What is Gödel’s incompleteness theorem, and why is it so infamous in Mathematics?
In: Mathematics
There are actually two incompleteness theorems. They are in the realm of logic, which attempts to axiomatize mathematics. Both theorems are about *formal systems*, which you can think of a set of rules for inferring from axioms. Kind of like meta math in a way.
The **first incompleteness theorem** says that all *consistent* (you can think of this as there are no contradictions) formal systems of mathematics that can carry out *Peano arithmetic* (just normal addition/multiplication, basically, with integers that are ordered) are *incomplete*. Here, incomplete means that there are statements you can formulate but you can’t prove.
The **second incompleteness theorem** says that no consistent formal systems of mathematics that can carry out Peano arithmetic can prove their own consistency.
I don’t think they are really “infamous”, by the way—Gödel is very respected, and no serious mathematician thinks these theorems are wrong. But maybe they can be considered depressing or disheartening. The first theorem says that there are some mathematical questions we don’t know the answer to and that we *can’t* know the answer to! To mathematicians whose whole purpose is to get those answers, it seems horrifying. And the second theorem says that you can’t really “know for sure you’re correct” since the formal system can’t prove its own consistency.
Latest Answers