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TABLE OF CONTENTS
Alrli gresehrtvsed
Prinitnted h Uen itSedt aotfAes m erica
. . . . . . . . . . . . . . . . .
ISB(PeNrf ec0t-):8 191-3869-X I INTRODUCTION 1
ISB(NO o.th�)-:8 191-3868-1
. . . . . . . . . . .
5
II SYSTEM A: ADDITION
III SYSTEM B: LOWER ARITHMETIC ...17
. . . . . . . . .
IV SYSTEM C: ARITHMETIC 29
V ID NUMBERS AND . . . . . . . . . .
THE GODEL FORMULA 41
. . . . . . . . . . . .
57
VI FURTHER QUESTIONS
VII CONSTRUCTING THE
71
GODEL FORMULA'S FA'I'HER .....
iii
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TABLE OF CONTENTS
Alrli gresehrtvsed
Prinitnted h Uen itSedt aotfAes m erica
. . . . . . . . . . . . . . . . .
ISB(PeNrf ec0t-):8 191-3869-X I INTRODUCTION 1
ISB(NO o.th�)-:8 191-3868-1
. . . . . . . . . . .
5
II SYSTEM A: ADDITION
III SYSTEM B: LOWER ARITHMETIC ...17
. . . . . . . . .
IV SYSTEM C: ARITHMETIC 29
V ID NUMBERS AND . . . . . . . . . .
THE GODEL FORMULA 41
. . . . . . . . . . . .
57
VI FURTHER QUESTIONS
VII CONSTRUCTING THE
71
GODEL FORMULA'S FA'I'HER .....
iii
CHAPTER I
INTRODUCTION
In 1931 Kurt Godel showed that it is impossi
ble to reduce basic arithmetic to a system. This
surprising result ranks as one of the great discov
eries of the century; and Godel's argumentation is
fascinating and ingenious. Unfortunately though,
the proof as usually presented is immensely diffi
cult and demands a good deal of mathematical
sophistication.
My goal in this little book is to explain and
prove this important but difficult theorem in as
simple and direct a way as possible, so that more
people may come to under,3tand and appreciate the
theorem. Most of the book presumes only an under
standing of high school algebra. Occasionally I
use some symbolic logic; but I explain this as I go
along and most of the book should be understandable
to a person with little or no background in logic.
The only really technical part is Chapter VII; to
understand this, you need a good knowledge of 1uan
tificational logic and a lot of patience. But you
may omit this chapter if you wish -- you will still
grasp the main argument (although you won't then
know how certain things are worked out from a t3ch
nical standpoint). In addition, I should mention
that I restrict myself to showing that a particular
set of axioms does not formalize arithmetic com
pletely; but the same argument could be used
regardless of the specific axioms chosen.
Godel's theorem establishes the following:
ARITHMETIC CANNOT BE REDUCED
TO AHY AXIOMATIC SYSTEM.
To explain what this means, we need to answer three
questions: (1) "What is arithmetic?," (2) "What is
an axiomatic system?," and (3) "What would it mean
to say that arithmetic is reduced to a given axio
matic system?"
1
CHAPTER I
INTRODUCTION
In 1931 Kurt Godel showed that it is impossi
ble to reduce basic arithmetic to a system. This
surprising result ranks as one of the great discov
eries of the century; and Godel's argumentation is
fascinating and ingenious. Unfortunately though,
the proof as usually presented is immensely diffi
cult and demands a good deal of mathematical
sophistication.
My goal in this little book is to explain and
prove this important but difficult theorem in as
simple and direct a way as possible, so that more
people may come to under,3tand and appreciate the
theorem. Most of the book presumes only an under
standing of high school algebra. Occasionally I
use some symbolic logic; but I explain this as I go
along and most of the book should be understandable
to a person with little or no background in logic.
The only really technical part is Chapter VII; to
understand this, you need a good knowledge of 1uan
tificational logic and a lot of patience. But you
may omit this chapter if you wish -- you will still
grasp the main argument (although you won't then
know how certain things are worked out from a t3ch
nical standpoint). In addition, I should mention
that I restrict myself to showing that a particular
set of axioms does not formalize arithmetic com
pletely; but the same argument could be used
regardless of the specific axioms chosen.
