F(x, Y(x)) holds for x ∈ dom Y .Proof Clearly 0 ∈ dom Y .If x ∈ dom Y is standard and y = Y(x), Lemma 4.3 gives an interval I = (x − e, x + e) (or I = [0, e)) such that x k ∈ I implies y k ∈ [y − d, y + d], hence y k is limited and Y(sh(x k )) = sh(y k ) is defined.If u ∈ I is standard, u = sh(x k ) holds for some x k ∈ I .Hence Y(u) ∈ [y − d, y + d] is defined for all standard u ∈ I , and by Transfer, the same holds for all u ∈ I .For the proof of continuity at a standard x ∈ dom Y let I be as above, ϵ > 0 be standard and δ = ϵ/(M + 1).If z ∈ I is standard and |x − z| < δ , then there are k and ℓ such that x ≈ x k and z ≈ x ℓ ; moreover, Y(x) ≈ y k and Y(z) ≈ y ℓ .We have |Y(x)−Y(z)| ≈ |y ℓ −y k | and |y k −y ℓ | ≤ (M +ε)•|x k −x ℓ |, where |x ℓ −x k | ≈ |x −z| < δ .It follows that |Y(x) − Y(z)| ≤ (M + 1) • δ = ϵ.As usual, Transfer gives continuity for all x ∈ dom Y .
i , Y(x i )) • h = ℓ−1 i=k (F(x i , y i ) + δ i ) • hwhere δ i ≈ 0 for k ≤ i < ℓ.The relation ≈ in (2) follows from the nonstandard theory of integration (see Definition 3.1) and the fact that F(t, Y(t)) is continuous on I .The relation = in (2) is justified as follows: Let x * = sh(x i ) and y * = sh(y i ); then Y(x * ) ≈ y * by the definition of Y and Y(x i ) ≈ Y(x * ) by the continuity of Y .The continuity of F then givesF(x i , y i ) ≈ F(x * , y * ) ≈ F(x i , Y(x i )).The formulas (1) and (2) imply Y(z) − Y(x) ≈ z x F(t, Y(t)) dt, hence Y(z) − Y(x) = z x F(t, Y(t)) dt as both sides are standard.By Transfer, the relationship holds for all x, z ∈ I .It remains to apply the Fundamental Theorem of Calculus.Let [0, a), a > 0, be the connected component of the domain of Y containing 0.

1 n
, a) holds for all standard n > 0. It follows that x p ≈ a; we let b = sh(y p ).By the definition of Y then ⟨a, b⟩ ∈ Y , and hence a ∈ dom Y , contradicting the fact that [0, a) is a connected component of the domain of Y .Conclusion of proof of Theorem 4.1 Let Y be the function defined by formula ( * * ).The proof of Theorem 4.1 is now concluded by letting y = Y ↾ [0, a).We write y ε when it is necessary to indicate the dependence of y on the perturbation ε.

•
The axiomatic system SPOT enables us to use infinitesimal methods without the underlying assumption of the existence of nonprincipal ultrafilters or any other strong form of AC. • We construct global, ie, noncontinuable, solutions rather than local solutions.• Traditional proofs of the existence of noncontinuable solutions typically depend on ADC; see Remark 7.1.By contrast, our proof does not assume any form of AC at all.