Mathematical Logic

Acknowledgments.

PART I: BOOLEAN LOGIC.

1. The Beginning.

1.1 Boolean Formulae.

1.2 Induction on the Complexity of WFF: Some Easy Properties of WFF.

1.3 Inductive definitions on formulae.

1.4 Proofs and Theorems.

1.5 Additional Exercises.

2. Theorems and Metatheorems.

2.1 More Hilbertstyle Proofs.

2.2 Equational-style Proofs.

2.3 Equational Proof Layout.

2.4 More Proofs: Enriching our Toolbox.

2.5 Using Special Axioms in Equational Proofs.

2.6 The Deduction Theorem.

2.7 Additional Exercises.

3. The Interplay between Syntax and Semantics.

3.1 Soundness.

3.2 Post’s Theorem.

3.3 Full Circle.

3.4 Single-Formula Leibniz.

3.5 Appendix: Resolution in Boolean Logic.

3.6 Additional Exercises.

PART II: PREDICATE LOGIC.

4. Extending Boolean Logic.

4.1 The First Order Language of Predicate Logic.

4.2 Axioms and Rules of First Order Logic.

4.3 Additional Exercises.

5. Two Equivalent Logics.

6. Generalization and Additional Leibniz Rules.

6.1 Inserting and Removing "(∀x)".

6.2 Leibniz Rules that Affect Quantifier Scopes.

6.3 The Leibniz Rules "8.12".

6.4 More Useful Tools.

6.5 Inserting and Removing "(∃x)".

6.6 Additional Exercises.

7. Properties of Equality.

8. First Order Semantics -- Very Naïvely.

8.1 Interpretations.

8.2 Soundness in Predicate Logic.

8.3 Additional Exercises.

Appendix A: Gödel's Theorems and Computability.

A.1 Revisiting Tarski Semantics.

A.2 Completeness.

A.3 A Brief Theory of Computability.

A.3.1 A Programming Framework for Computable Functions.

A.3.2 Primitive Recursive Functions.

A.3.3 URM Computations.

A.3.4 Semi-Computable Relations; Unsolvability.

A.4 Godel's First Incompleteness Theorem.

A.4.1 Supplement: øx(x) " is first order definable in N.

References.

Index.