Download Formalizing the Liar`s Paradox

History of mathematical logic
Gödel’s Incompleteness Theorems
Formalizing the Liar’s Paradox
Selwyn Ng
University of Wisconsin
April 2010
Selwyn Ng
Formalizing the Liar’s Paradox
Logic and mathematics
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
What is mathematical logic?
Mathematical logic, also known as symbolic logic, is a
subfield of mathematics with close connections to
computer science and philosophical logic.
Formal logic and other areas of mathematics.
Theme: expressive power of formal systems and the
deductive power of formal proof systems.
Divided further in the subfields: set theory, model theory,
proof theory, computability theory.
⇔ Definability.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Brief history of foundations
Many ancient civilizations had their own intricate systems
of reasonings, notably Indian logic, Chinese logic and
Greek philosophy.
Prehistory - Egyptians clearly figured certain empirical
facts about geometry (pyramids).
Etymology: logic – from the Greek logos, meaning
discourse or sentence.
First appears in the writings of Alexander of Aphrodisias
(3rd century AD).
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Brief history of foundations
Greeks: Linked empirical truth with demonstrative science.
(6th BC): Pythagorean school. Basic principles of geometry
are:
Certain propositions must be accepted as true without
demonstration, which we call axioms.
All other propositions of the system are derived from these.
The derivation must be formal, i.e. independent of the
particular subject matter in question.
Fragments of early “proofs" are preserved in the works of
Plato and Aristotle.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Brief history of foundations
(5th BC): A more sophisticated argument pattern is found
in Reductio ad absurdum used by Zeno of Elea.
Known today as “proof by contradiction" – this draws an
obviously false, absurd or impossible conclusion from an
assumption.
(4rd BC): None of the surviving works of Plato include any
formal logic, but philosophical logic:
What is it that can properly be called true or false?
Connection between the assumptions of a valid argument
and its conclusion?
What is the nature of definition?
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
The Organon (tools)
(1st BC) Greeks: Aristotle gave a complete study of logic
(Aristotelian logic)
He know connectives, however wasn’t symbolic. Far
reaching acceptance in ancient science and math.
Interpretation (Latin: De Interpretatione, Greek
Perihermenias) introduces proposition and judgment.
Relations between affirmative, negative, universal and
particular propositions.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Future contingents
Problem of future contingents: These are statements about
events in the future that are neither necessarily true nor
necessarily false.
Suppose that a sea-battle will not be fought
tomorrow. Then it was also true yesterday (and
the week before, and last year). But all past truths
are necessary truths. Therefore it was
necessarily true in the past that the battle will not
be fought. Thus that the statement that it will be
fought is necessarily false. This conflicts with the
idea of our own freedom.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Future contingents
Aristotle suggested this problem: It’s not true that either a
battle will happen, or it will not.
He added a third term, called contingent, for something
which may happen.
The first two are mutually exclusive. The third term is
consistent with the other two.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Modern mathematical logic
(50 BC – 14th AD) Developed by medieval Islamic and
European logicians.
(14th AD – 19th AD) Regarded by logicians to be a barren
era of neglect.
(Mid 19th AD) Revived in Europe – logical calculus
(Leibniz) and symbolic logic (Boole).
Boolean logic: George Boole observed a deep analogy
between the symbols of algebra and logical forms.
He was restricted to the two quantities, 0 and 1. By
attaching meaning to symbols, one can build up other
symbols by algebraic expressions.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Modern mathematical logic
(50 BC – 14th AD) Developed by medieval Islamic and
European logicians.
(14th AD – 19th AD) Regarded by logicians to be a barren
era of neglect.
(Mid 19th AD) Revived in Europe – logical calculus
(Leibniz) and symbolic logic (Boole).
Boolean logic: George Boole observed a deep analogy
between the symbols of algebra and logical forms.
He was restricted to the two quantities, 0 and 1. By
attaching meaning to symbols, one can build up other
symbols by algebraic expressions.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Modern mathematical logic
(Early 20th AD) Regarded as the golden era of foundations
studies.
