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Finite axiomatizability of theories in the predicate calculus using additional predicate symbols. Two papers on the predicate calculus

About this Title

S. C. Kleene

Publication: Memoirs of the American Mathematical Society
Publication Year: 1952; Number 10
ISBNs: 978-0-8218-1210-5 (print); 978-0-8218-9889-5 (online)
DOI: https://doi.org/10.1090/memo/0010
MathSciNet review: 0051186

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Table of Contents

Chapters

  • Permutability of inferences in Gentzen’s calculi $LK$ and $LJ$
  • 1. The formal system $G$
  • 2. Relationships in a proof in $G$
  • 3. Interchange of adjacent logical inferences
  • 4. Permutability of logical inferences
  • Finite axiomatizability of theories in the predicate calculus using additional predicate symbols
  • Part I. Construction of the formal system, and proof of completeness
  • 1. $H$-numbers
  • 2. A primitive recursive enumeration of $H$-numbers
  • 3. The system $S_1$
  • 4. The systems $S_2, S_3$
  • 5. The systems $S_0,S$
  • Part II. Proof of consistency
  • 6. A non-constructive proof of consistency for the classical case
  • 7. Effect of the equality axioms with a new equality symbol
  • 8. Effect of the axioms $S_1$
  • 9. Effect of the axiom $S_2$
  • 10. Effect of the axioms of $S_3$
  • 11. Conclusion of the consistency proof