maths/pure/ AbstractSetTheory101

Here is a rough overview of the basics of AbstractSetTheory, in particular ZFC and how we construct things like numbers and functions.

What is 'abstract' about Abstract Set Theory?

We never worry about, or ask about, what sets actually are. In the formal language of SetTheory, there is only one non-logical symbol: ∈. Abstract set theory is all about the relation ∈. We begin with a set of a priori assumptions about properties that any sensible notion of ∈ should satisfy. We call these axioms. There are a few different sets of axioms for Set Theory, but the most common is ZFC (Zermelo-Fraenkel with the AxiomOfChoice).

Axioms

1. Extensionality: two sets are equal if and only if they have the same elements. ff0cfe41ea95dff085ea9116d05e646908bab8323fe4c02f5f5b2473fa236ed6d4735e3a265e16eee03f59718b9b5d03019c07d8b6c51f90da3a666eec13ab35
2. Regularity: every non-empty set contains an element ff0cfe41ea95dff085ea9116d05e646908bab8323fe4c02f5f5b2473fa236ed64b227777d4dd1fc61c6f884f48641d02b4d121d3fd328cb08b5531fcacdabf8a disjoint from ff0cfe41ea95dff085ea9116d05e646908bab8323fe4c02f5f5b2473fa236ed6ef2d127de37b942baad06145e54b0c619a1f22327b2ebbcfbec78f5564afe39d. ff0cfe41ea95dff085ea9116d05e646908bab8323fe4c02f5f5b2473fa236ed64e07408562bedb8b60ce05c1decfe3ad16b72230967de01f640b7e4729b49fce