Like any math newbie, Godel's Incompleteness Theorems are easy to understand in general layman's terms, but difficult to understand beyond the typical "liar's paradox" and "barber's paradox" type examples.
But then I started thinking, are axioms examples of the truths of mathematics that can't be proven? For example, Peano's Postulates: a very popular "starting point" for deducing other mathematical truths. One of the postulates is, "0 is a natural number." Well wait a second: what is "0"? What is a "natural number"? In order to take that axiom as a truth, there must be a definition for what 0 is, and what a natural number is. But even if we assign them definitions in English, can those definitions be proven?
If I follow this train of thought, I eventually find myself completely outside of mathematics, and more into philosophy --- can we even prove what a "natural number" actually is? If I'm not careful, eventually I end up in lala land thinking about the meaning of existence and reality itself.
Does this even make sense? My mind has been blown so many times on this topic, I spend more time scooping my brain off the floor than forming coherent questions.