1
INTRODUCTION
CONCRETE MATHEMATICAL INCOMPLETENESS
0.1. General Incompleteness.
0.2. Some Basic Completeness.
0.3. Abstract and Concrete Mathematical Incompleteness.
0.4. Reverse Mathematics.
0.5. Incompleteness in Exponential Function Arithmetic.
0.6. Incompleteness in Primitive Recursive Arithmetic,
Single Quantifier Arithmetic, RCA
0
, and WKL
0
.
0.7. Incompleteness in Nested Multiply Recursive
Arithmetic, and Two Quantifier Arithmetic.
0.8. Incompleteness in Peano Arithmetic and ACA
0
.
0.9. Incompleteness in Predicative Analysis and ATR
0
.
0.10. Incompleteness in Iterated Inductive Definitions and
Π
1
1
CA
0
.
0.11. Incompleteness in Second Order Arithmetic and ZFC\P.
0.12. Incompleteness in Russell Type Theory and Zermelo Set
Theory.
0.13. Incompleteness in ZFC using Borel Functions.
0.14. Incompleteness in ZFC using Discrete Structures.
0.15. Detailed overview of book contents.
0.16. Some Open problems.
This Introduction sets the stage for the new advances in
Concrete Mathematical Incompleteness presented in this
book.
Readers can proceed immediately to the overview of the
contents of the book by reading the brief overview in
section 0.14C, and the fully detailed overview in section
0.15. These are self contained and do not rely on the rest
of the Introduction.
In this Introduction, we give a general overview of what is
known concerning Incompleteness, with particular emphasis
on Concrete Mathematical Incompleteness. The emphasis will
be on the discussion of examples of concrete mathematical
theorems  in the sense discussed in section 0.3  which
can be proved only by using unexpectedly strong axioms.
This growing body of results shows rather explicitly what
is to be gained by strengthening axiom systems for
mathematics.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
2
Of course, there is an even greater loss realized by
strengthening a consistent axiom system to an inconsistent
one. The issue of why we believe, or why we should believe,
that the relevant axiom systems used in this book are
consistent  or even that they prove only true arithmetic
sentences  is an important one, and must lie beyond the
scope of this book.
Since this Introduction is to be viewed as clarifying
background material for the six Chapters, many of the
proofs are sketchy, and in many cases, we have only
provided sketches of the ideas behind the proofs. We have
also included results from the folklore, results that can
be easily gleaned from the literature, and results of ours
that we intend to publish elsewhere. We have provided many
relevant references. No results presented in this
Introduction will be used subsequently in the book (except
for Gödel's second incompleteness theorem).
0.1. General Incompleteness.
General Incompleteness was initiated by Gödel's landmark
First and Second Incompleteness Theorems, which apply to
very general formal systems. The original reference is
[Go31].
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '08
 JOSHUA
 Math, Set Theory, Exponential Function, Mathematical logic, Model theory, Gödel's incompleteness theorems, Godel

Click to edit the document details