In this thesis, we introduce the axiomatic set theory known as New foundations. Firstly, we define the basic axioms and properties and introduce some important constructions. Next, we define cardinal and ordinal numbers and state some results regarding them. We consider the axiom of counting, prove the axiom of infinity and disprove the axiom of choice. We prove that the category of sets is not Cartesian closed. Lastly, we prove the consistency of a variant of New foundations with urelements.
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