Godel's theorem establishes the following:
ARITHMETIC CANNOT BE REDUCED
TO AHY AXIOMATIC SYSTEM.
To explain what this means, we need to answer three
questions: (1) "What is arithmetic?," (2) "What is
an axiomatic system?," and (3) "What would it mean
to say that arithmetic is reduced to a given axio
matic system?"
1
(1) What is arithmetic? Very roughly, some arithmetic truths are not be provable or else
"arithmetic" here is high school algebra (but lim some arithmetic falsehoods are provable -- there
ited to positive whole numbers) and includes such are either unprovable truthsor else provable fal
truths as "2+2=4," "If x+y=z then y+x=z," and "If sehoods -- either your system fails to prove
xy=l8 and x=2y then x=6 and y=3." More techni "2+2=4" (or some more complex arithmetic truth) or
cally, "arithmetic" is defined as the set of truths it proves "2+2=5" (or some more complex arithmetic
and falsehoods expressible using a certain speci falsity):
fied vocabulary: symbols for positive whole numbers
(1, 2, 3, .•. ), "plus," "times," "to the power of,"
...;.:t
parentheses, "equals," variables (x, y, z, .•• ), AXIOMATIC an arithmetic truth
and some simple logical concepts ["not " "and " 0 r
"or," "if-then," "for every (number)," 'and "f�r SYSTEM � an arithmetic falsity
some (number)" J.
(2) What is an axiomatic system? Very Our proof tries to show this result in detail for a
roughly, an "axiomatic system" is a system in which specific axiomatic system to be called "SYSTEM C."
some principles (axioms) are just taken to be true
while other things (theorems) are proven to be true We will start slowly to work toward the proof
using these principles; high school geometry is an of GodeJ. 's theorem. Chapter II will present an
example of such an axiomatic system. More techni extremely simple axiomatic system -- SYSTEM A -
cally, an "axiomatic system" is a specified set of which gives a way of symbolizing and proving formu
formulas (axioms) together with a specified set of las involving the addition of specified "positive
logical operations (inference rules) for deriving whole numbers ("addition formulas"). SYSTEM A uses
further formulas (theorems). [I regard as implicit symbols for positive whole numbers, "plus," paren
here another technical qualification: that in an theses, and "equals." SYSTEM A's proof mechanism
axiomatic system there must be a finite and mechan contains one axiom and two inference rules and suf
ical notational procedure for checking whether a fices to prove each truth (but no falsehood)
string of symbols is an instance of an axiom or an expressible in the system. Hence the arithmetic of
inference rule or a proof.] addition is reducible to SYSTEM A. [Note that
Godel's theorem does not say "No part of arithmetic
(3) What would it mean to say that arithmetic can be reduced to any axiomatic system"; rather it
is reduced � a given axiomatic system? It would says "Arithmetic (as a whole) cannot be reduced to
mean that the axiomatic system contains as a theo any axiomatic system." J Part of the goal of Chap
rem each arithmetic truth but no arithmetic false ter II is to give an intuitive feel for what axio
hood -- that any arithmetic truth is derivable from matic systems are and how they work.
the axioms but that no arithmetic falsehood is so
derivable -- that for any arithmetic truth or fal Chapter III will present a richer axiomatic
sehood A: system -- SYSTEM B -- which incorporates the symbo
lism and proof mechanism of SYSTEM A and more
besides. SYSTEM B adds symbols for "times" and "to
If A is true, then AXIOMATIC SYS'l'EM the power of" and gives a way to symbolize "lower
' arithmetic" (formulas involving the addition, mul
tiplication, and exponentiation of specified posi
tive whole numbers). SYSTEM B's proof mechanism
If A is false, then AXIOMATIC SYSTEM adds four inference rules and suffices to prove
each truth (but no falsehood) expressible in the
system. Hence lower arithmetic is reducible to
Can the goal of reducing arithmetic to an axiomatic SYSTEM B.
