Incompleteness

DC Nerd Nite

@jamesdabbs

Disclaimers

True or False

1 + 1 = 2
1+1=21 + 1 = 2

True or False

1 + 1 = 2
1+1=21 + 1 = 2

True or False

\forall n \in \mathbb{Z}. n^2 > 0
∀n∈Z.n2>0\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

  1. Prove or disprove the Continuum Hypothesis
  2. Prove that math is consistent
  3. ...

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
∀n∈Z.n2>0\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}|
∣X∣>∣Q∣⇒∣X∣≥∣R∣|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(2)=4BB(2) = 4
BB(3) = 6
BB(3)=6BB(3) = 6
BB(4) = 13
BB(4)=13BB(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(5)≥4098BB(5) \geq 4098
BB(6) > 3.5*10^{18,267}
BB(6)>3.5∗1018,267BB(6) > 3.5*10^{18,267}
BB(7) > 10^{10^{10^{10^{1870535313}}}}
BB(7)>101010101870535313BB(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
BB(8)∈R2∖D2BB(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)
BB(1919)BB(1919)

is undecidable

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