Set Theory

A foundation for mathematics, set theory asserts everything may be encoded using sets (like how on computers everything is encoded using bits). Although mathematicians believe they use sets as their foundations, few can really enumerate their axioms. Realistically, it seems the foundations employed in practice is some amalgam of set theory equipped with a soft type system, in "$(1+ε)$-order logic" for \(0\lt\varepsilon\ll1\).

We might also want to use a "many-sorted logic" to treat proper classes as a different "sort" than proper sets. Confusingly, this treatment is referred to as "second-order set theory" in the literature (as opposed to "two-sorted set theory", which would make more sense).

We could instead start with some family of collections, "Coll", which obeys the obvious ZF axioms. Then we form the NBG completion of this theory, giving us "Coll+". But we could iterate this, getting "Coll++", …, "Coll + \(n\)", for example. Taking the limit of this, well that's precisely what Tarski–Grothendieck set theory does, and it treats all these collections as sets. It's useful for handling large categories.

The formal definition of Tarski–Grothendieck set theory is just ZF + Universes. That is to say, given any set \(X\), we can find a Grothendieck universe \(\mathcal{U}\) containing \(X\). What is a "Grothendieck universe"? It is a set \(U\) such that:

1. It is transitive: for each \(X\in U\) and \(x\in X\), we have \(x\in U\)
2. For each \(X\in U\), its power set \(\mathcal{P}(X)\in U\)
3. It contains the empty set: \(\emptyset\in U\)
4. Let \(I\in U\), let \(\{X_{\alpha}\}_{\alpha\in I}\) be a family of elements of \(U\). Then their union \(\bigcup_{\alpha\in I}X_{\alpha}\in U\).

The exact axioms vary. Some omit the inclusion of the empty set, others require the inclusion of the natural numbers instead. But with the axioms we have given, we see every Grothendieck universe is a model for ZF set theory.

See:

• Grothendieck Universe on nLab
• Grothendieck Universe on Wikipedia
• Tarski–Grothendieck set theory on Wikipedia

• Thomas Jech,
  Set Theory: The Third Millennium Edition.
  Springer, 2003.
• Thomas Jech and K. Hrbáček,
  Introduction to Set Theory.
  Marcel Dekker, 3rd edition, 1999.
• Mike Shulman,
  "Set theory for category theory".
  arXiv:0810.1279