DC Nerd Nite

@jamesdabbs

# True or False

1 + 1 = 2
$1 + 1 = 2$

# True or False

1 + 1 = 2
$1 + 1 = 2$

# True or False

\forall n \in \mathbb{Z}. n^2 > 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

The sky is blue

The sky is blue

# True or False

Godfather III is the best Godfather

# True or False

Godfather III is the best Godfather

# 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)

# 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
$\forall n \in \mathbb{Z}. n^2 > 0$

## 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}|
$|X| > |\mathbb{Q}| \Rightarrow |X| \geq |\mathbb{R}|$

Hilbert (1900): prove or disprove this

# The Continuum Hypothesis

• Gödel (1940) – cannot be disproven
• Cohen (1964) - cannot be proven

# 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)

# 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

X

Blue

# 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

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
$BB(2) = 4$
BB(3) = 6
$BB(3) = 6$
BB(4) = 13
$BB(4) = 13$

# The Busy Beaver

Define BB(n) to be the biggest number that a Turing Machine of size n can compute

BB(5) \geq 4098
$BB(5) \geq 4098$
BB(6) > 3.5*10^{18,267}
$BB(6) > 3.5*10^{18,267}$
BB(7) > 10^{10^{10^{10^{1870535313}}}}
$BB(7) > 10^{10^{10^{10^{1870535313}}}}$

# 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
$BB(8) \in \mathbb{R}_2 \setminus \mathbb{D}_2$

# The Busy Beaver

Define BB(n) to be the biggest number that a Turing Machine of size n can compute

BB(1919)
$BB(1919)$

is undecidable

By James Dabbs

# Incompleteness

A presentation for DC Nerd Nite

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