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A CONSISTENCY PROOF FOR ELEMENTARY ALGEBRA AND GEOMETRY
by
Harvey M. Friedman
Department of Mathematics
Ohio State University
August 10, 1999
August 23, 1999
ABSTRACT
We give a consistency proof within a weak fragment of
arithmetic of elementary algebra and geometry. For this
purpose, we use EFA (exponential function arithmetic), and
various first order theories of algebraically closed fields
and real closed fields.
We actually prove in EFA that RCF (real closed fields) is
consistent and complete. The completeness proof is an
adaptation of known constructions.
We also obtain a proof in EFA that every quantifier free
formula provable in RCF has a quantifier free proof in the
theory of real fields and in the theory of ordered fields.
As a Corollary, we obtain an interpretation of RCF into EFA
in the sense of model theory, as well as interpretations of
finitely axiomatized extensions of RCF into EFA.
The development can be used to provide fixed length iterated
exponential estimates in connection with Hilbert’s
seventeenth problem. This application will appear elsewhere.
INTRODUCTION
to be written. discusses Hilbert’s program briefly. discusses
systems of arithemetic briefly. introduces EFA = exponential
function arithmetic. reference to Hajek/Pudlak. disclaimer
that we make no attempt here to obtain more precise
information.
1. PRELIMINARIES ABOUT LOGIC
In the free variable predicate calculus with equality, we
start with a signature consisting of a set of constant,
relation, and function symbols. Formulas are built up in the
usual way using the variables v
1
,v
2
,..., connectives not, or,
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and, implies, iff, =, and the symbols in . There are no
quantifiers.
The variables are literally the v
i
, i
≥
1. We use this to
define the universal closure of a formula
j
. This is obtained
from
j
by successively universally quantifying by the free
variables of
j
in reverse order of their subscripts. If
j
is
a sentence then the universal closure of
j
is
j
.
The logical axioms are
1. All tautologies in the language.
The equality axioms are
2. x = x, where x is a variable.
3. x = y implies (
j
implies
j
’), where x,y are variables,
j
,
j
’ are atomic formulas, and
j
’ is the result of replacing
some occurrences of x in
j
by y.
The rules of inference are
4. From
j
and (
j
implies
y
) derive
y
.
5. From
j
derive any term substitution of
j
in the language.
Suppose we are given a set T of proper axioms; i.e., a set of
quantifier free formulas in the language. Then a proof in T
is a finite sequence of formulas, every one of which is
either a logical axiom, an equality axiom, a proper axiom, or
follows from previous formulas in the finite sequence by one
or more rules of inference. The proof in T is said to be a
proof in T of the last formula. We say that the last formula
is free variable provable in T.
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 Fall '08
 JOSHUA
 Math, Algebra, Geometry

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