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Axioms for Arbitrary Object Theory

Luca Steinkrauss, Leon Horsten

TL;DR

AOT formalizes arbitrary object theory within a classical first-order framework, introducing a structured ontology of particular objects, arbitrary objects, states, and sets, and an abstraction mechanism via a two-place $F$ and a three-place $Val$ that connect particular object systems to their abstracted arbitrary-object systems. The theory enforces a partition of the domain, a comprehension-like abstraction, and extensionality principles to avoid duplication, while demonstrating consistency through simple models and offering a definitional core using Ur-element set theory. Through simple examples and properties, the paper shows how arbitrary-object systems can be systematically constructed from particular objects and clarified by a mathematical backbone, while acknowledging current scope limitations and outlining paths for future enrichment, including reductions and higher-order variants.

Abstract

We formulate and discuss a general axiomatic theory of arbitrary objects. This theory is expressed in a simple first-order language without modal operators, and it is governed by classical logic.

Axioms for Arbitrary Object Theory

TL;DR

AOT formalizes arbitrary object theory within a classical first-order framework, introducing a structured ontology of particular objects, arbitrary objects, states, and sets, and an abstraction mechanism via a two-place and a three-place that connect particular object systems to their abstracted arbitrary-object systems. The theory enforces a partition of the domain, a comprehension-like abstraction, and extensionality principles to avoid duplication, while demonstrating consistency through simple models and offering a definitional core using Ur-element set theory. Through simple examples and properties, the paper shows how arbitrary-object systems can be systematically constructed from particular objects and clarified by a mathematical backbone, while acknowledging current scope limitations and outlining paths for future enrichment, including reductions and higher-order variants.

Abstract

We formulate and discuss a general axiomatic theory of arbitrary objects. This theory is expressed in a simple first-order language without modal operators, and it is governed by classical logic.
Paper Structure (11 sections, 14 theorems, 14 equations, 5 figures)

This paper contains 11 sections, 14 theorems, 14 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Generic attribution
  3. Axioms
  4. Some basic elements of H-theory
  5. Language and definitions
  6. Background axioms
  7. Abstraction
  8. Simple examples
  9. Basic properties
  10. A definitional approach
  11. In conclusion

Key Result

Proposition 1

Suppose that a totally arbitrary object exists, and at least one object exists. Then where $d$ is any individual constant for an object, and where $\vdash_{CL}$ is derivability in classical first-order logic (CL).

Figures (5)

  • Figure 1: The partition of the domain of discourse by $P,S,A,Set$. (The hatched area is empty.)
  • Figure 2: Illustration of the structure postulated by Axiom \ref{['abstractionaxiom']}
  • Figure 3: Illustration of Axiom 10.
  • Figure 4: Representation of Example 1.
  • Figure 5: A 1-0-diagonal arbitrary object system of size $n$.

Theorems & Definitions (33)

  • Definition 1: PGA
  • Proposition 1
  • Definition 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Definition 6
  • Definition 7
  • Definition 8
  • Proposition 2
  • ...and 23 more