I have recently been studying Godel's Incompleteness Theorem. I am completely new to the study of logic, so I have been working to break down each component. Assuming I've understood the theorem correctly there exists true yet unprovable statements within any system. So to ask my question please picture this: (Bear with me, as I am not very familiar with set notation as a high school student)
There is a set that holds all natural numbers. The rule for this system is that adding one to any integer will result in the integer immediately after it. Within this system, how would you find a particular statement that is true yet unprovable? Does it have to do with the innumerability of infinity? Meaning after numbers are uncountable, the rule still stands except it can't be represented?
I would appreciate any input into the subject. Please feel free to correct any discrepancies in my understanding.