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appearing in S. Then a follows from S* algebraically iff a
follows from S algebraically.
We can compare the least size of a Herbrand proof of a from
S* and from S. There is a necessary and sufficient iterated
exponential blowup in passing from S* to S.
This corresponds to the situation with cut elimination in
mathematical logic.
2. THE INCOMPLETENESS THEOREMS.
Gödel’s first incompleteness theorem asserts that in any
consistent recursively axiomatized formal system whose axioms
contain a certain minimal amount of arithmetic, there exist
sentences that are neither provable nor refutable. (This is
actually a sharpening of Gödel’s original theorem due to
Rosser). This can be made more friendly by
1) looking only at systems with finitely many axioms;
2) making the “minimal amount of arithmetic” very friendly.
However, formal systems still remain. So we wish to go
further and capture the mathematical essence without using
formal systems.
To maximize friendliness, we incorporate work of Matiyasevich/Robinson/Davis/Putnam on Hilbert’s 10th problem.
THEOREM 7. Let S be a finite set of statements in free
variable logic, including the ring axioms, that hold
universally in some algebra. There is a ring inequation that
holds universally in the ring of integers but does not follow
from S algebraically.
Gödel’s second incompleteness theorem is more delicate than
his first incompleteness theorem.
It asserts that for any consistent recursively axiomatized
formal system whose axioms contain a certain minimal amount
of arithmetic, that system cannot prove its own consistency. 7
(This is actually a sharpening of Gödel’s original theorem
due to several people).
Again, this can be made more friendly as before by
1) looking only at systems with finitely many axioms;
2) making the “minimal amount of arithmetic” very friendly.
However, formal systems still remain, as well as issues
concerning appropriate formalizations of consistency.
So we go further and capture the mathematical essence without
using formal systems.
For this purpose, we introduce the concept of an interpretation. This concept, formalized by Tarski, is normally presented in terms of the first order predicate calculus with
equality.
Here we only use inte...
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This note was uploaded on 08/05/2011 for the course MATH 366 taught by Professor Joshua during the Fall '08 term at Ohio State.
 Fall '08
 JOSHUA
 Math, Logic

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