Neil Barton’s Invited Lecture: Independence and Maximality in Set Theory

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!

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