Gödel’s incompleteness theorems

Gödel’s incompleteness theorems
From The Advent of the Algorithm: The 300-Year Journey from an Idea
to the Computer
Prof. Sérgio Souza Costa1
1Coordenação da Engenharia da Computação
Universidade Federal do Maranhão
GELF, 2017.1
Prof. Sérgio Souza Costa ( Coordenação da Engenharia da Computação Universidade Federal do Maranhão )Gödel’s incompleteness theorems GELF, 2017.1 1 / 34

Content
Axiomatic system
The propositional calculus
The formalization of arithmetic
Peano’s Axioms
Frege contributions
Russell’s Paradox
Richard’s Paradox
Hilbert’s Program
Gödel’s Incompleteness Theorems
Introduction
The language
The Godel Number
The Proof
Theorems

First of all
These slides are the highlights from several references, mainly from the
"The advent of the algorithm", where the goal is to understand the
content.

Axiomatic system
A long time ago ...
Elements of Euclid, was one of the earliest formal systems and is still a
reference for the sciences and mathematics: "Euclidean geometry proceeds
from a ﬁnite set of axioms; from these, the mathematician derives various
geometrical conclusions or theorems."[1]

Axiomatic system: The propositional calculus
"The propositional calculus is the simplest imaginable formal system, and
as its name suggests, it is a system in which whole propositions (or
sentences) come to traﬃc with one and other". [1] Pg. 50
1 A propositional symbol standing alone is grammatical or well-formed.
2 If any formula A (such as (P ∨ Q)) is well-formed, then so is its
negation ¬A,(¬(P ∨ Q) in this case).
3 If A and B are well-formed formulas, then so are (A ∧ B), (A ∨ B),
and (A → B).
Axioms:
1 P → (Q → P)
2 S → (P → Q) → ((S → P) → (S → Q))
3 (¬P → ¬Q) → (Q → P)

Axiomatic system: The propositional calculus
The propositional calculus is decidable complete, consistent and
Non-trivial.
Consistent: The theory does not contradict itself. That is, we cannot
use our axioms to derive A and ¬A.”
Complete: From the axioms, we can derive all mathematical truths.
Non-Trivial: We don’t have something like “Every true statement is
an axiom!”
Decidable: There is a ﬁnite, explicit, and eﬀective scheme for deter
mining whether an arbitrary formula is a theorem within the formal
system

The Formalization of Arithmetic: Peano’s Axioms
In 1889, Peano published ﬁve axioms:
1 0 is a number
2 The successor of any number is a number
3 If a and b are numbers, and if their successors are equal, then a and b
are equal.
4 0 is not the successor of any number
5 If S is a set of numbers containing 0, and if the successor of any
number win S is contained in S as well, then S contains all the
numbers. (advent of algorithm)

The Formalization of Arithmetic: Peano’s axioms
In his Work, Russell presents some limitations in Peano’s axioms:
"Peano’s three primitive ideas namely, 0, number, and successor are
capable of an infinite number of different interpretations, all of which will
satisfy the five primitive propositions"
Then, he gives some examples, like:
Let 0 be taken to mean loo, and let number be taken to mean the numbers
from 100 onward in the series of natural numbers. Then all our primitive
propositions are satisfied, even the fourth, for, though 100 is the successor
of 99, 99 is not a number in the sense which we are now giving to the word
number"

The Formalization of Arithmetic - Frege contributions
Frege’s ambition to snap this chain of contingencies by discovering within
logic itself powers suﬃcient to encompass arithmetic. However, the calculus
of propositional is entirely beside the point to describe the arithmetic.
What is the use, after all, of a system in which 2 + 2 = 4 and
√
36 = 6 are
simply swallowed up as P and Q?
Frege, propose the predicate calculus, a possible language to describe the
arithmetic. He published his masterpiece, The Foundations o f Arithmetic
in 1884; within its pages, arithmetic ﬁnds expression in what Frege
considered purely logical terms

