Goedel, Escher, Bach - Lecture 3

A guy named Euclid.

Justin Curry

Curry briefly explaines the concepts:
- consistency
- completeness
- and geometry.

A consistent system leads to conclusions that are not contradictory in any sense. A statement is either true or false, and never both true and false.
A system is complete if everything that is true in the context of that system can be derived from the axioms.
Regarding geometry he mentioned that there are Euclidean non-Euclidean geometries.

Then he attempts to explain Goedel's Incompleteness Theorems.
1. Any system as powerful as number theory which can prove its own consistency is necessarily inconsistent.
2. Any system as powerful as number theory is necessarily incomplete.
He explains that Goedel managed to transform the idea of provability to a property of numbers by introducing his Goedel numbers.
He says that students should now have a notion of the Goedel theorems and promises that this is just a first glance at Goedel's theorem. ( Not sure if he meant he would come back at Goedel in this lecture series. )

Trying to explain Goedel

He then talks about Euclid and his postulates.
(1) Any 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, a circle can be drawn having the segment as radius and the
(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

He explains that the 5th postulate is consistent in Euclidean Geometry but not in spherical and hyperbolic geometry.

Hofstadter Dialog - Little Harmonic Labyrinth is removed from the video due to copyright concerns. It is part of what makes GEB such a difficult book. Here is part of it.

The Tortoise and Achilles are spending a day at Coney Island. After buying a couple of cotton candies, they decide to take a ride on the Ferris wheel.
Tortoise: This is my favorite ride. One seems to move so far, and yet in
reality one gets nowhere.
Achilles: I can see why it would appeal to you. Are you all strapped in?
Tortoise: Yes, I think I've got this buckle done. Well, here we go. Whee!
Achilles: You certainly are exuberant today.
Tortoise: I have good reason to be. My aunt, who is a fortune-teller, told me that a stroke of Good Fortune would befall me today. So I am tingling with anticipation.
Achilles: Don't tell me you believe in fortune-telling!
Tortoise: No . . . but they say it works even if you don't believe in it.

About 1% of the Little Harmonic Labyrinth dialog.

He explains the cardinal arithmetic, the arithmetic of infinities.
An interesting definition of infinity is that a set can be mapped to a subset of itself. I.e. the natural numbers can be bijectively mapped to the even numbers. The points on the real line can be bijectively mapped to the points on the line between 0 and 1.

Non-Euclidean geometries