Eli5: can someone explain to me the difference between what godels system and the Turing machine proved? Aren’t they both basically just proving that x can’t equal (x≠x)

Question

Eli5: can someone explain to me the difference between what godels system and the Turing machine proved? Aren’t they both basically just proving that x can’t equal (x≠x)

In: Mathematics

Answer 1

x is always equal to x by definition of “equal”.

Turing machines are theoretical models of computers. Everything a normal computer can do can be done with a Turing machine and vice versa (if we have enough storage), but Turing machines are pretty simple in their design so they are often easier to analyze. You can use a Turing machine to prove various statements (e.g. “4 is even”), just like you can do it with a normal computer.

Are you asking about Gödel’s incompleteness theorems? They say (roughly) that not all true statements have a proof.