The Formalization of Arithmetic - Frege contributions
The purpose of Frege was to assimilate arithmetic itself, and everything
that followed from arithmetic, to a form of logic that included the
elementary concepts of set theory. However, in 1903, he was set to publish
the second volume of The Basic Laws o f Arithmetic, The book was already
in press, when he received a letter from the young Bertrand Russell. Russell
asked Frege, to consider sets that do not contain themselves as members

The Formalization of Arithmetic - Russell’s paradox
A solution ?
Type theory, proposed by Russel
Zermelo–Fraenkel set theory, proposed by Ernst Zermelo and Abraham
Fraenkel.
"Russell, da sua parte, tentou reparar a lacuna na construção de Frege dos
números naturais a partir de conjuntos. Sua ideia foi restringir o tipo de
propriedade que podia ser usada para deﬁnir um conjunto. É claro que
precisava achar uma prova de que esse tipo restrito de propriedade jamais
levasse a um paradoxo."[7]

The Formalization of Arithmetic - Russell’s paradox
"Eles apresentaram sua abordagem numa densa obra em três volumes, o
Principia Mathematica , de 1910-13. A deﬁnição do número 2 é quase no
ﬁnal do primeiro volume, e o teorema 1 + 1 = 2 é provado na página 86
do volume dois. Todavia, o Principia Mathematica não encerrou o debate
sobre as fundações. A teoria dos tipos era controversa por si mesma. Os
matemáticos queriam algo mais simples e mais intuitivo".[7]

The Formalization of Arithmetic - Richard’s paradox
Was described by the French mathematician Jules Richard in 1905. A
variation of the paradox uses integers instead of real-numbers, while
preserving the self-referential character of the original. Consider a language
(such as English) in which the arithmetical properties of integers are
deﬁned.
(1) E(X) x is odd number
(2) E(X) x is is a prime number
(3) E(X) x is a even number
Then ... Is the ﬁrst expression true for 1? Is the second expression true for
2? Is the third expression true for 3? Is the nth expression true for N?

Consider the 100th expression that pick out the natural numbers that are
not satisﬁed by their correlative expressions, like the third in this example:
(1) E(X) x is odd number
(2) E(X) x is is a prime number
(3) E(X) x is a even number
...
(100) the xth expression is not true for X
Is the Hundredth expression true for 100 ?

Diremos que um número é Richardiano se ele NÃO possui a propriedade
denotada por ele. Por exemplo: suponha que 15 é o número associado à
expressão “é primo”. Então 15 não é Richardiano, pois 15 não é primo.
Seja N o número correspondente à expressão “é Richardiano”. Pergunta-se
N é Richardiano? [2]
Supondo que N é Richardiano, concluímos, pela deﬁnição da propriedade
“Richardiano”, que N não possui a propriedade denotada por ele. Sendo
assim, N não é Richardiano.
Supondo que N não é Richardiano, concluímos, pela deﬁnição da
propriedade “Richardiano”, que N possui a propriedade denotada por ele.
Sendo assim, N é Richardiano.

“Richard’s example pertains to linguistics, not to mathematics” and this
statement opens up the distinction between set-theoretic or mathematical
antinomies and semantical antinomies: the weak point in Richard’s
definition is that to some extent it is symbolic and formal, but it also
makes use of the natural language (“lingua commune”); this contains ideas
that are quite familiar but nevertheless are not sharply defined and are
ambiguous (Peano 1906, p. 357–358). For instance, there is no precise
criterion for deciding whether a given expression of the natural language
represents a rule uniquely defining a number". [3]

The Formalization of Arithmetic - Hilbert’s Program
Hilbert and his followers held that mathematicians should seek to express
mathematics in the form of a complete, consistent, decidable formal
system. The project of formulating mathematics in this way became known
as the “Hilbert program.” [4]
"The task of mathematical logic is not just to develop appropriate
inferential calculi, but also to secure the foundations of mathematics, in
particular the theory of the natural numbers, against the paradoxes. ‘From
this point of view a genuine consistency proof for the axioms of number
theory would be desirable; but nobody has yet been able to provide one’".
[3]

