# 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 ${\displaystyle x}$ such that a formula in first-order logic ${\displaystyle \phi (y)}$ holds for all elements ${\displaystyle y}$ in ${\displaystyle x}$, i.e., ${\displaystyle x=\{y\,|\,\phi (y)\}}$.

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 ${\displaystyle R=\{x\,|\,x\notin x\}}$ is a set. Which is contradictory since ${\displaystyle R\in R}$ if and only if ${\displaystyle R\notin R}$. 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.

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