3 ms·

Gödel tells us that every formal system contains contradictions and unprovable statements (save for very simple systems). I’ve never heard a summary remotely ak

• greiskul 3y ago
  > contains contradictions It definitely does not say that. Only that we cannot make a proof of consistency of them. It is possible, and believed, that the Zermelo–Fraenkel axioms we use for set theory for example are consistent. But we cannot prove that because of Godel.
  • markisus 3y ago
    I think the statement may be corrected by replacing the “and” with “or”. Does it work to say that “Every sufficiently powerful formal system is incomplete or has a contradiction.”?
• sgt101 3y ago
  I was introduced to incompleteness in the context of Turing Machines and the halting problem in a CS course. I think that if you are introduced to it as part of a foundations of mathematics course then the focus is on what it implies about mathematical reasoning. As ever if you can find the right bit of Wikipedia then there's a fascinating explanation clearly written by someone who really knows about it that I don't understand but hints at worlds of thought that I will spend time exploring when I don't have to get this fucking demo to work or deal with my boss. So maybe never... https://en.wikipedia.org/wiki/Foundations_of_mathematics#Philosophical_consequences_of_G%C3%B6del's_completeness_theorem https://en.wikipedia.org/wiki/Foundations_of_mathematics#Phi...