Logicism: Frege, Russell, Dedekind and Whitehead
believes that logic is the foundations of mathematics.
Dedekind reduces theory of real numbers to the theory of
rationals.
Frege started to reduce everything to his “naive" set
theory; Russell famously sent him a letter explaining a
paradox he discovered.
This was resolved by Zermelo’s axioms, evolving to
ZF /ZFC that we study today.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Modern mathematical logic
(Early 20th AD) Regarded as the golden era of foundations
studies.
Logicism: Frege, Russell, Dedekind and Whitehead
believes that logic is the foundations of mathematics.
Dedekind reduces theory of real numbers to the theory of
rationals.
Frege started to reduce everything to his “naive" set
theory; Russell famously sent him a letter explaining a
paradox he discovered.
This was resolved by Zermelo’s axioms, evolving to
ZF /ZFC that we study today.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Modern mathematical logic
Meta-mathematical period (1910-1930): David Hilbert
(finitism). This is the belief that a mathematical object does
not exist unless it can be constructed from natural numbers
in a finite number of steps.
The most famous proponent of finitism was Leopold
Kronecker, who said:
“God created the natural numbers, all else is the
work of man."
Hilbert’s program: Axiomatize all of mathematics (in a
single system). Dealt a death blow by Gödel’s
incompleteness theorems.
We’ll come back to Hilbert’s vision later...
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Modern mathematical logic
Meta-mathematical period (1910-1930): Other work of
Löwenheim, Skolem, Tarski and Gödel.
Tarski and semantics: The concept of truth in formalized
languages. A sentence such as “snow is white" is true if
and only if snow is white.
Final period in history: Post Gödel’s Incompleteness
Theorems.
Mathematical logic branched into the four separate areas
of research: model theory, proof theory, set theory and
computability theory.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Algorithms
Etymology: Al-Khwā-rizmī, Persian astronomer and
mathematician. He wrote a treatise in 825 AD, “On
Calculation with Hindu Numerals"
The Latin translation is “Algoritmi de numero Indorum"
There is no generally accepted formal definition of
"algorithm"
What we intuitively mean is there is a mechanical
procedure (devoid of intelligence), and gives the desired
result after a finite number of steps.
Notice the word “finite".
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Algorithms
Etymology: Al-Khwā-rizmī, Persian astronomer and
mathematician. He wrote a treatise in 825 AD, “On
Calculation with Hindu Numerals"
The Latin translation is “Algoritmi de numero Indorum"
There is no generally accepted formal definition of
"algorithm"
What we intuitively mean is there is a mechanical
procedure (devoid of intelligence), and gives the desired
result after a finite number of steps.
Notice the word “finite".
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Algorithms
Selwyn Ng
Formalizing the Liar’s Paradox
Logic and mathematics
History of mathematical logic
Gödel’s Incompleteness Theorems
Algorithms
Selwyn Ng
Formalizing the Liar’s Paradox
Logic and mathematics
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Algorithms
In these cases you specify an input, or set of ingredients.
The algorithm applies a mechanical method to get the
desired result.
Euclid’s algorithm for finding greatest common divisor:
Input: a pair of numbers (1001,357).
1001 = 357 · 2 + 287
357 = 287 · 1 + 70
287 = 70 · 4 + 7
70 = 7 · 10
Output: the gcd(1001,357)= 7
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Algorithms
In these cases you specify an input, or set of ingredients.
The algorithm applies a mechanical method to get the
desired result.
Euclid’s algorithm for finding greatest common divisor:
Input: a pair of numbers (1001,357).
1001 = 357 · 2 + 287
357 = 287 · 1 + 70
287 = 70 · 4 + 7
70 = 7 · 10
Output: the gcd(1001,357)= 7
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Into the 20th century
David Hilbert had a grand plan to finitely “mechanize" all of
mathematics.
Based on the idea that in mathematics there should be no
"ignorabimus" (statement that the truth can never be
known),
A machine, which you can feed
Input: a statement about mathematics
Process: the machine uses a reasonable formal system to
generate “proofs"
Output: True or False.
