# Advanced Mathematical Logic Track

## University Mathematics

Formal mathematical logic is the foundation on which all of mathematics and mathematical reasoning is built. This track provides a rigorous, university-level treatment of this area of mathematics.

Students who complete the entire IMACS Advanced Mathematical Logic track typically will have an "unfair advantage" with a mathematical foundation that will make all technical classes significantly easier. Former students remark on this effect in courses ranging from physics to philosophy to computer science to pre-law.

This sequence of courses begins with the subject matter of the logic courses that are a required part of a college major in mathematics, engineering, computer science or philosophy, and goes on to introduce the techniques in logic and reasoning that underpin research and development in mathematics.

Students are introduced to the branches of mathematics called "propositional logic", "predicate logic" and "set theory". The emphasis throughout is on developing a true understanding for the logical underpinning of mathematics. The track consists of the following three classes:

**Introduction to Logic I**
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This course introduces students to Propositional Logic, a branch of
modern mathematics that provides the foundation for formal and
rigorous mathematical proofs.

Introduction

Logic Puzzles

Logical Thinking

Problem Set LM1.1

The Formal Language

When Is An Argument A Proof?

Analyzing An Argument

Truth and Falsity: Simple Statements

Truth and Falsity: Compound Statements

The Language of the Propositional Calculus

Well-formed Formulas

Interpreting Well-formed Formulas

Decomposing Well-formed Formulas

Instances of Well-formed Formulas

Introduction to Truth Tables

The Truth Table for Conjunction

Building Truth Tables

The Truth Table for Disjunction

The Truth Table for Implication

Problem Set LM1.2

Tautologies

Equivalent Well-formed Formulas

The Biconditional

Problem Set LM1.3

Transitivity of Equivalence

Converse and Contrapositive

Equivalence and Tautologies

Wffs, Meta-wffs, and Instances

Lifting Well-formed Formulas

Truth Table Templates

The Tautology Principle

Problem Set LM1.4

Review for Test LM1.1

Introducing Demonstrations

Applying Modus Ponens

Examining an Argument

What is a Demonstration?

Demonstrations and Program Outlines

Problem Set LM1.5

Conjunctive Inference

Conjunctive Simplification

Problem Set LM1.6

Unnecessary Hypotheses

Contrapositive Inference and Modus Tollens

Applying Modus Tollens

Problem Set LM1.7

Syllogistic Inference

Inference by Cases

Problem Set LM1.8

Modus Ponens for the Biconditional

Commutativity and Transitivity of the Biconditional

Contrapositive Inference and Modus Tollens for the Biconditional

Biconditional Inference

The Substitution Principle

Problem Set LM1.9

Tautologies and Demonstrations

The Substitution Principle and Equivalence

Consequences of the Substitution Principle

Rules of Inference: A Summary

Problem Set LM1.10

Review for Test LM1.2

Working With Demonstrations

Using The Deduction Theorem

Problem Set LM1.11

More Uses of the Deduction Theorem

Problem Set LM1.12

Justifying the Deduction Theorem

Problem Set LM1.13

Reflecting on the Deduction Theorem

Language and Metalanguage

Indirect Inference

Problem Set LM1.14

Justifying Indirect Inference

Using Indirect Inference

Review for Test LM1.3

**Introduction to Logic II**
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This course introduces students to Predicate Logic, a so-called "first-order logic"
sufficient to formalize all of *set theory*, which provides the basic
language in which most mathematical texts are written.

A Logic for Set Theory

A New Formal Language

Interpreting Terms and Formulas

Free and Bound Occurrences

Open and Closed Terms and Formulas

Problem Set LM2.1

Simplifying the Notation

Rebuilding Terms and Formulas

Problem Set LM2.2

Metaterms, Metaformulas, and Instances

Introducing Tautologies

Demonstrations

Rules of Inference: A Summary

A Sample Demonstration

Problem Set LM2.3

Axioms for the Predicate Calculus

Problem Set LM2.4

Using Axioms in Demonstrations

Rules of Inference for Axioms

Demonstrations Revisited

Using IU and PGU

Problem Set LM2.5

Review for Test LM2.1

Theorems and Metatheorems

Problem Set LM2.6

Using Metatheorems in Demonstrations

Streamlining Demonstration Outlines

The Existential Quantifier

Properties of the Existential Quantifier

Problem Set LM2.7

The Substitution Principle

Naming Objects that Exist

Inference from an Existential

Rules and Regulations

Problem Set LM2.8

Demonstrations Revisited Again

Review for Test LM2.2

**Set Theory**
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This course introduces students to the axiomatic set theory of John von Neumann,
Paul Bernays and Kurt Godel ("NBG"), which plays a central role in modern
mathematics and is fundamental to understanding math at its most sophisticated levels.
(*Note: A small, select group of graduates of this class may be invited to take a sequence
of extraordinarily advanced courses based upon the highly rigorous* Elements of
Mathematics *curriculum.*)

The Theory of Sets and Classes

Properties of Equality

Using Theorems as Metatheorems

Substitution of Equals

Problem Set LM3.1

Subclasses

Unique Existence

Axioms for Described Terms

Problem Set LM3.2

Working With Described Terms

Sets

Class Symbols

The Comprehension Principle

Problem Set LM3.3

Using The Comprehension Principle

Some Special Classes

Russell's Paradox

Problem Set LM3.4

The Empty Set

Rules and Regulations

Review for Test LM3.1

Complements and Differences

Unions and Intersections

Power Classes

Problem Set LM3.5

Singletons and Doubletons

Manifold Union and Intersection

Problem Set LM3.6

The Whole Numbers

Mathematical Induction

The Peano Postulates

Ordering the Whole Numbers

The Well-ordering Principle

The Axiom of Regularity

Problem Set LM3.7

Review for Test LM3.2