What are the problems and theorems that are derived from it?
In: 80
In the early 1900s, mathematicians such as David Hilbert were trying to develop a system of axioms (starting assumptions or statements accepted as true without proof) from which they could define all of mathematics in a consistent way (meaning you can’t contradict yourself by proving something both true and false). Godel ended these efforts by putting forth not one, but two, incompleteness theorems. They were:
1. There is no* system for which you can prove every true statement. That is, for any set of axioms, there will be some statements that are true but cannot be proved from those axioms.
2. No* consistent system can be used to prove its own consistency. This means that you cannot prove a system has no contradictions from within the system.
* It’s not strictly true that the incompleteness theorems say that *no* theorems can meet these conditions. It’s more accurate to say that *no useful, non-trivial* system can do so.
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