Slides for the talk I gave on the incompleteness theorem and how it relates to logic / introduction to classical and intuitionistic logic.

1. Introduction to Logic:
The quest for truth
Bernhard Huemer

2. The Seven Days Of Creation
Computer scientists’ version Mathematicians’ versions
“God threw something together
under a 7-day deadline. He’s still
debugging.”
“God laid down axioms, and all
else followed trivially.”

3. Mathematics at the
beginning of the 20th century
Some mathematicians
began an ambitious project:
to prove everything.

4. Sentiment of the time:
Analytical machine
Ada Lovelace

5. Russell’s paradox
Suppose there is a town with just one barber, who is male. In
this town, every man keeps himself clean-shaven, and he does
so by doing exactly one of two things:
• shaving himself; or
• being shaved by the barber.
Also, "The barber is a man in town who shaves all those, and
only those, men in town who do not shave themselves."

6. • “an attempt to describe a set
of axioms and inference rules
in symbolic logic from which
all mathematical truths could
in principle be proven”
• Construction of paradoxical sets can be avoided
by adding types
Principia Mathematica

7. .. but then something happened

8. Kurt Gödel
• Mathematician from Austria
• 1906 - 1978 († by starvation)
• Completeness theorem
• close friends with Einstein,
also published papers on
relativity

10. Euclid’s postulates
1. A straight line segment can be drawn joining any two points.
2. Any straight line segment can be extended indefinitely in a straight
line.
3. Given any straight line segment, a circle can be drawn having the
segment as radius and one endpoint as center.
4. All right angles are congruent.
5. If two lines are drawn which intersect a third in such a way that the
sum of the inner angles on one side is less than two right angles,
then the two lines inevitably must intersect each other on that side
if extended far enough.

11. • If two lines are drawn which intersect a third in such a way that the
sum of the inner angles on one side is less than two right angles,
then the two lines inevitably must intersect each other on that side
if extended far enough.
• Conversely, will these two lines ever cross each other?
Euclid’s parallel postulate

12. Hyperbolic / non-euclidean
geometry

13. Three traditional laws of
thought
• The law of identity
• The law of contradiction
• The law of the excluded middle
http://en.wikipedia.org/wiki/Law_of_thought#The_three_traditional_laws

14. http://www.thenewsh.com/~newsham/formal/curryhoward/
“Just as there are geometries in
which Euclid's fifth postulate is not
assumed to be true, there are logic
systems in which the double-
negation rule is not assumed to be
true.”

15. Inference rules

16. Modus ponens / tollens
If it is raining, the streets are wet.
The streets are not wet.
Therefore, it is not raining.
All men are mortal.
Socrates is a man.
Therefore, Socrates is mortal.

17. • Proving the Law of the excluded middle by
contradiction, i.e. assume ¬(p ∨ ¬p) and look for
a contradiction
Proving by contradiction

18. Example
There exists irrational numbers
and such that is rational.x y xy

19. is rational
is not rational
Classical proof
2
2
2
2
x = 2
y = 2
x = 2
2
y = 2
2
2⋅ 2
= 2
• Does not tell us which of the two is true
.. not particularly useful :(

20. Constructive proof
x = 2
y = 2⋅log2 (3)

21. Intuitionistic logic:
Definition
Taken from: Lectures on the Curry-Howard Isomorphism, Volume 149 (Studies in Logic and the Foundations of Mathematics)

22. Construction rules
Taken from: Lectures on the Curry-Howard Isomorphism, Volume 149 (Studies in Logic and the Foundations of Mathematics)

23. Classic tautologies that can
be proved constructively
Taken from: Lectures on the Curry-Howard Isomorphism, Volume 149 (Studies in Logic and the Foundations of Mathematics)

24. Classical tautologies that
aren’t constructive too
Taken from: Lectures on the Curry-Howard Isomorphism, Volume 149 (Studies in Logic and the Foundations of Mathematics)

25. Curry-Howard
Correspondence
• Direct relationship between computer programs
and mathematical proofs
• e.g. tautologies and identity functions
ρ → ρ

26. Curry-Howard
Correspondence

28. Category theory:
The rosetta stone
Taken from the paper “Physics, Topology, Logic and Computation: A Rosetta Stone”

29. More to come ..

30. Hilbert’s
Entscheidungsproblem
• 1936 both Alan Turing and Alonzo Church
published papers showing that this was impossible
• If the Halting Problem wasn’t in fact a problem, one
could theoretically guess the solution for Fermat’s
Last Theorem (i.e. mechanization of mathematics)
an
+ bn
= cn

31. Q&A
“Either mathematics is too big
for the human mind or the
human mind is more than
a machine.”
- Kurt Gödel