Gödel’s incompleteness theorem
Adam Lund Madsen, Aktuel Matematik, blok 1, 2008
Statement: Gödel’s (first) Incompleteness Theorem roughly says that a consistent logical system,
containing the basic arithmetic of natural numbers, cannot be complete.
This means that there will be statements in the system that cannot be proven and nor will it be
possible to prove their negation.
Origin of the problem: The Theorem was proved in 1931 and therefore, its historical context is
that of the discussion of the foundations of mathematics.
In the beginning of the 20th century, mathematicians like Hilbert, Russel and Brouwer tried to find a
firm foundation of mathematics.
Brouwer founded the school “Intuitionism”. The intuitionists believe that mathematics is a created
by the human mind and they build up mathematics from our understanding of the number 1.
“Logicism” was started by Frege and is based on set theory. In the 20th century, Russel and
Whitehead continued Frege’s program and tried to find a foundation of mathematics based on set
theory. They did this in the work Principia Mathematica (published 1910-13), where they tried to
avoid Russel’s Paradox.
Hilbert founded the school “Formalism”. Formalists believe that mathematics is a construction
based on language and logic and that mathematics is nothing more than “a game”. Mathematics is
just a closed system based on a set of rules we decide.
The foundation schools tried to find a firm foundation of mathematics, but none of them were
successful, especially thanks to Gödel’s Incompleteness Theorem (in fact Gödel proved two
incompleteness theorems which were both hard strikes against the foundation schools).
The problem of finding a firm foundation of mathematics has yet not been solved, so Gödel’s
theorem is still very important in pointing out the weaknesses of ZFC, which is the system of
axioms we use as foundation of mathematics today.
Though some people believe that the second incompleteness theorem was most fatal for the
Formalism and Logicism, the subject of this paper is Gödel’s first Incompleteness Theorem, which
was also very important in the discussion of a foundation of mathematics.
The Incompleteness Theorem:
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Now, the problem is to determine whether the system N (a system which contains the basic
arithmetic)1 is complete (completeness means that it is always possible to prove a statement or its
negation). Gödel proves that it is not.
The way he does it is by showing the existence of a closed wf. (“well formed formula”)2 U in N, so
that neither U nor the negation ~U is a theorem in N. He does this by expressing U explicitly and
then showing that both U and ~U leads to a contradiction under the assumption that U and ~U are
theorems that can be proven.
Gödel considers the system N which is the first order arithmetic of positive integers and the number
0 with addition and multiplication. The way Gödel expresses it is that we have a constant 0, a
successor function with which we can describe every natural number, the addition and
multiplication functions, the relations “=” and “>”, the quantifiers “  ” and “  ”, the parentheses
“(“ and “)”, logical symbols like “~”, “  ”, “  ” and the symbols “ x1 ”, “ x2 ”, “ x3 ” and so on for
free variables.
To carry out the proof, Gödel introduces a few new terms. These are “Gödel numbers”, “recursive
functions” and “expressibility”.
The use of Gödel numbers is a coding system, where you can express every wf. and every sequence
of wf.’s uniquely as a number (integer). There is a one-to-one correspondence between the wf.’s
and the code numbers, so it is easy to obtain the specific wf. if the related code number is given.
Gödel uses the function g to assign a unique odd number to every symbol, relation and wf. in first
order language. For instance, g ())  3 means that the symbol “(” has 3 as its Gödel number. Every
symbol gets a coding number so that g (()  5 , g (, )  7 and so on. g ( xk )  7  8k , k=1,2,…, and so
on. This means, that every coding number is odd. For example, given a number greater than 13, you
must divide by 8 and then examine the remainder, to find out which unique wf. the number refers
to. To relate a wf. to a given number Gödel uses that every number has a unique factorization of
powers of primes. To describe a string of the symbols u1 , u2 ,..., uk we use g so that
g (u1 , u2 ,..., uk )  2 g (u1 )  3g (u2 )  pk g (uk ) . This means that, for instance, the string of symbols f11 ( x1 )
has the code number g ( f11 ( x1 ))  2 g ( f1 )  3g (()  5g ( x1 )  7 g ())  259  33  515  75 .
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By this procedure, it is possible to express first order language L by code numbers.
Referred to as ”N written”.
A wf. is a sentence that observe the rules of the symbols in a given system. For instance the sentence “ x   ” is not a
wf. in N though “ x ” is.
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I will not comment further on the details of the “recursive functions” and “expressibility” but just
mention that the study of recursive functions allows Gödel to translate relations from DN to N and
decide whether a given relation in DN (the Domain of Natural Numbers) is expressible in N (and
the other way around).
I will not go into details with the proof but only try to explain the basic idea and structure of it.
The basic idea of the proof is, by use of Gödel numbers, to express a wf. in N which asserts its own
unprovability. In “normal” language, this wf. can be related to the liar paradox “this sentence is
false”. What Gödel does is that he constructs a wf. U that can be interpreted as “For every number
n, n is not the Gödel number of a proof in N of the wf. U”3. Now, this would not be a problem if N
was not consistent, because then every wf. would be a theorem. Therefore it has to be assumed, that
the system N is consistent (Gödel uses the term  -consistent which is a stronger version of the
regular consistency).
It is (according to Hamilton) easily shown that N is  -consistent. So now the proof of the
incompleteness of N can be made, using the assumption that N is  -consistent.
The proof uses a lot of “translation” between DN and N (founded on the study of recursive
functions and expressibility) and is based on contradiction following from the assumption that U is
a theorem in N with a Gödel number q related to the proof of U in N. This gives rise to a
contradiction to the consistency of N and so U cannot be a theorem of N. But if U is not a theorem,
then a certain sentence W in DN related to U doesn’t hold.4 This sentence can be expressed in N
again, and the assumption of  -consistency of N then tells us that ~U is not a theorem of N. And so
it is proven that neither U nor ~U are theorems of N.
The Continuum Hypothesis: Today, the most firm foundation of mathematics is the system ZFC,
which is a list of axioms in set theory. The most famous undecidable problem in ZFC, is the
Continuum Hypothesis (CH), which was presented by Georg Cantor (1845-1918).
The CH concerns infinite sets. It states that there exists no set with cardinality larger than the
natural numbers and smaller than the cardinality of the real numbers. Cardinality is a term which
describes the “size” of an infinite set. For instance, the set of natural numbers has a lower
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See Hamilton page 151 (in Hamilton: Logic for Mathematicians, Cambridge University Press, 1988).
The wf. U is based on a wf. in N, which is connected to the relation W in DN (see Hamilton page 149 for the
definition of W) .
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cardinality than the real numbers. We say that the set of natural numbers is countable, because we
can count the elements by starting at 1, then 2, then 3 and so on. We cannot do the same thing with
the real numbers, because we can never determine “the next number” after a given number. It is
very simple to show that both of the sets Z and Q are countable and therefore have the same
cardinality as the natural numbers.
Cantor believed that the CH was true, but he was unable to prove it. It was proven by Gödel that the
CH does not lead to a contradiction in ZFC. Later it was proven that also the negation of the CH
does not lead to a contradiction. Therefore, the CH is an example of an unprovable statement which
demonstrates the incompleteness of ZFC.
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