Incompleteness
DC Nerd Nite
@jamesdabbs
Disclaimers
True or False
1 + 1 = 2
True or False
1 + 1 = 2
True or False
\forall n \in \mathbb{Z}. n^2 > 0
True or False
If n is any integer, n² is greater than 0
True or False
If n is any integer, n² is greater than 0
True or False
If n is any integer, n² is greater than or equal to 0
True or False
If n is any integer, n² is greater than or equal to 0
True or False
This slide is false
True or False
This slide is false
True or False
The sky is blue
True or False
The sky is blue
True or False
Godfather III is the best Godfather
True or False
Godfather III is the best Godfather
A Portrait of the Speaker as a Young Man
Math: a Naïve View
- Anything that's true can be proved
- There's only one version of math
- Mathematicians all agree about what's true
Math: a Naïve View
- Complete – every true statement can be proven
- Consistent - no false statement can be proven
There is the problem. Seek its solution. You can find it by pure reason, for in mathematics there is no "we will not know".
– David Hilbert (1900)
David Hilbert
Hilbert's Problems
- Prove or disprove the Continuum Hypothesis
- Prove that math is consistent
- ...
Kurt Gödel
Gödel's Incompleteness Theorems
-
(1930) In any* axiomatic system, there will be statements that are true but unprovable
-
(1931) No* axiomatic system can prove its own consistency
Gödel's Incompleteness Theorems
This statement cannot be proven from the axioms
Gödel's Incompleteness Theorems
If this statement is not provable is true
this statement is not provable has no proof
⇒ true statement with no proof
Gödel's Incompleteness Theorems
If this statement is not provable is false
this statement is not provable has a proof
⇒ false statement with a proof
Gödel's Incompleteness Theorems
This statement cannot be proven from the axioms
The sky is blue
Gödel's Incompleteness Theorems
\forall n \in \mathbb{Z}. n^2 > 0
Gödel's Incompleteness Theorems
Gödel's Incompleteness Theorems
Any* axiomatic system is either inconsistent or incomplete
The Continuum Hypothesis
If a set is bigger than the rationals, it's at least as big as the reals
|X| > |\mathbb{Q}| \Rightarrow |X| \geq |\mathbb{R}|
Hilbert (1900): prove or disprove this
The Continuum Hypothesis
- Gödel (1940) – cannot be disproven
- Cohen (1964) - cannot be proven
WAT
ZFC
Zermelo–Fraenkel set theory with the Axiom of Choice
- ZFC
- ZFC + CH
- ZFC - CH
- ZFC + V=L
- ZFC + ◊
- ZFC + ♣ - ◊
- ZF (= ZFC - C)
The Axiom of Choice
If you have a bunch of sets, you can pick one thing out of each of them
Cannot be proven or disproven (from ZF)
Physics
The more important fundamental laws and facts of physical science have all been discovered
– Morely (1903)
Time moves slower if you go real fast
– Einstein (1905)
It is impossible to know both how fast something is going and where it is
– Heisenberg (1927)
Computer
Science
Alan Turing
Turing Machines
There is no algorithm to decide if a Turing machine will halt or if it will run forever
– Turing (1936)
The Halting Problem
This program runs forever if you feed it a program that halts
The Halting Problem
This program runs forever if you feed it a program that halts
Art
X
Blue
Postmodernism
Take
Aways
Take Aways
There are fuzzy, unresolvable problems everywhere
Try to savor them
Take Aways
You're not going to have it all figured out
Don't try to figure it all out
Embrace your
Incompleteness
Incompleteness
DC Nerd Nite
@jamesdabbs
The Busy Beaver
Define BB(n) to be the biggest number that a Turing Machine of size n can compute
BB(2) = 4
BB(3) = 6
BB(4) = 13
The Busy Beaver
Define BB(n) to be the biggest number that a Turing Machine of size n can compute
BB(5) \geq 4098
BB(6) > 3.5*10^{18,267}
BB(7) > 10^{10^{10^{10^{1870535313}}}}
The Busy Beaver
Define BB(n) to be the biggest number that a Turing Machine of size n can compute
BB(8) \in \mathbb{R}_2 \setminus \mathbb{D}_2
The Busy Beaver
Define BB(n) to be the biggest number that a Turing Machine of size n can compute
BB(1919)
is undecidable
Incompleteness
By James Dabbs
Incompleteness
A presentation for DC Nerd Nite
