Infinitesimal analysis without the axiom of choice: Effective infinitesimals

See the article 21e in Annals of Pure and Applied Logic by Karel Hrbacek and Mikhail Katz.

See also "Effective infinitesimals in ℝ" here: 23d

See also "Peano and Osgood theorems via effective infinitesimals" here: 23e

It turns out that analysis with infinitesimals does not require the axiom of choice any more than traditional non-infinitesimal analysis. There are two popular approaches to Robinson's mathematics (including analysis with infinitesimals): model-theoretic and axiomatic/syntactic. The model-theoretic approach (including the construction of the ultrapower) typically relies on strong forms of the axiom of choice. The axiomatic/syntactic approach turns out to be more economical in the use of foundational material, and exploits a richer st-∈-language, as explained below. This enables effective infinitesimals in ℝ.

The traditional set-theoretic foundation for mathematics is Zermelo-Fraenkel set theory (ZF). The theory ZF is a set theory in the ∈-language. Here "∈" is the two-place membership relation. In ZF, all mathematical objects are built up step-by-step starting from ∅ and exploiting the one and only relation ∈.

For instance, the inequality 0<1 is formalized as the membership relation ∅∈{∅}, the inequality 1<2 is formalized as the membership relation {∅}∈{∅,{∅}}, etc. Eventually ZF enables the construction of the set of natural numbers ℕ, the ring of integers ℤ, the field of real numbers ℝ, etc.

The following questions are meant as motivation for the sequel:

1. Why should all of mathematics be based on a single relation?

2. Could such a foundational choice have been the result of historical contingency rather than mathematical necessity?

3. Do other set-theoretic foundational possibilities exist?

For the purposes of mathematical analysis, a more congenial set theory SPOT has been developed in the more versatile st-∈-language, exploiting a predicate st in addition to the relation ∈. The theory SPOT is proved to be a conservative extension of ZF. Thus SPOT does not require any additional foundational commitments beyond ZF. In particular, the axiom of choice is not required, and nonprincipal ultrafilters are not required. Here "st" is the one-place predicate standard so that st(x) is read "x is standard".

Since SPOT incorporates the axioms of ZF, the natural numbers ℕ and the real numbers ℝ are developed as usual.

The predicate st formalizes the distinction already found in Leibniz between assignable and inassignable numbers. An inassignable (nonstandard) natural number μ∈ℕ is greater than every assignable (standard) natural number in ℕ.

The reciprocal of such a μ, denoted ε=1/μ∈ℝ, is an example of a positive infinitesimal (smaller than every positive standard real). Such an ε is a nonstandard real number. A real number smaller in absolute value than some standard real number is called limited, and otherwise unlimited. SPOT proves that every nonstandard natural number is unlimited.

The theory SPOT enables one to take the standard part, or shadow, of every limited real number r, denoted sh(r). This means that the difference r - sh(r) is infinitesimal.

Then the derivative of f (x) is sh((f(x+ε) - f(x))/ε). In more detail, a standard number L is the slope of f at a standard point x if (∀ⁱⁿε) (∃ⁱⁿ ℓ) f(x+ε) - f(x) = (L+ℓ)ε. Here ∀ⁱⁿ and ∃ⁱⁿ denote quantification over infinitesimals.

The definite integral of f over [a,b] (with a, b standard) is the shadow of the infinite sum Σ_{_i=1}^μ f (x_i) ε as i runs from 1 to μ, where the x_i are the partition points of an equal partition of [a,b] into μ subintervals.

We will present the axioms that enable this effective approach (conservative over ZF) to analysis with infinitesimals. SPOT is a subtheory of axiomatic (syntactic) theories developed in the 1970s independently by Hrbacek and Edward Nelson. In addition the the axioms of ZF, SPOT has the following three axioms:

T (Transfer) Let φ be an ∈-formula with standard parameters. Then ∀^st x φ(x) → ∀x φ(x).

The intuitive meaning of the transfer axiom is that a statement known to be true for all standard inputs of the variables, should be true for all inputs. Thus, knowing that a relation cos²x+sin²x=1 holds for all standard x should entail its validity for all x. In Leibnizian terminology, whatever is true in the finite realm, should remain true in the infinite realm (in the context of axiomatic systems such as SPOT, it is preferable to use the term "unlimited integer" rather than "infinite integer").

O (Nontriviality) ∃ν ∈ ℕ ∀^st n ∈ ℕ (n ≠ ν).

Nontriviality asserts that an unlimited natural number exists (or equivalently that a nonzero infinitesimal exists).

SP (Standard Part) Every limited real is infinitely close to a standard real.

Equivalently, SP' (Standard Part) ∀A ⊆ ℕ ∃^st B ⊆ ℕ ∀^st n ∈ ℕ (n ∈ B ↔ n ∈ A).

Intuitively, the equivalence of SP and SP' can be understood in terms of the binary expansion of a real number x. If A denotes the set of ranks at which the corresponding digit of x is 1, and B denotes the set of ranks at which the corresponding digit of sh(x) is 1, then the fact that x and sh(x) are infinitely close corresponds to the fact that A and B agree at all standard (limited) ranks.

The theory SCOT is a stronger theory than SPOT. SCOT is conservative over ZF + ADC, where ADC is the axiom of dependent choice. Neither SPOT nor SCOT proves the existence of nonprincipal ultrafilters over ℕ. SCOT enables a construction of the Lebesgue measure via infinitesimal partitions and counting measures. The axiom system of Nelson's Radically Elementary Probability Theory is a subsystem of SCOT.

A nonstandard integer (greater than any naive integer such as 1,2,3) can be called a ringinal (to distinguish it from the infinite numbers of Cantorian set theory, such as ordinal or cardinal).