Equivalently, can you have a mechanical procedure that
“enumerates" all the truths in a system (e.g. number
theory)?
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Into the 20th century
David Hilbert had a grand plan to finitely “mechanize" all of
mathematics.
Based on the idea that in mathematics there should be no
"ignorabimus" (statement that the truth can never be
known),
A machine, which you can feed
Input: a statement about mathematics
Process: the machine uses a reasonable formal system to
generate “proofs"
Output: True or False.
Equivalently, can you have a mechanical procedure that
“enumerates" all the truths in a system (e.g. number
theory)?
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Into the 20th century
The formal proof system should consist of reasonable
axioms such as ∀x(x + 1 exists).
Other obvious facts such as ∀x∀y (x + y = y + x).
Principle of induction
Modus ponens: If we have P and P ⇒ Q then we will have
Q.
However Hilbert’s plan was destroyed by...
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Into the 20th century
The formal proof system should consist of reasonable
axioms such as ∀x(x + 1 exists).
Other obvious facts such as ∀x∀y (x + y = y + x).
Principle of induction
Modus ponens: If we have P and P ⇒ Q then we will have
Q.
However Hilbert’s plan was destroyed by...
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Into the 20th century
Selwyn Ng
Formalizing the Liar’s Paradox
Logic and mathematics
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Into the 20th century
Gödel proved his two famous Incompleteness Theorems.
First Incompleteness Theorem: Any sufficiently strong
formal system of axioms has a statement P for which
neither P nor ¬P can be proven.
Furthermore if you add P to the system, there will still be
another statement P 0 independent from the augmented
system.
Second Incompleteness Theorem: Any such system of
axioms cannot prove the statement “I am consistent",
unless it is itself inconsistent.
The collective intuition of generations of mathematicians
were wrong.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Into the 20th century
Gödel proved his two famous Incompleteness Theorems.
First Incompleteness Theorem: Any sufficiently strong
formal system of axioms has a statement P for which
neither P nor ¬P can be proven.
Furthermore if you add P to the system, there will still be
another statement P 0 independent from the augmented
system.
Second Incompleteness Theorem: Any such system of
axioms cannot prove the statement “I am consistent",
unless it is itself inconsistent.
The collective intuition of generations of mathematicians
were wrong.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Into the 20th century
Selwyn Ng
Formalizing the Liar’s Paradox
Logic and mathematics
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Incompleteness Theorems
To try and understand a bit more of these two theorems,
we look at first order logic.
A formal language has propositional connectives (¬, ∨, ∧),
quantifier symbols (∀, ∃), and the symbols for arithmetic
(+, −, ×, ÷, <).
A sentence is any coherent combination of these symbols.
e.g. ϕ(x) = ∃z∃y (y · z = x ∧ (z, y < x)). Then ϕ(a) is
true of the natural numbers iff a is not a prime number.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Incompleteness Theorems
A proof of ϕ in the formal system is a sequence of
sentences ϕ0 , ϕ1 , · · · , ϕn , such that each sentence ϕn+1 is
either an axiom, a hypothesis, or follows logically from the
previous ϕ, and ϕn = ϕ.
In this case we say that Γ ` ϕ, where Γ is the set of
hypotheses.
For example, “the sum of two even numbers is even".
If Γ is the set of axioms for free abelian groups, then Γ `
“every element is torsion-free".
If Γ is the set of axioms for the algebraically closed fields,
then Γ ` “every irreducible polynomial is of degree 1".
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Incompleteness Theorems
Let PA denote “Peano’s Arithmetic". This is a set of axioms
consisting of the arithmetical properties of N, such as
∀x∀y (x + y = y + x).
Also the induction axioms
ϕ(0) ∧ ∀x(ϕ(x) → ϕ(x + 1)) → ∀xϕ(x).
PA is an example of a “sufficiently strong theory" satisfying
the premises of the Incompleteness theorems. For our
purposes we think of PA and all its theorems as exactly all
the statements true of the natural numbers.
Another example of a sufficiently strong theory is set
theory (ZF/ZFC).
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Incompleteness Theorems
The first step to Gödel ’s proof is to arithmetize syntax.
