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THE INEVITABILITY OF LOGICAL STRENGTH
by
Harvey M. Friedman
Department of Mathematics
The Ohio State University
http://www.math.ohiostate.edu/%7Efriedman/
May 31, 2005
An extreme form of logic skeptic claims that "the present
formal systems used for the foundations of mathematics are
artificially strong, thereby causing headaches such as the
Godel incompleteness phenomena". The skeptic continues by
claiming that "logician's systems always contain overly
general assertions, and/or assertions about overly general
notions, that are not used in any significant way in normal
mathematics. For example, induction for all statements, or
even all statements of certain restricted forms, is far too
general  mathematicians only use induction for natural
statements that actually arise. If logicians would tailor
their formal systems to conform to the naturalness of
normal mathematics, then various logical difficulties would
disappear, and the story of the foundations of mathematics
would look radically different than it does today". Here we
present some specific results that sharply refute aspects
of this viewpoint.
1. EFA = I
S
0
(exp) and logical strength.
For our purposes, we say that a theory in many sorted free
logic has
logical strength
if and only if the system EFA =
I
S
0
(exp) is interpretable in it. I will elaborate after EFA
is presented.
We introduced EFA = exponential function arithmetic, in
[Fr80]. It was also used in the exposition of my work on
Translatability and Relative Consistency, in [Sm82].
Since then, this system has often been referred to as
I
S
0
(exp). See, e.g., [HP98], p. 62. Also see [HP98], p. 405,
second paragraph, referring to our presentation. We will
use the terminology EFA here.
First we present PFA = I
S
0
. Here PFA stands for “polynomial
function arithmetic”.
The language of PFA is based on 0,S,+,•,
£
,=. The variables
are intended to range over nonnegative integers.
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The
S
0
formulas are the formulas of PFA defined as follows.
i) every atomic formula of PFA is
S
0
;
ii) if
j
,
y
are
S
0
then so are
j
,
j
y
,
y
y
,
j
y
,
j
y
;
iii) if
j
is
S
0
and x is a variable not in the term t of
PFA, then (
$
x
£
t)(
j
) and (
"
x
£
t)(
j
) are
S
0
.
In iii), the expressions are treated as abbreviations.
The nonlogical axioms of PFA are as follows.
1. The axioms of Q.
2. (
j
[x/0] (
"
x)(
j
j
[x/Sx]))
j
, where
j
is
S
0
.
The nonlogical axioms of Q are
Q1. Sx = 0.
Q2. Sx = Sy
x = y.
Q3. x ≠ 0
(
$
y)(x = Sy).
Q4. x + 0 = x.
Q5. x + Sy = S(x + y).
Q6. x • 0 = 0.
Q7. x • Sy = (x • y) + x.
Q8. x
£
y
(
$
z)(z + x = y).
This presentation is slightly different than that given in
[HP98]. There
£
is not taken as a primitive, but instead is
defined by Q8. Also there the terms t in bounded
quantification are required to be variables.
We now present EFA. The language of EFA is 0,S,+,
•
,
£
,*,=.
The new symbol * is a binary function symbol standing for
exponentiation.
The
S
0
(exp) formulas are the formulas of EFA defined as
follows.
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 Fall '08
 JOSHUA
 Math, Logic

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