Title: Independence and Maximality in Set Theory

It is well-known that the standard axioms of set theory (ZFC) do not

suffice to settle many natural set-theoretic questions (and some

category-theoretic ones for that matter). One way that many set

theorists (including Godel and contemporary scholars) have sought to

resolve this independence is through the use of *maximality*

principles: argue that our set concept should engender the

structure(s) satisfying it with many and varied sets (in some

mathematically precise sense). This series of lectures aims to

introduce students to the phenomenon of set-theoretic independence and

explore principles aiming to capture maximality.

In Lecture 1 (Independence and Large Cardinals) we’ll explore the

techniques and philosophical issues surrounding two kinds of

independence: statements (such as *Projective Determinacy*) that are

*responsive* to the addition of large cardinal axioms (which we’ll

also examine), and statements *unresponsive* to the addition of large

cardinals (such as the *Continuum Hypothesis*). We’ll show how strong

large cardinal principles can be characterised through elementary

embeddings, and then make some remarks as to how far this can be

extended, looking at where and how they become inconsistent, and

making some philosophical observations.

In Lecture 2 (Reflection and Independence) we’ll examine one way the

existence of large cardinals has been motivated: through the use of

*reflection principles*, which aim to say (in some mathematically

precise sense) that a structure satisfying our concept of set should

be indistinguishable from one of its initial segments. We note a

challenge posed by Koellner: reflection principles seem either *weak*

or *inconsistent*. We’ll then explore two attempts to surmount

Koellner’s challenge: the addition of embeddings and the use of a

second-order satisfaction predicate.

In Lecture 3 (Absoluteness Principles) we’ll look at a third kind of

principle that has been said to capture the maximality of

set-theoretic structures, through asserting the absoluteness between

certain structures. In this regard we’ll look at two approaches:

forcing axioms and the Inner Model Hypothesis. Again we’ll see where

and how these kinds of principles go inconsistent, but what can be

accomplished with them. We’ll discuss a third problem in the

discussion of maximality: how to synthesise mutually inconsistent

principles. We’ll conclude that there’s lots of interesting

philosophical and mathematical work to be done in the area!