What is gödel’s theorem?
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The way math works if you start with a set of rules and axioms, and then you construct statements with those rules and axioms.
For example, a statement for arithmetic might be 1+1=2. Or “all integers have a unique set of prime factors” (since this is ELI5, I wrote that as words, but there is a way to write it with math). Before Gödel mathematics thought that math would be a completely enumerable system, which a concrete and finite set of these rules and axioms, from which all valid math statements would derive.
What Gödel showed was that there exist true statements within the language of the rules and axioms, that cannot be proved with the rules and axioms.
This doesn’t mean the statements can’t be proved; it just means you need a different or expanded set of rules and axioms to prove it. It’s really quite a deep result. In a way it shows that math is infinite; there is always another layer of the onion to peel back, so to speak.
Unfortunately there are no trivial examples of a true statement that cannot be proved with a simple set of
Rules and axioms.
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