Infinitesimal analysis without the axiom of choice


See the article 21d by Karel Hrbacek and myself.

Analysis with infinitesimals does not require the axiom of choice any more than traditional non-infinitesimal analysis.

The canonical 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 theory SPOT is proved to be conservative over 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". 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 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 finite, and otherwise infinite. Every nonstandard natural number is infinite.

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

The derivative of f (x) is then sh((f(x+ε) - f(x))/ε). In more detail, a standard number L is the derivative of f at a standard point x if (∀in ε) (∃0in ℓ) f(x+ε) - f(x) = (L+ℓ)ε. Here ∀in denotes quantification over nonzero infinitesimals, whereas ∃0in denotes quantification over all infinitesimals (including 0).

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

We will present the axioms that enable this effective (i.e., ZF-based) approach to analysis with infinitesimals. SPOT is a subtheory of axiomatic (syntactic) theories developed in the 1970s independently by Hrbacek and Nelson.











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