Express what you do in the meta-language within the
formal language.
Assign to every formula ϕ, a code number, also called a
Gödel number #ϕ.
So within the language, you can have formulas ϕ which
talks about other formulas (e.g. ψ, indirectly using #ψ).
You can even have ϕ which talks about ϕ itself!
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Incompleteness Theorems
Crucial Lemma:
Lemma (Fixed Point Lemma)
For any sentence β(z), there is another sentence σ such that
σ ↔ β(#σ).
In other words, σ says that “β is true of myself".
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Liar’s Paradox
Lemma (Tarski’s Undefinability Lemma)
There is no formula P(x) which defines PA. That is, there is no
formula P(x) such that for every formula ϕ,
P(#ϕ) ↔ ϕ.
Proof.
Suppose formula P exists.
The proof uses the formalized Liar’s paradox to get a
contradiction.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Liar’s Paradox
Proof continued.
By the fixed point lemma applied to ¬P, we have some σ such
that
σ ↔ ¬P(#σ)
But by definition of P, σ actually says “I am false".
Since we get a paradox, hence the contradiction shows that P
cannot exist, otherwise we could formalize the Liar’s paradox.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
First Incompleteness Theorem
Theorem (Gödel ’s First Incompleteness Theorem)
Any sufficiently strong formal system of axioms (such as PA)
has a statement ϕ for which neither ϕ nor ¬ϕ can be proven.
Rough sketch.
If PA is complete, i.e. for every statement, either ϕ or ¬ϕ can be
proven from PA, then PA is decidable.
If PA is decidable, there is a finite list of instructions which tell us
how to decide each ϕ.
We can then write down a formula P which defines PA,
contradicting the Tarski’s undefinability lemma.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Consequences of the First Incompleteness Theorem
Let’s return to Hilbert’s (Second) problem: can you have a
mechanical procedure that “enumerates" all the truths in a
system (e.g. number theory)?
Gödel ’s theorem says NO: a complete finite list of axioms
can never be created, nor even an infinite list that can be
enumerated by a computer program.
Each time a new statement is added as an axiom, there
are other true statements that still cannot be proved.
If an axiom is ever added that makes the system complete,
it does so at the cost of making the system inconsistent
(i.e. proves a contradiction).
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
The Second Incompleteness Theorem
The second incompleteness theorem, is often viewed as
making the problem (of Hilbert) impossible.
One can find a formula Con(PA) expressing the fact that
“PA is consistent".
This formula expresses the property that “there does not
exist a natural number coding a sequence of formulas,
such that each formula is either an axioms of PA or an
immediate consequence of preceding formulas, and such
that the last formula is a contradiction".
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
The Second Incompleteness Theorem
Theorem (Gödel ’s Second Incompleteness Theorem)
For any sufficiently strong formal system of axioms T (such as
PA),
T ` Con(T )iff T is inconsistent.
As Georg Kreisel remarked, “it would actually provide no
interesting information if a theory T proved its consistency."
A consistency proof of T in T would give us no clue as to
whether T really is consistent.
E.g. a person proving or demonstrating that he/she isn’t
drunk. Or telling you that he/she is honest.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
The Second Incompleteness Theorem
The interest in consistency proofs lies in the possibility of
proving the consistency of T in some other T 0 which is less
dubious than T itself.
Again, the honesty of the person.
For example if T 0 is less dubious in the sense that
T ` Con(T 0 ), then T 0 6` Con(T ).
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
Independence
Some systems are decidable. For example, real closed
fields, Euclidean geometry.
Alfred Tarski and quantifier elimination.
Gödel’s example was artificial.
Are there statements which matter in working
mathematics?
Yes... from Peano’s arithmetic (PA).
Kruskal’s Tree Theorem, Goodstein sequences, etc.
Selwyn Ng
Formalizing the Liar’s Paradox
History of mathematical logic
Gödel’s Incompleteness Theorems
Logic and mathematics
The End
I hope many of you will want to do mathematical logic.
Thank you.
Selwyn Ng
Formalizing the Liar’s Paradox