I. Statement Systems and Propositional Calculus.- 1. Statement Systems.- 1.1. Statement Systems.- 1.2. Language of a Statement System.- 1.3. Names for Swffs.- 2. Propositional Calculus.- 2.1. Well-Formed Formulas.- 2.2. Parentheses.- 2.3. Main Connective of Wffs.- 2.4. Names for Wffs; Principal Connective of Name.- 2.5. Valuations.- 3. Provable Wffs.- 3.1. ?-Interpreters.- 3.2. True Wffs.- 3.3. Proofs and Provable Wffs.- 3.4. Rules of Inference.- 3.5. Equivalent Wffs.- 4. Substitution Theorems.- 4.1. Subwffs, Components, and Wff-Builders.- 4.2. Substitution Theorem for Wffs.- 5. Duality.- 5.1. Normal Form.- 5.2. Syntactical Transforms.- 5.3. Normal Transforms.- 5.4. Duality.- 6. Deducibility and Completeness.- 6.1. More Provable Wffs.- 6.2. Conjunctive Normal Form.- 6.3. Completeness.- 6.4. Deducibility.- 6.5. Consistent Sets and Contradictory Sets.- 6.6. Maximal-Consistent Sets.- 6.7. Strong Completeness Theorem.- II. Semantical Systems and Predicate Calculus.- 7. Semantical Systems.- 7.1. Relational Systems.- 7.2. Semantical Systems.- 7.3. Language of a Semantical System.- 7.4. Extensions; Elementary Extensions.- 8. Predicate Calculus.- 8.1. Well-Formed Formulas.- 8.2. Parentheses.- 8.3. Main Connective of Wffs.- 8.4. Names for Wffs; Principal Connective of Name.- 8.5. Syntactical Transforms.- 8.6. Interchange and Substitution Transforms.- 8.7. Valuations.- 9. Provable Wffs.- 9.1. ?-Interpreters.- 9.2. True Wffs.- 9.3. Proofs and Provable Wffs.- 9.4. Equivalent Wffs.- 9.5. Rules of Inference.- 9.6. A Fact about the Interchange Transform.- 10. Substitution Theorems.- 10.1. Subwffs, Components, and Wff-Builders.- 10.2. Substitution Theorem for Wffs.- 11. Duality.- 11.1. Normal Form.- 11.2. Normal Transforms.- 11.3. Duality.- 11.4. Prenex Normal Form.- 12. Deducibility and Completeness.- 12.1. Deducibility.- 12.2. Consistent Sets and Contradictory Sets.- 12.3. Strong Completeness Theorem.- 12.4. Maximal-Consistent Sets.- 12.5. ?-Complete Sets.- 12.6. Proof of the Strong Completeness Theorem.- III. Applications.- 13. Nonstandard Analysis.- 13.1. Extended Natural Number System.- 13.2. Extended Real Number System.- 13.3. Properties of ?.- 13.4. Paradoxes.- 13.5. The Limit Concept.- 13.6. Continuity; Uniform Continuity.- 13.7. Principles of Permanence.- 14. Normal Semantical Systems.- 14.1. Equality Relations.- 14.2. Normal Semantical Systems.- 14.3. Löwenheim-Skolem Theorem.- 14.4. Theories.- 15. Axiomatic Set Theory.- 15.1. Introduction.- 15.2. Axiom of Extensionality.- 15.3. Axiom Scheme of Replacement.- 15.4. Axiom of Power Set.- 15.5. Axiom of Sum Set.- 15.6. Axiom of Infinity.- 15.7. Nonstandard Set Theory.- 15.8. Axiom of Regularity.- 15.9. Axiom of Choice.- 16. Complete Theories.- 16.1. Vaught's Test.- 16.2. Diagrammatic Sets.- 16.3. Simplifying the Concept of a Model.- 16.4. Robinson's Test.- Symbol Index.
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.