A short introduction to Kurt Gödel's concept of incompleteness, and its implications in philosophy.

1. The Incompleteness
of
Reason
Subhayan Mukerjee 18 March 2013

2. Mathematics in the early
20th century
Bertand Russel Alfred North Whitehead

3. Mathematics in the early
20th century
● Attempting to find the theory of everything
● Bertrand Russell and Alfred North
Whitehead's Principia Mathematica
● Aim :
○ trying to reduce pure mathematics, particularly
number theory, to a formal axiomatic system.
○ Every true mathematical statement should be
completely provable. The proof should begin from
some basic axioms and follow some rigorous rules
to arrive at the level.

4. Enter Kurt Gödel

5. and, in 1931...
● Gödel dealt a death blow to these attempts,
by stating that such a theory of everything is
not possible.
● To be more correct, he proved that such a
theory could not exist.

6. Typographical Number
Theory
● Invented by Douglas Hofstadter.
● Way of writing every mathematical statement about
natural numbers a string.
● Uses :
○ basic math symbols + - * /
○ logical symbols ~ (not) V (or) E (there exists) and A
(for all)
○ variables a a' a'' ...
○ numbers 0 or S (meaning successor)
■ ie, 0 = 0
■ 1 = S0
■ 2 = SS0 and so on.
● Only positive numbers allowed

7. Writing a TNT statement
● There exists no natural number whose
square is 2, can be written in TNT as
~Ea : a*a = SS0

8. Axioms in TNT
● Axiom 1: Aa : ~Sa = 0
● Axiom 2: Aa : (a+0) = a
● Axiom 3: Aa : Aa' : (a+Sa') = S(a+a')
● Axiom 4: Aa : (a*0) = 0
● Axiom 5: Aa : Aa' : (a*Sa') = ((a*a')+a)

9. Some rules for reduction
Aa : ~Sa = 0
~Ea : Sa = 0

10. A couple of assertions
● any statement you can make about natural numbers —
no matter how complex, no matter how long, no matter
how bizarre — can be written in a TNT string
● Two, if such a statement is true, its TNT string can be
derived as a theorem from the axioms. If the
statement is false, we can derive its converse from
the axioms. (Meaning, the same string with a ~ symbol
in front of it.)

11. Consider sentence G
Sentence G : This statement is not a theorem
of TNT

12. Is sentence G true of false?
We cannot say!
Why? Think about it.

13. Incompleteness!
This is what Gödel proved.
He showed that TNT, although it may be
perfectly consistent and always correct, cannot
possibly prove every true statement about
number theory; there is always something
which is true, which the system cannot prove.

14. The one lacuna
TNT applies to natural numbers.
How do know that TNT can be applied to a
string that is about a TNT statement? Like
Sentence G?
After all, Sentence G is not about natural
numbers!
Or, is it?

15. Gödelizing a string
● merely a change in notation.
● use three random numbers to represent all
symbols!
● For example 0 == 666; S == 111; = 123
● every number has three digits, and
● no two numbers are the same.

16. Writing a Gödelized string
● TNT statement: ~Ea : a*a = SS0
● Gödelized:
223333262636262236262111123123666

17. Now change some rules.
● Whenever a number is a multiple of 1000,
you can add 5 to it,
is same as writing
● Whenever a string ends in the symbol "000",
you can replace that symbol with "005".

18. Enter, theoremhood of
numbers
● Writing a Gödelized expression where 1 is represented
by 444 and = by 333
444333444
● this stands for 1 = 1
● Now it is true. So we say, this Gödelized expression
has theoremhood.
● It's like saying the number 7 has primeness
● Theoremhood is thus a property of a Gödelized
expression by virtue of which its corresponding
mathematical statement is true.

19. Now iterate!
● '444333444 has theoremhood’ can be
rewritten as TNT just as 7 is prime can be
written as TNT.
● We can then Gödelize that TNT and get
another Gödelized expression!
● This can continue for ever...

20. The big thing
● Each Gödelized expression asserts the truth
of the previous Gödelized expression!
● In other words, we can now write TNT forms
of other TNT statements.
● Which means ...

21. We can indeed write the TNT form of the
Sentence G!

22. In philosophy
● Every system, no matter how rigorous or
how complete, is unable to completely prove
itself.
● TNT is unable to prove the truth of Sentence
G which is a part of TNT.

23. A system, using itself, and solely itself
cannot prove itself!

24. Consider any object, say a
cycle
● Can the cycle, using only itself, explain its
existence?
● NO!
● We needs a third person to explain the
cycle.
● The third person must not be a part of the
cycle.
● What can the third person be?
● So many things, the factory, the mechanics,
etc.

25. But, the cycle cannot explain its existence on
its own.

26. This is incompleteness.

27. Another way of looking at it
“Anything you can draw a circle around cannot
explain itself without referring to something
outside the circle – something you have to
assume but cannot prove.”
● You can draw a circle around a bicycle but the
existence of that bicycle relies on a factory that is
outside that circle.
● Similarly, you can draw a circle around TNT, but the
completeness of TNT, depends on the third party
observer, who is outside the circle.
● Gödel proved that there are always more things that are
true than you can prove.

28. Two types of reasoning
● Deductive reasoning or reasoning inward
from a larger circle to a smaller circle is
“deductive reasoning.”
● Example of a deductive reasoning:
1. All men are mortal
2. Socrates is a man
3. Therefore Socrates is mortal

29. Two types of reasoning
● Inductive reasoning or Reasoning outward
from a smaller circle to a larger circle is
“inductive reasoning.”
1. All the men I know are mortal
2. Therefore all men are mortal

30. The important ideas
● We cannot prove natural laws.
● We can only verify them!
● Any system can completely be explained by
something outside that system.

31. And given the biggest
circle that we can draw...
● There has to be something outside that circle.
Something which we have to assume but cannot prove

32. Biggest circle - around the
universe?
● The universe as we know it is finite – finite matter, finite
energy, finite space and 13.7 billion years time
● The universe is mathematical. Any physical system
subjected to measurement performs arithmetic. (The
moment you are counting or measuring something, you
are subjecting that physical entity to arithmetic.)
● The universe (all matter, energy, space and time)
cannot explain itself

33. what is outside the biggest
circle?
● whatever is outside the biggest circle - ie
that around the universe - is boundless!
● Why? Because if it wasn't, we would have
included that in our biggest circle.

34. what is outside the biggest
circle?
● is it matter? NO. Matter is part of the
universe.
● is it energy? Space? Time? NO! These are
all parts of the universe.

35. what is outside the biggest
circle?
It is not matter, not energy, not space, not
time.
It is immaterial!

36. what is outside the biggest
circle?
● not a system - because we can draw a circle
around a system.

37. Yet!
● It is not nothing!
● Why?
● Because it is that which mantains the
consistency of the universe.
● In fact,
○ It is the equivalent of the outside observer who
proves that unprovable Sentence G that the system
(the universe) cannot prove!

38. enter information
● In the history of the universe we also see the
introduction of information, some 3.5 billion years ago. It
came in the form of the Genetic code, which is symbolic
and immaterial.
● The information had to come from the outside, since
information is not known to be an inherent property of
matter, energy, space or time
● All codes we know the origin of are designed by
conscious beings.
● Therefore whatever is outside the largest circle is a
conscious being.

39. the startling revelation
That which is outside the universe is
○ infinite
○ immaterial
○ conscious

40. Thank
you!