Keisler's Function Extension Axiom

Roughly speaking, Keisler's Function Extension Axiom says that all real functions have extensions to the hyperreal numbers and these "natural" extensions obey the same identities and inequalities as the original function.  Some familiar identities are

Sin[α + β] = Sin[α] Cos[β] + Cos[α] Sin[β]

Log[x · y] = Log[x] + Log[y]

The log identity only holds when x and y are positive.  Keisler's Function Extension Axiom is formulated so that we can apply it to the Log identity in the form of the implication

(x>0&y>0) ⇒ Log[x] and Log[y] are defined and Log[x · y] = Log[x] + Log[y]

The Function Extension Axiom guarantees that the natural extension of Log[·] is defined for all positive hyperreals and its identities hold for hyperreal numbers satisfying x>0 and y>0.

We can state the addition formula for sine as the implication

(α = α& β = β) ⇒
    Sin[α], Sin[β], Sin[α + β], Cos[α], Cos[β] are defined
        and
    Sin[α + β] = Sin[α] Cos[β] + Cos[α] Sin[β]

Logical Real Expressions

Logical real expressions are built up from numbers and variables using functions.

(a) A real number is a real expression.

(b) A variable standing alone is a real expression.

(c) If E_1,E_2,··· ,E_n are a real expressions and f[x_1, x_2, · · · , x_n] is a real function of n variables, then f[E_1, E_2, · · · , E_n] is a real expression.

Logical Real Formulas

A logical real formula is one of the following:

(i) An equation between real expressions, E_1 = E_2

(ii) An inequality between real expressions, E_1 < E_2,E_1 ≤ E_2,E_1 > E_2,E_1 ≥ E_2, or E_1 ≠ E_2=E2.

(iii) A statement of the form "E is defined or of the form E is undefined."A statement of the form “E is defined”oroftheform“Eis undefined.” A statement of the form “E is defined”oroftheform“Eis undefined.” A statement of the form “E is defined”oroftheform“Eis undefined.”

Logical Real Statements

Let S and T be finite sets of real formulas. A logical real statement is an implication of the form,

S⇒T

The functional identities for sine and log given above are logical real statements.

Keisler's Function Extension Axiom

Every real function f[x_1, x_2, · · · , x_n] has a "natural" extension to the hyperreals such that every logical real statement that holds for all real numbers also holds for all hyperreal numbers when the real functions in the statement are replaced by their natural extensions.

There are two general uses of the Function Extension Axiom that underlie most of the theoretical problems in calculus. These involve extension of the discrete maximum and extension of finite summation. The proof of the Extreme Value Theorem below uses a hyperfinite maximum, while the proof of the Fundamental Theorem of Integral Calculus uses hyperfinite summation and a maximum.

Equivalence of infinitesimal conditions and the “epsilon - delta” real number conditions are usually proved by using an auxiliary real function as in the following proof.

Theorem: Simple Equivalency of Limits and Infinitesimals

Let f [x] be a real valued function defined for 0 < | x −a | <Δ with Δ a fixed positive real number. Let b be a real number. Then the following are equivalent:

(a) Whenever the hyperreal number x satisfies a≠x≈a, the natural extension function satisfies

f[x] ≈b

(b) For every real accuracy tolerance θ there is a sufficiently small positive real number γ such that if the real number x satisfies 0< | x − a | <γ,then

| f[x] −b | <θ

Condition (b) is the familiar Weierstrass "epsilon-delta" condition (written with θ and γ.)  Notice that the condition f[x] ≈b is NOT a logical real statement because the infinitesimal relation is NOT included in the formation rules for forming logical real statements.

Proof

We show that (a) ⇒ (b) by proving that not (b) implies not (a), the contrapositive. Assume (b) fails. Then there is a real θ>0 such that for every real γ>0 there is a real x satisfying 0 < | x − a | <γ and | f[x] − b | ≥θ. Let X[γ] = x be a real function that chooses such an x for a particular γ. Then we have the equivalence

γ>0⇔ (X[γ] is defined, 0< | X[γ] −a | <γ, | f[X[γ]] −b | ≥θ)

By the Function Extension Axiom this equivalence holds for hyperreal numbers and the natural extensions of the real functions X[·] and f[·]. In particular, choose a positive infinitesimal γ and apply the equivalence. We have 0 < | X[γ] − a | <γ and | f[X[γ]] −b | >θ and θ is a positive real number. Hence, f[X[γ]] is not infinitely close to b, proving not (a) and completing the proof that (a) implies (b).

Conversely, suppose that (b) holds. Then for every positive real θ, there is a positive real γ such that 0 < | x − a | <γ implies | f[x] − b | <θ. By the Function Extension Axiom, this implication holds for hyperreal numbers. If ξ ≈ a,then 0< | ξ−a | <γ for every real γ, so | f[ξ] −b | <θ for every real positive θ. In other words, f[ξ] ≈b, showing that (b) implies (a) and completing the proof of the theorem.

Other examples of uses of the Function Extension Axiom are given below.  We use Keisler's foundations to complete the proofs of the basic results of Section 1.

The Differential Approximation is NOT a logical real expression

The implication

δx≈0 ⇒ (f[x + δx] - f[x])/(δx) ≈f^′[x]

or the differential approximation

f[x + δx] - f[x] = f ' [x] · δx + ε · δx

with ε≈0 whenever δx≈0 are not real expressions because they involve the infinitesimal relation.

Created by Mathematica  (September 22, 2004)