Mathematical Logic

Before his death in March, 1976, A. H. Lightstone delivered the manu script for this book to Plenum Press. Because he died before the editorial work on the manuscript was completed, I agreed (in the fall of 1976) to serve as a surrogate author and to see the project through to completion. I have changed the manuscript as little as possible, altering certain passages to correct oversights. But the alterations are minor; this is Lightstone's book. H. B. Enderton vii Preface This is a treatment of the predicate calculus in a form that serves as a foundation for nonstandard analysis. Classically, the predicates and variables of the predicate calculus are kept distinct, inasmuch as no variable is also a predicate; moreover, each predicate is assigned an order, a unique natural number that indicates the length of each tuple to which the predicate can be prefixed. These restrictions are dropped here, in order to develop a flexible, expressive language capable of exploiting the potential of nonstandard analysis. To assist the reader in grasping the basic ideas of logic, we begin in Part I by presenting the propositional calculus and statement systems. This provides a relatively simple setting in which to grapple with the some times foreign ideas of mathematical logic. These ideas are repeated in Part II, where the predicate calculus and semantical systems are studied.

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I. Statement Systems and Propositional Calculus.- 1. Statement Systems.- 2. Propositional Calculus.- 3. Provable Wffs.- 4. Substitution Theorems.- 5. Duality.- 6. Deducibility and Completeness.- II. Semantical Systems and Predicate Calculus.- 7. Semantical Systems.- 8. Predicate Calculus.- 9. Provable Wffs.- 10. Substitution Theorems.- 11. Duality.- 12. Deducibility and Completeness.- III. Applications.- 13. Nonstandard Analysis.- 14. Normal Semantical Systems.- 15. Axiomatic Set Theory.- 16. Complete Theories.- Symbol Index.