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{{Reading level|Advanced}}
{{Prerequisite|Logic|Formal Logic}}
{{PDF version}}

'''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 [[w:Naive_set_theory|Naive Set Theory]], there is an axiom which is known as the [[w:Axiom_schema_of_specification#Unrestricted_comprehension|unrestricted comprehension schema axiom]]. It states that there exists a set <math>x</math> such that a formula in first-order logic <math>\phi(y)</math> holds for all elements <math>y</math> in <math>x</math>, i.e., <math>x=\{y\,|\,\phi(y)\}</math>.

In 1901, [[w:Bertrand_Russell|Bertrand Russel]] found this to be inconsistent. This inconsistency is now known as [[w:Russell%27s_paradox|Russel's Paradox]]. Russel claimed that if it were consistent, then <math>R=\{x\,|\,x \notin x\}</math> is a set. Which is contradictory since <math>R \in R</math> if and only if <math>R \notin R</math>. Thus, this theory was found to be inconsistent. ('''Fun fact''': apparently [[w:Ernst_Zermelo|Zermelo]] discovered this inconsistency in 1899, but did not publish <ref>[[w:Russell%27s_paradox]]</ref>.)

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

::<small>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.</small>

==Table of Contents==
<big>Chapter</big>

[[/Introduction/]] {{stage short|0%|}}
#[[/The Language of Set Theory/]] {{stage short|100%|}}
#[[/Zermelo-Fraenkel (ZF) Axioms/]] {{stage short|75%|}}
#[[/Relations/]] {{stage short|50%|}}
#[[/Constructing Numbers/]] {{stage short|25%|}}
#[[/Orderings/]] {{stage short|0%|}}
#[[/Zorn's Lemma and the Axiom of Choice/]] {{stage short|0%|}}
#[[/Ordinals/]] {{stage short|0%|}}
#[[/Cardinals/]] {{stage short|0%|}}
<big>Appendix</big> 
#[[/Naive Set Theory/]] {{stage short|0%|}}
#[[/Sets/]] {{stage short|0%|}}
[[/Review/]] {{stage short|0%|}}

==Links==
*[[/Zermelo‒Fraenkel set theory/]] {{stage short|100%|}}
*[[/Countability/]] {{stage short|0%|}}
*[[/Systems of sets/]] {{stage short|0%|}}

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

{{Shelves|Discrete mathematics|Mathematical logic|University level mathematics books}}
{{Alphabetical|S}}
{{status|25%}}
{{BookCat}}