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.
0.17. Concreteness in the Hilbert Problem List.
This Introduction sets the stage for the new advances in
Concrete Mathematical Incompleteness presented in this
book.
The remainder of this book can be read without relying on
this Introduction. However, we advise the reader to peruse
this Introduction in order to gain familiarity with the
larger context.
Readers can proceed immediately to the overview of the
contents of the book by first reading the brief account in
section 0.14C, and then 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.
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The incompletess phenomenon, in the sense understood today,
was initiatied by Kurt Gödel with his first incompleteness
theorem, where he essentially established that there are
sentences which cannot be proved or refuted using the usual
axioms and rules of inference for mathematics, ZFC
(assuming ZFC is free of contradiction). See [Go31], and
[Go8603], volume 1.
Gödel also established in [Go31] that this gap is not
repairable, in the sense that if ZFC is extended by
finitely many new axioms (or axiom schemes), then the same
gap remains (assuming the the extended system is free of
contradiction).
With his second incompleteness theorem, Gödel gave a
critical example of this incompleteness. He showed that the
statement
Con(ZFC) = "ZFC is free of contradiction"
is neither provable nor refutable in ZFC (assuming ZFC is
legitimate in the sense that it proves only true statements
in the ring of integers). Again, see [Go31], and [Go8603],
volume 1.
Although Con(ZFC) is a natural statement concerning the
axiomatization of abstract set theory, it does not
represent a natural statement in the standard subject
matter of mathematics.
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 Fall '08
 JOSHUA
 Math, Exponential Function, Mathematical logic, Model theory, Firstorder logic, Proof theory, Kurt Gödel

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