logicless non-standard analysis: an axiom...

Logicless Non-Standard Analysis: AnAxiom System

Abhijit Dasgupta

University of Detroit Mercy

June 3, 2008

Abhijit Dasgupta Logicless Non-Standard Analysis: An Axiom System

Getting reals from rationals: Construction vs Axiomatic setup

Reals from rationals: ConstructionDedekind’s method of cuts (order-completion), orCantor’s method of using equivalence classes of Cauchysequences of rationals (metric completion)Provides existence proof, and classic techniques

But once the construction is done, no use is ever made of howthe reals are constructed! And all we need in practice are theaxioms for a complete ordered field:

Reals from rationals: Axiomatic setupAxioms for complete ordered fieldsProvides rigorous framework for real numbersAvoids getting bogged down with the construction of realsPrimary approach in many modern real analysis textbooks

Getting hyperreals from reals: Construction

Or more generally: Obtaining proper elementaryextensions of the structure of all functions and relations ona set AUseful in developing infinitesimals rigorously without logic,as in some modern calculus texts (Keisler, Crowell)

How to construct proper elementray extensions

Logical methods (Lowenheim-Skolem / compactnessarguments): Not appropriate for non-logicians

The ultrapower construction (over non-principal ultrafilters):1 Avoids logic2 Sufficiently algebraic (?) for non-logicians (cf. quotient field

from a commutative ring over a maximal ideal)

Some Basic Terminology

Total and Partial functions, Projections, Compositionf is an n-ary total function on A ↔ f : An → Af is an n-ary partial function on A ↔ f : D → A, D ⊆ An

f is the k-th n-ary projection over A (1 ≤ k ≤ n) ↔f : An → A and f (x1, . . . , xn) = xkGeneral compositions (substitutions) of partial functions:Example: If φ(x, y, z, w) ≡ f (x, g(y, z), h(w)), then φ is acomposition of f , g, h

The setup for axiomatic approach to elementary extensions

Extending the collection of all partial functions on a setA : A fixed set, together with the collection of all partialfunctions on AB : A proper superset of A, i.e. A ( BThe transform: To every partial function f on A, there isassociated a partial function ∗f on B with the same arity,called the transform of f

The Axioms

The Five AxiomsAxiom 1 (Projection Function Axiom). If f is a projectionover A, then ∗f is the corresponding projection over BAxiom 2 (Constant Function Axiom). If f is a constantfunction over A, then ∗f is the constant function over B withthe same arity and taking same constant value as fAxiom 3 (Composition Axiom). Composition of partialfunctions are preserved ∗(f ◦ g) = ∗f ◦ ∗g, where f and g arepartial functions on A; and similarly for more general formsof compositionAxiom 4 (The Domain Axiom). If the domain of a partial(n + 1)-ary function f is itself a partial (n-ary) function g,then dom(∗f ) = ∗gAxiom 5 (The Finiteness Axiom). Finite functions areinvariant: If dom(f ) is finite then ∗f = f

The Axioms

Transforms of relations

Defining transforms of relationsFix a ∈ AGiven a relation R on A (i.e. R ⊆ An), identify R with thepartial constant function fR having domain R and taking theconstant value aLet ∗R be defined as the domain of ∗fRThis definition of ∗R is independent of the choice of theelement a ∈ A, assuming that axioms 1–5 hold

Main Result

An axiomatic approach to full elementary extensionsLet

1 LA = The language which consists of all relations andfunctions on A

2 A = The structure over A where each symbol of LA isinterpreted as itself

3 B = The structure over B where each symbol of LA isinterpreted as its transform

Then, under axioms 1–5, we have: A 4 B, i.e. A must be anelementary substructure of B.

logicless non-standard analysis: an axiom...

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