Chapter 1: Introduction to Hermeneutics
1.1 What is hermeneutics?
1.2 Why study hermeneutics?

Chapter 2: Back to Basics: Natural Numbers
2.1 Peano axioms
2.2 Addition
2.3 Multiplication

Chapter 3 Set Theory
3.1 Basics of Set Theory
3.2 Russell's Paradox
3.3 Function
3.4 Up and down
3.5 Cartesian product
3.6 Set size

Chapter 4 Integers and Rational Numbers
4.1 Integer
4.2 Rational numbers
4.3 Absolute value and powers
4.4 Intervals between rational numbers

Chapter 5 Connectivity
5.1 Cauchy sequence
5.2 Equivalents of the Cauchy sequence
5.3 Composition of errors
5.4 Order of errors
5.5 Least upper bound property
5.6 Real Numbers Exponentiated I

Chapter 6 Limits of Sequences
6.1 Convergence and Limit Rules
6.2 Extended real number system
6.3 Upper and lower bounds of a sequence
6.4 Upper limit, lower limit, and accumulation point
6.5 Various extreme examples
6.6 Subsequences
6.7 Real Numbers Exponentiated Ⅱ

Chapter 7 Series
7.1 Finite series
7.2 Infinite series
7.3 Sum of non-negative numbers
7.4 Rearrangement series
7.5 Root Judgment and Non-Judgment Methods

Chapter 8 Infinite Sets
8.1 Additivity
8.2 Sum of infinite sets
8.3 Uncountable sets
8.4 Axiom of Choice
8.5 Ordered sets

Chapter 9 Continuous Functions in R
9.1 Subsets of the real line
9.2 Operations on real-valued functions
9.3 Limits of functions
9.4 Continuous functions
9.5 Left and right limits
9.6 Maximum Principle
9.7 Median Theorem
9.8 Monotonic functions
9.9 Equal Continuous
9.10 Limits at Infinity

Chapter 10 Differentiation of Functions
10.1 Basic Definitions
10.2 Maxima, Minima, and Derivatives
10.3 Monotonic functions and derivatives
10.4 Inverse functions and derivatives
10.5 L'Hopital's Rule

Chapter 11 Riemann Integral
11.1 Split
11.2 Functions that are constant for each piece
11.3 Riemann upper and lower integrals
11.4 Basic properties of the Riemann integral
11.5 Riemann integrability of continuous functions
11.6 Riemann integrability of monotonic functions
11.7 Non-Riemann integrable functions
11.8 Riemann-Stieltszes integral
11.9 Fundamental Theorem of Differential and Integral Calculus
11.10 Applications of the Fundamental Theorem of Differential and Integral Calculus

Appendix A: Foundations of Mathematical Logic
A.1 Mathematical propositions
A.2 Implications
A.3 Proof Structure
A.4 Variables and Quantifiers
A.5 Nested quantifiers
A.6 Examples of proofs and quantifiers
Equation A.7


Appendix B Decimal System
B.1 Decimal representation of natural numbers
B.2 Decimal numbers as real numbers

A different approach to hermeneutics than traditional textbooks! An introductory book on hermeneutics that provides a rigorous understanding of mathematical concepts.

There are many books on the market on the subject of hermeneutics.
Typically, the course begins with a definition of limit using the epsilon-delta (ε-δ) argument and then revisits differential and integral calculus. However, despite being the first major subject taught in mathematics and mathematics education departments, not many students understand the analysis textbook. Terence Tao, who taught analysis at UCLA, raised questions about this point.
In typical lectures, it is assumed that students already 'know' the basic concepts, but I noticed that in reality, those students do not clearly understand the concepts.
The book that emerged from these considerations is [TAO Hermeneutics I (4th Edition)].
The author explains in his own easy-to-understand and friendly style how to derive rigorous logic from familiar concepts.
If you study with [TAO Analysis II (4th Edition)], you will be able to clearly understand various concepts, starting from the basics of mathematics to the general topics of analysis.