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The disjunction property implies the numerical existence property.

H Friedman1

  • 1Department of Mathematics, State University of New York at Buffalo, 4246 Ridge Lea Road, Amherst, N.Y. 14226.

Proceedings of the National Academy of Sciences of the United States of America
|August 1, 1975
PubMed
Summary

Recursively enumerable extensions of intuitionistic arithmetic that satisfy the disjunction property also satisfy the numerical existence property. Such systems prove their disjunction property if and only if they prove their own inconsistency.

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Area of Science:

  • Mathematical logic
  • Foundations of mathematics
  • Proof theory

Background:

  • Intuitionistic arithmetic is a foundational system in constructive mathematics.
  • The disjunction property and numerical existence property are key features of intuitionistic logic.
  • Recursively enumerable theories are central to computability and formal systems.

Purpose of the Study:

  • To investigate the relationship between the disjunction property and the numerical existence property in extensions of intuitionistic arithmetic.
  • To explore the provability of these properties within recursively enumerable theories.
  • To establish conditions under which these properties are equivalent to proving inconsistency.

Main Methods:

  • Axiomatic system analysis

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  • Proof-theoretic techniques
  • Recursive enumerability considerations
  • Model-theoretic interpretations (if applicable, though not explicitly stated in abstract)

    • Main Results:

    • Demonstration that any recursively enumerable extension of intuitionistic arithmetic obeying the disjunction property also obeys the numerical existence property.
    • Proof that such extensions prove their own disjunction property if and only if they prove their own inconsistency.

    Conclusions:

    • The disjunction property and numerical existence property are closely linked in these formal systems.
    • The ability of a theory to prove its own disjunction property serves as a marker for its potential inconsistency.
    • These findings contribute to understanding the self-referential capabilities and limitations of formal arithmetic systems.