system be reached? Godel's theorem says NO -- the
Chapter IV will give an even richer axiomatic
goal is impossible. For each axiomatic system you
system SYSTEM C which incorporates the
construct, either of two evils happen to it: either
3
2
(1) What is arithmetic? Very roughly, some arithmetic truths are not be provable or else
"arithmetic" here is high school algebra (but lim some arithmetic falsehoods are provable -- there
ited to positive whole numbers) and includes such are either unprovable truthsor else provable fal
truths as "2+2=4," "If x+y=z then y+x=z," and "If sehoods -- either your system fails to prove
xy=l8 and x=2y then x=6 and y=3." More techni "2+2=4" (or some more complex arithmetic truth) or
cally, "arithmetic" is defined as the set of truths it proves "2+2=5" (or some more complex arithmetic
and falsehoods expressible using a certain speci falsity):
fied vocabulary: symbols for positive whole numbers
(1, 2, 3, .•. ), "plus," "times," "to the power of,"
...;.:t
parentheses, "equals," variables (x, y, z, .•• ), AXIOMATIC an arithmetic truth
and some simple logical concepts ["not " "and " 0 r
"or," "if-then," "for every (number)," 'and "f�r SYSTEM � an arithmetic falsity
some (number)" J.
(2) What is an axiomatic system? Very Our proof tries to show this result in detail for a
roughly, an "axiomatic system" is a system in which specific axiomatic system to be called "SYSTEM C."
some principles (axioms) are just taken to be true
while other things (theorems) are proven to be true We will start slowly to work toward the proof
using these principles; high school geometry is an of GodeJ. 's theorem. Chapter II will present an
example of such an axiomatic system. More techni extremely simple axiomatic system -- SYSTEM A -
cally, an "axiomatic system" is a specified set of which gives a way of symbolizing and proving formu
formulas (axioms) together with a specified set of las involving the addition of specified "positive
logical operations (inference rules) for deriving whole numbers ("addition formulas"). SYSTEM A uses
further formulas (theorems). [I regard as implicit symbols for positive whole numbers, "plus," paren
here another technical qualification: that in an theses, and "equals." SYSTEM A's proof mechanism
axiomatic system there must be a finite and mechan contains one axiom and two inference rules and suf
ical notational procedure for checking whether a fices to prove each truth (but no falsehood)
string of symbols is an instance of an axiom or an expressible in the system. Hence the arithmetic of
inference rule or a proof.] addition is reducible to SYSTEM A. [Note that
Godel's theorem does not say "No part of arithmetic
(3) What would it mean to say that arithmetic can be reduced to any axiomatic system"; rather it
is reduced � a given axiomatic system? It would says "Arithmetic (as a whole) cannot be reduced to
mean that the axiomatic system contains as a theo any axiomatic system." J Part of the goal of Chap
rem each arithmetic truth but no arithmetic false ter II is to give an intuitive feel for what axio
hood -- that any arithmetic truth is derivable from matic systems are and how they work.
the axioms but that no arithmetic falsehood is so
derivable -- that for any arithmetic truth or fal Chapter III will present a richer axiomatic
sehood A: system -- SYSTEM B -- which incorporates the symbo
lism and proof mechanism of SYSTEM A and more
besides. SYSTEM B adds symbols for "times" and "to
If A is true, then AXIOMATIC SYS'l'EM the power of" and gives a way to symbolize "lower
' arithmetic" (formulas involving the addition, mul
tiplication, and exponentiation of specified posi
tive whole numbers). SYSTEM B's proof mechanism
If A is false, then AXIOMATIC SYSTEM adds four inference rules and suffices to prove
each truth (but no falsehood) expressible in the
system. Hence lower arithmetic is reducible to
Can the goal of reducing arithmetic to an axiomatic SYSTEM B.
