A Comparison Process for Mouse Pairs

This book proves important new theorems in the theory of canonical inner models for large cardinal hypotheses in set theory.

John R. Steel (Author)

9781108840682, Cambridge University Press

Hardback, published 24 November 2022

550 pages
23.5 x 15.7 x 3.7 cm, 0.97 kg

'The book starts slowly with revisions of various inner model-theoretic and fine-structural notions that fit better in the context of mouse pairs and associated iteration trees. Such revisions prove to be useful in particular when it comes to producing induced iteration strategies for the levels of background constructions which have suitable condensation properties and sufficiently close connection to the iteration strategies of the background universes in which those background constructions are carried out. These properties are crucial, as background constructions are the very source for producing mouse pairs.' Martin Zeman, MathSciNet

This book proves some important new theorems in the theory of canonical inner models for large cardinal hypotheses, a topic of central importance in modern set theory. In particular, the author 'completes' the theory of Fine Structure and Iteration Trees (FSIT) by proving a comparison theorem for mouse pairs parallel to the FSIT comparison theorem for pure extender mice, and then using the underlying comparison process to develop a fine structure theory for strategy mice. Great effort has been taken to make the book accessible to non-experts so that it may also serve as an introduction to the higher reaches of inner model theory. It contains a good deal of background material, some of it unpublished folklore, and includes many references to the literature to guide further reading. An introductory essay serves to place the new results in their broader context. This is a landmark work in inner model theory that should be in every set theorist's library.

1. Introduction
2. Preliminaries
3. Background-induced iteration strategies
4. More mice and iteration trees
5. Some properties of induced strategies
6. Normalizing stacks of iteration trees
7. Strategies that condense and normalize well
8. Comparing iteration strategies
9. Fine structure for the least branch hierarchy
10. Phalanx iteration into a construction
11. HOD in the derived model of a HOD mouse
References
Index.

Subject Areas: Set theory [PBCH], Mathematical logic [PBCD]