Gödel’s Incompleteness Theorems: Introduction
"The most comprehensive current formal systems are the system of
Principia Mathematica (PM) on the one hand, the Zermelo-Fraenkelian
axiom-system of set theory on the other hand. These two systems are so
far developed that you can formalize in them all proof methods that are
currently in use in mathematics, i.e. you can reduce these proof methods
to a few axioms and deduction rules ... We will show that this is not
true, but that there are even relatively easy problem in the theory of
ordinary whole numbers that can not be decided from the axioms. This is
not due to the nature of these systems, but it is true for a very wide
class of formal systems ... The analogy of this conclusion with the
Richard-antinomy leaps to the eye; there is also a close kinship with the
liar-antinomy "[5]

Gödel’s Incompleteness Theorems in four steps
1 Reduce mathematics to arithmetic; arithmetic was considered the safe
part of mathematics because it was concrete and eﬁnitarian. Every
proposition of mathematics should be written as a proposition of
arithmetic.
2 List the mathematical propositions; classiﬁcation would reduce both
mathematical expressions and meta-mathematical expressions to
numbers. Any of these expressions would be regarded as a number.
3 Use the enumeration obtained to reproduce a self-referential
sentence.Informally in Richard’s Paradox. Take the number for a
negative proposition and apply the proposition to the number denoting
it.
4 Finally, demonstrate that the self-referential sentence can not be
proved in the system

Gödel’s Incompleteness Theorems: The language
The primitive vocabulary of formal arithmetic includes the primitive
vocabulary of the predicate calculus but goes somewhat farther. Herewith a
complete list of symbols organized into clusters.
i Logical symbols: ¬,∀,→,∨,∧,(,),S,0,=,.,+
ii Propositional symbols:P, Q, R, S
iii Individual variables: x, y, z
iv Predicate symbols: E,F ,G H, ...

Gödel’s Incompleteness Theorems: The Godel Number
Numbers are now assigned to the symbols of this system. The logical
symbols get the numbers from one to twelve.
Logical symbols
SYMBOL ¬ ∀ → ∨ ∧ ( ) S 0 = . +
NUMBER 1 2 3 4 5 6 7 8 9 10 11 12
The propositional symbols are assigned numbers greater than ten but
divisible by three.
Propositional symbols
P Q R S
12 15 18 21

Individual variables are assigned a number greater than ten, leaving a
remainder of one when divided by three.
Individual variables
v x y
13 16 19
Predicate symbols, ﬁnally, are assigned a number greater than ten which
leaves a remainder of two when divided by three:
Predicate symbols
E F G
14 17 20

The numbering system not only assigns tags to symbols, it assigns tags to
formulas and sequences of formulas as well.For example, the numbers
corresponding to symbols in P → P are 12, 3, 12. The formula as a whole
is now given the number 212
33
512
, where 2, 3, and 5 are the ﬁrst three
prime numbers. This number is now assigned to P → P as a whole.

The number to the right of each formula is its Godel number.
Sequece of formulas
Formula Gödel Number
1. ∀x(x + S(0) = S(x + 0) m1
2. ∀x(x + 0 = x) m2
3. ∀x(x + S(0) = S(x)) m3
4. ∀x(x + 1 = S(x)) m4
The sequence of four formulas is now assigned the number 2m1
3m2
5m3
7m4
.
A very large but perfectly determinate number, one capturing information
in a powerful and compact arithmetical form.

