Gödel’s incompleteness theorem and Russell’s paradox
4/18/21 My favorite picture lately: the blue sunset on Mars! I heard that the Uranus’s sunset is turquoise!
Some of the fun intellectual moments: listen to smart people to explain difficult movies, and discuss math theorems/astrophysics.
I recently watched Tenet, and I was not able to figure out the relativity quick enough to follow the movie. Christopher Nolan became more and more abstract nowadays: Inception and Prestige are complex but manageable, Interstellar is good and I even like Dunkirk quite a lot. Then came this super insane Tenet. To give you a taste: “Everything that has happened will happen, and will always have happened” #^S*&%???
One of my favorite theorems is Gödel’s incompleteness theorem. Gödel has two incompleteness theorems, and I am talking about the first one today. This theorem said that in any consistent axiomatic systems, there are truths about arithmetics that cannot be proved or disapproved.
(1) an axiom: a statement taken to be true
(2) an axiomatic system: a set of axioms and the truths derived/proved from them
(3) consistent: no contradictory (among the truths derived from the system)
(4) truth about arithmetics: arithmetics on natural numbers (maybe beyond ?)
(5) prove: deductive reasoning (can be coded in algorithm procedures?)
The implications are: (1) the math is limited; there are truths that cannot be proved/disproved by maths (anecdote: some mathematicians quitted from their math career as a result of knowing maths is limited) (2) Roger Penrose and J.R. Lucas: human intelligence is able to recognize inconsistencies, which under Gödel’s theorem, is impossible for Turing machines. Thus, computer cannot reproduce human intelligence.
Russell (earlier than Gödel)’s paradox is an indication of Gödel’s incompleteness theorem. Let R be the set of all sets that are not members of themselves. If R is not a member of itself, then its definition dictates that R must contain itself, and if it contains itself, then it contradicts its own definition as the set of all sets that are not members of themselves. An informal version is “barber’s paradox”: the barber is the “one who shaves all those, and those only, who do not shave themselves”. Does the barber shave himself?