2009 seminar talk: First-order theories without the independence property
Talk held by Hans Adler (KGRC) at the KGRC seminar on 2009-11-26.
Roughly speaking, a first-order formula has the independence property (relative to a complete theory T) if it cannot be used to represent arbitrary subsets of an infinite set in a model of T. T itself is said to have the independence property if one of its formulas has it. The independence property defines a very significant dividing line in the sense of Shelah. In the last years theories without the independence theory (also known as NIP theories) have received wider attention as a potentially fruitful generalisation of stable theories.
I will give an elementary introduction into this field, with a special focus on some of Shelah's remarkable results related to indiscernible sequences.