This numbering scheme determines that every symbol, formula, and
sequence of formulas has a unique Godel number, and it determines that
every Godel number uniquely determines some sequence, formula, or
symbol of the system. This is a way of statements of arithmetic to
comment about themselves. What he has to say, he might say to the same
eﬀect by using Godel numbers instead of words:
2m1
3m2
5m3
7m4
is a proof of m4
This arithmetical relationship may be reﬂected completely from within
ordinary arithmetic by an arithmetical predicate— call it PR— such that:
PR(2m1
3m2
5m3
7m4
,m4)

Gödel’s Incompleteness Theorems: The proof
Godel proposed to construct a sentence of formal arithmetic that when
given its ordinary meaning says of itself that it is not provable.

Let us consider formulas A(v) of formal arithmetic in which just one free
variable v ﬁgures. The formula or expression “v is a prime number” is an
example, one true if v is given the interpretation of 3 or any other prime
number, and false otherwise. But V is free in A(v), and so we do not
know the depth of its generality or whether it is intended to hold for all
numbers, or for some, or for none at all.

Suppose, for example, that the 93rd formula on the list is Ev — nothing
more. Its Gödel number is just 214
313
The logician now invests this formula
with its usual meaning, discovering that Ev expresses the arithmetical
proposition that v is an even number.

The predicate A(v,x) is brought into play. The variable v is the Gödel
number of some formula on the list in which v is free. In this case,
v = 214
313
And the variable x is the Gödel number of a proof of that
formula, when v is removed from the formula and replaced by 214
313
. In
that case, A(v,x) says that X is the Gödel number of a proof of the
formula that results when the numerals expressing the number 214
313
are
substituted for v.

The logician next applies quantiﬁcation to this perfectly ordinary formula of
formal arithmetic, obtaining ∀x¬A(v,x). This formula, when properly
interpreted says just what it seems to say. Nothing is a proof of the
formula whose Godel number is v when the free variable in that formula is
replaced by a numeral naming the Godel number of that formula.

The proof
The formula ∀x¬A(v,x) has just one free variable. It follows that it, too,
must appear on the master list of formulas with just one free variable. Say
that its Godel number is P. And now, following the recipe that has
governed the construction of the predicate A, let us substitute the numeral
naming P for v in ∀x¬A(v,x) , yielding ∀x¬A(P,x).
The formula ∀x¬A(P,x) says that nothing is a proof of the formula that
results when P is substituted for v in the formula ∀x¬A(v,x),in another
word, says o .bif itself that it is not provable.

Gödel’s Incompleteness Theorems: Theorems
First incompleteness theorem
Any consistent formal system F within which a certain amount of
elementary arithmetic can be carried out is incomplete; i.e., there are
statements of the language of F which can neither be proved nor disproved
in F. [6]
Second incompleteness theorem
For any consistent system F within which a certain amount of elementary
arithmetic can be carried out, the consistency of F cannot be proved in F
itself. [6]

References I
D. Berlinski.
The advent of the algorithm: the 300-year journey from an idea to the
computer.
Houghton Miﬄin Harcourt, 2001.
I. Cafezeiro and E. H. Heausler.
Computabilidade: Um pouco de história...... um pouco de matemática.
A. Cantini and R. Bruni.
Paradoxes and contemporary logic.
In E. N. Zalta, editor, The Stanford Encyclopedia of Philosophy.
Metaphysics Research Lab, Stanford University, spring 2017 edition,
2017.
L. Floridi.
The Blackwell guide to the philosophy of computing and information.
John Wiley & Sons, 2008.

References II
K. Gödel.
On formally undecidable propositions of principia mathematica and
related systems i, and on completeness and consistency.
1931.
P. Raatikainen.
Gödel’s incompleteness theorems.
In E. N. Zalta, editor, The Stanford Encyclopedia of Philosophy.
Metaphysics Research Lab, Stanford University, spring 2015 edition,
2015.
I. Stewart.
Em busca do Inﬁnito: Uma historia da Matemática dos primeiros
números a teoria do Caos.
Zahar, 2014.

Gödel’s incompleteness theorems

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