system be reached? Godel's theorem says NO -- the
Chapter IV will give an even richer axiomatic
goal is impossible. For each axiomatic system you
system SYSTEM C which incorporates the
construct, either of two evils happen to it: either
3
2
symbolism and proof mechanism of SYSTEM B and more
besides. SYSTEM C adds symbols for variables (x,
y, z, ••• ) and some simple logical concepts ["not,"
"and," "or," "if-then," "for every (number)," and
CHAPTER II
"for some (number)"]; hence SYSTEM C is able to
express any formula of "arithmetic," as we defined
"arithmetic" above. SYSTEM C's proof mechanism
SYSTEM A: ADDITION
adds ten axioms and two inference rules but does
not suffice to prove each truth (but no falsehood)
expressible in the system. In Chapter V we will
Our first system -- the very simple SYSTEM A
try to show this: that arithmetic cannot be reduced
covers addition formulas (formulas expressible
to SYSTEM C or ( by ext ens ion of the argument) to
using only symbols for positive whole numbers,
any axiomatic system at all. This is what Godel's
"plus," parentheses, and "equals"), The set of
theorem establishes.
truths and falsehoods expressible in such addition
formulas is called "the arithmetic of addition."
Chapter VI will discuss some broader ques
tions relating to Godel's theorem. Finally, Chap
SYSTEM A formulas uses these five symbols:
ter VII will fill in some technical details that
were conveniently skipped in the argumentation of
SYMBOL MEANING
Chapter Vin order to make this argumentation more
understandable.
I one
+ plus
Godel's theorem -- that .!!2 possible axiomatic ( (left parenthesis)
system suffices to prove each truth (but no false ) (right parenthesis)
hood) of arithmetic -- at first may seem difficult
equals
to believe. Arithmetic seems to be an area where
everything is cut i:i,nd dried, where ideas are pre A formula of SYSTEM A is defined as any string
cise and everything can be proven decisively one (whether meaningful or not) of one or more of these
way or the other. But, according to Godel's theo five symbols, So "=/=" ("equals one equals") is a
rem, what "seems" to be the case here is simply not formula of SYS'rEM A, even though it doesn't make
so. While some limited parts of arithmetic (e.g. any sense. The legitimate grammatical strings of
SYSTEMS A and B) are cut and dried in this way, SYSTEM A (the formulas which do malce sense) are
arithmetic itself (in e.g. SYSTEM C) is not and called "numerals" and "terms" and "wffs" and are
never will be so cut and dried. Godel's theorem
specified by grammar rules (1) to (3),
shatters our preconceptions.
The first grammar rule gives a way to con
struct numerals (which are strings of /'s):
( 1) A string consisting in one or more
instances of "/" is a NUMERAL and is also
a TERM.
Examples include the following:
I
("1" or "one")
II ("2" or "one one")
Ill ("3" or "one one one")
Ill/ ("4" or "one one one one")
//Ill ("5" or "one one one one one")
and so forth
5
4
symbolism and proof mechanism of SYSTEM B and more
besides. SYSTEM C adds symbols for variables (x,
y, z, ••• ) and some simple logical concepts ["not,"
"and," "or," "if-then," "for every (number)," and
CHAPTER II
"for some (number)"]; hence SYSTEM C is able to
express any formula of "arithmetic," as we defined
"arithmetic" above. SYSTEM C's proof mechanism
SYSTEM A: ADDITION
adds ten axioms and two inference rules but does
not suffice to prove each truth (but no falsehood)
expressible in the system. In Chapter V we will
Our first system -- the very simple SYSTEM A
try to show this: that arithmetic cannot be reduced
covers addition formulas (formulas expressible
to SYSTEM C or ( by ext ens ion of the argument) to
using only symbols for positive whole numbers,
any axiomatic system at all. This is what Godel's
"plus," parentheses, and "equals"), The set of
theorem establishes.
truths and falsehoods expressible in such addition
formulas is called "the arithmetic of addition."
Chapter VI will discuss some broader ques
tions relating to Godel's theorem. Finally, Chap
SYSTEM A formulas uses these five symbols:
ter VII will fill in some technical details that
were conveniently skipped in the argumentation of
SYMBOL MEANING
Chapter Vin order to make this argumentation more
understandable.
