# Set Theory

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**Set Theory** is the study of *sets*. Essentially, a *set* is a collection of mathematical objects. Set Theory forms the foundation of *all* of mathematics.

In Naive Set Theory, there is an axiom which is known as the unrestricted comprehension schema axiom. It states that there exists a set such that a formula in first-order logic holds for all elements in , i.e., .

In 1901, Bertrand Russel found this to be inconsistent. This inconsistency is now known as Russel's Paradox. Russel claimed that if it were consistent, then is a set. Which is contradictory since if and only if . Thus, this theory was found to be inconsistent. (**Fun fact**: apparently Zermelo discovered this inconsistency in 1899, but did not publish ^{[1]}.)

This motivated Zermelo to Axiomatize Set Theory. And motivates why we, too, should study this.

- This is an undergraduate book, but will also include some graduate level topics. But mainly, anyone with basic
*mathematical maturity*can engage in this book.

- This is an undergraduate book, but will also include some graduate level topics. But mainly, anyone with basic

## Table of Contents

[edit | edit source]Chapter

- The Language of Set Theory
- Zermelo-Fraenkel (ZF) Axioms
- Relations
- Constructing Numbers
- Orderings
- Zorn's Lemma and the Axiom of Choice
- Ordinals
- Cardinals

Appendix

## Links

[edit | edit source]## Further reading

[edit | edit source]- Discrete Mathematics/Set theory
- Krzysztof Ciesielski,
*Set Theory for the Working Mathematician*(1997) - P. R. Halmos,
*Naive Set Theory*(1974) - Karel Hrbacek, Thomas J. Jech,
*Introduction to set theory*(1999) - Thomas J. Jech,
*Set Theory*3rd Edition (2006) - Kenneth Kunen,
*Set Theory: an introduction to independence proofs*(1980) - Judith Roitman,
*Introduction to Modern Set Theory*(1990) - John H. Conway, Richard Guy
*The Book of Numbers*- chapter 10 - Tobias Dantzig, Joseph Mazur
*Number: The Language of Science*