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Weak set theories in foundational debates.

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Summary

This study explores weaker formal logic systems, introducing PROVI set theory to support constructibility and forcing techniques. New results on illfounded models and formal systems are presented, advancing proof theory research.

Keywords:
mathematical logicproof theoryweak set theory

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Area of Science:

  • Mathematical Logic
  • Set Theory
  • Proof Theory

Background:

  • The historical pursuit of a single logical system for all mathematics has yielded to the need for weaker, context-specific formal systems.
  • Logic's interdisciplinary connections with mathematics, philosophy, computer science, biology, economics, engineering, and physics are evolving.
  • Current research focuses on developing flexible logical frameworks applicable across diverse scientific domains.

Purpose of the Study:

  • To introduce the novel, very weak set theory PROVI.
  • To demonstrate PROVI's capacity to support foundational techniques like constructibility and forcing.
  • To explore new results concerning illfounded models and formal systems in the style of Quine.

Main Methods:

  • Introduction of the PROVI set theory.
  • Outline of PROVI's support for constructibility and forcing techniques.
  • Analysis of illfounded [Formula: see text]-models and development of new formal systems.

Main Results:

  • PROVI is shown to support constructibility and forcing, key techniques in mathematical logic.
  • New findings regarding illfounded models of PROVI and related systems are presented.
  • Novel formal systems, inspired by Quine's work, are described.

Conclusions:

  • Weaker formal systems like PROVI offer versatile applications across various scientific disciplines.
  • The study contributes to modern perspectives in proof theory by exploring new formal systems and models.
  • This research highlights the ongoing evolution of logic and its foundational role in science.