I one
+ plus
Godel's theorem -- that .!!2 possible axiomatic ( (left parenthesis)
system suffices to prove each truth (but no false ) (right parenthesis)
hood) of arithmetic -- at first may seem difficult
equals
to believe. Arithmetic seems to be an area where
everything is cut i:i,nd dried, where ideas are pre A formula of SYSTEM A is defined as any string
cise and everything can be proven decisively one (whether meaningful or not) of one or more of these
way or the other. But, according to Godel's theo five symbols, So "=/=" ("equals one equals") is a
rem, what "seems" to be the case here is simply not formula of SYS'rEM A, even though it doesn't make
so. While some limited parts of arithmetic (e.g. any sense. The legitimate grammatical strings of
SYSTEMS A and B) are cut and dried in this way, SYSTEM A (the formulas which do malce sense) are
arithmetic itself (in e.g. SYSTEM C) is not and called "numerals" and "terms" and "wffs" and are
never will be so cut and dried. Godel's theorem
specified by grammar rules (1) to (3),
shatters our preconceptions.
The first grammar rule gives a way to con
struct numerals (which are strings of /'s):
( 1) A string consisting in one or more
instances of "/" is a NUMERAL and is also
a TERM.
Examples include the following:
I
("1" or "one")
II ("2" or "one one")
Ill ("3" or "one one one")
Ill/ ("4" or "one one one one")
//Ill ("5" or "one one one one one")
and so forth
5
4
11
Thus we have a simple way of expressing any posi [So "((/ +///)+(// +/ /)) is
11
tive whole number -- we just write out a string of equivalent to "//////// . ]
/'s of the corresponding length. People often use
this primitive way to express numbers when they So far in SYSTEM A we have seen examples of
tabulate the results of a vote or a questionnaire; terms (representing numbers) but not statements
sometimes they write 111'f.J./11 (instead of "/////") �esenting true or false claims about numbers).
for "5." our third rule gives the only way to form state
ments (or wffs) in SYSTEM A:
Every numeral is a term, but the next grammar
rule gives a way to construct como• lex terms that (3) A string consisting in a term and then "="
are not numerals: and then a term is a WFF ("well formed
formula").
(2) A string consisting in "(" and then a term
11
and then +" and then a term and then ")" So every SYSTEM A wff is an equation. Examples of
is itself a TERM. wffs would include the following (note that we do
not use a pair of parentheses with each 11="):
Applying rule (2) to numerals gives complex terms
like the following: /=/ ("l=l")
/ =// ("1 =2'')
(/+///) ("1+311 or "one plus one one one") (//+//)=//// ("2+2=4")
(//+//) ("2+211 or "one one plus one one") (/ /+//)=/ //// ("2+2=5")
Note that a pair of parentheses is always to be Here the first and third are true, while the second
used with each "+." Applying rule (2) to the terms and fourth are false.
constructed above gives this more complex term:
A simple way to determine whether a wff is
((/+/ //)+(//+//)) ["( 1+3)+(2+2)" J true or false is to rewrite the 11 wff, replacing
every instance of "(" or "+" or ") with a blank,
l's
Repeated uses of rule (2) would yield terms of and then rewrite the resulting string of to
greater and greater complexity. eliminate the blanks; if the resulting wff consists
in "=" flanked by an equal number of /'s on each
Every complex term is "equivalent" to some side, then the original wff is true -- otherwise it
numeral, in the sense that both refer to the same is false. Let me give two examples:
number -- for example, "(//+//)" ("2+2") is equiva
lent to "////" ("4"). One simple way to find the (I/+//)=/// / ("2+2=4")
numeral which is equivalent to a complex term is to II II =/Ill
rewrite the complex term, replacing every instance /!//=//// ("4=4")
of "(" or "+" or ")" with a blank, and then rewrite
the resulting string of l's to eliminate the [So "(//+// )=////" is true.]
blanks. Let me give two examples:
(//+/ /)=/// // ("2 +2=5")
(I /+I/) ("2+2") II II =//Ill
II II ' //// =///// ("4=5")
!Ill ("4 ")
[So "(//+//)=/////" is false.]
11
[So 1(1 //+//)" is equivalent to "////. ]
Notice how simple and straightforward it is to test
((/+///)+(//+//)) ["( 1+3)+(2+2)11 J whether a SYSTEM A wff is true or false!
I Ill II II
Now we begin to set up an axiomatic proof
I/II/Ill ( II 8")
mechanism for SYSTEM A. This will give us another
7
6
