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https://github.com/choukh/Set-Theory

A formalization of the textbook Elements of Set Theory
https://github.com/choukh/Set-Theory

coq formal-languages math set-theory theorem-proving

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A formalization of the textbook Elements of Set Theory

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README 

        
[中文](./README.zh-CN.md) 👈



# Set-Theory



This project is a Coq formalization of the textbook Elements of Set Theory - Herbert B. Enderton. It is basically written in the order of the textbook, without considering modularity. It is suitable as an aid to the learning of set theory, not as a general mathematical library.



## Requirement

```

Coq 8.13.2

```



## Build

```

make

```



## Meta.v

- Law of excluded middle

- Church's iota operator

- Informative excluded middle

- Decidable inhabitance of type



## ZFC0.v

- Axiom of extensionality

- Axiom of empty set

- Axiom of union

- Axiom of power set

- Axiom schema of replacement



## ZFC1.v

- Pair

- Singleton

- Binary union

- Union of a family of sets



## ZFC2.v

- Set comprehension

- Intersaction, binary intersaction

- Ordered pair

- Cartesian product



## ZFC3.v

- Axiom of infinity

- Axiom of choice



## EST2.v

- Complement

- Proper subset

- Algebra of sets



## EST3_1.v

- Relation, function

- Inverse, composition



## EST3_2.v

- Injection, surjection, bijection

- Left inverse and right inverse of function

- Restriction, image

- Function space

- Infinite Cartesian product



## EST3_3.v

- Binary relation

- Equivalence relation, equivalence class, quotient set

- Trichotomy, linear order



## EST4_1.v

- Natural number

- Induction principle

- Transitive set

- Peano structure

- Recursion theorem



## EST4_2.v

- Embedding of type-theoretic nat

- Natural number arithmetic: addition, multiplication, exponentiation



## EST4_3.v

- Linear ordering of ω

- Well ordering of ω

- Strong induction principle



## EST5_1.v

- Integer

- Integer arithmetic: addition, additive inverse



## EST5_2.v

- Multiplication of integers

- Order of integers

- Embedding of the natural numbers



## EST5_3.v

- Rational number

- Rational number arithmetic: addition, additive inverse, multiplication, multiplicative inverse



## EST5_4.v

- Order of rational numbers

- Embedding of the integers

- Algebra regarding to inverse



## EST5_5.v

- Real number (Dedekind cut)

- Order of real numbers

- Completeness of the real numbers

- Real number arithmetic: addition, additive inverse



## EST5_6.v

- Absolute value of real number

- Multiplication of non-negative real numbers

- Multiplicative inverse of positive real number



## EST5_6.v

- Arithmetic of rational numbers: multiplication, multiplicative inverse

- Embedding of the rational numbers

- Density of the real numbers



## EST6_1.v

- Equinumerous

- Cantor's theorem

- Pigeonhole principle

- Finite cardinal



## EST6_2.v

- Infinite cardinal

- Cardinal arithmetic: addition, multiplication, exponentiation



## EST6_3.v

- Dominate

- Schröder–Bernstein theorem

- Order of cardinals

- Aleph Zero



## EST6_4.v

- Systematic discussion on AC

  - Uniformization

  - Infinite Cartesian product of nonempty sets is nonempty

  - Choice function

  - Cardinal comparability

  - Zorn's lemma

  - Tukey's lemma

  - Hausdorff maximal principle

- Aleph Zero is the least infinite cardinal

- Dedekind infinite

- Infinite sum of cardinals

- Infinite product of cardinals



## EST6_5.v

- Countable set

  - Countable union of countable sets is countable



## EST6_6.v

- Algebra of infinite cardinals

  - Cardinal multiplied by itself equals to itself

  - Absortion law of cardinal addition and multiplication



## EST7_1.v

- Partial order, linear order

- Minimal, minimum, maximal, maximum

- Bound, supremum, infimum



## EST7_2.v

- Well order

- Transfinite induction principle

- Transfinite recursion theorem

- Transitive closure of set



## EST7_3.v

- Order structure

- Isomorphism

- Epsilon image



## EST7_4.v

- Ordinal

- Order of ordinals

- Burali-Forti's paradox

- Successor ordinal, limit ordinal

- Transfinite induction schema on ordinals



## EST7_5.v

- Hartog's number

- Equivalence among well order theorem, AC and Zorn's lemma

- von Neumann cardinal assignment

- Initial cardinal, successor cardinal



## EST7_6.v

- Transfinite recursion schema on ordinals

- von Neumann universe

- Rank

- Axiom of regularity



## EST8_1.v

- Ordinal class

- Ordinal operations

  - Subclass separation

  - Normal operation

- Aleph number

- Beth number



## EST8_2.v

- Properties of ordinal operations

- Veblen fixed-point theorem

  - Enumeration of fixed-point is normal operation

  - There exist fixed-point of fixed-point



## EST8_3.v

- Order types

- Addition of order types



## EST8_4.v

- Multiplication of order types

- Laws of order type arithmetic



## EST8_5.v

- Order type arithmetic on well-ordered structure



## EST8_6.v

- Ordinal Arithmetic (defined as order type arithmetic)

  - Addition, multiplication



## EST8_7.v

- Ordinal Arithmetic (defined by recursion)

  - Addition, multiplication, exponentiation

- Tetration, epsilon numbers



## EX{n}.v

- Solution to exercises of Chapter n