1
CONTEMPORARY PERSPECTIVES ON HILBERT'S
SECOND PROBLEM AND THE GÖDEL
INCOMPLETENESS THEOREMS
AMS/ASL PANEL DISUCSSION
Moderator: Akihiro Kanamori
Panelists: Harvey M. Friedman, David E. Marker,
Michael Rathjen
January 6, 2007
1. First Incompleteness Theorem.
2. First Incompleteness Theorem in ( ,<,0,1,+,•). (!!!)
3. Second Incompleteness Theorem.
4. Is there any real logical strength?
5. Strict Reverse Mathematics.
It is not yet clear just what the most illuminating ways of
rigorously stating the Incompleteness Theorems are. This is
particularly true of the Second. Also I believe that there
are more illuminating proofs of the Second that have yet to
be uncovered.
NOTE: See “Formal Statements of Godel’s Second
Incompleteness Theorem”, http://www.math.ohio
state.edu/%7Efriedman/
There is also a very interesting viscously anti
foundational argument which suggests that mathematics can
be developed in a way that can be proved to be free of
contradiction in Peano Arithmetic, or even in weak
fragments such as Exponential Function Arithmetic = EFA =
I
S
0
(exp) – thereby suggesting that the Incompleteness
Theorems are an irrelevant and misleading distraction.
Refutation of this mind numbing heresy is ongoing and leads
to some very interesting formal work, called Strict Reverse
Mathematics.
NOTE: See “The Inevitability of Logical Strength”,
February, 2007, recently submitted for publication.
http://www.math.ohiostate.edu/%7Efriedman/
1. FIRST INCOMPLETENESS THEOREM.
R.M. Robinson’s Q. L(Q) = 0,S,+,•, with =, and
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1. Sx 0.
2. Sx = Sy x = y.
3. x 0 (
$
y)(x = Sy).
4. x+0 = x.
5. x+Sy = S(x+y).
6. x•0 = 0.
7. x•Sy = (x•y)+x.
THEOREM 1.1. Let T be a consistent many sorted theory with
finitely many axioms, and
p
be an interpretation of Q in T.
Then there is a sentence
j
of L(Q) such that
p
(
j
) is neither
provable nor refutable in T.
Note that Theorem 1.1 is extremely clean and fully
rigorously stated. There is an important extension that is
not so clean.
THEOREM 1.2. Let T be a consistent many sorted theory, and
p
be an interpretation of Q into T.
Assume that the set of
axioms of T is recursively enumerable.
Then there is a
sentence
j
of L(Q) such that
p
(
j
) is neither provable nor
refutable in T.
Now the statement involves Gödel numberings of syntax.
Questions of robustness occur, which are nowadays
considered trivial.
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 Fall '08
 JOSHUA
 Math, Mathematical logic, Model theory, Strict Reverse Mathematics, finitely many axioms

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