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I am self-learning about Godel's incompleteness theorems and think that this analogy is good:

When trying to prove that infinite primes exist we assume finite number of them exist and then we arrive at a number that cannot be built by those primes.

Similarly we assume a finite number of axioms exist and arrive at a statement that cannot be proved by those axioms.

So axioms in a theory are small indivisible units that can be used to build bigger things but for a complete picture of the theory we might need infinite number of them.

PS: This question might be vague, but I want to know if this analogy is erroneous.

rajashekar
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    This analogy is erroneous in my opinion. For example, finiteness of the set of axioms is irrelevant: Godel applies to any recursively enumerable set of axioms satisfying some additional conditions. Also, there isn't really a good sense in which axioms are "indivisible units." – Noah Schweber Oct 06 '21 at 05:44
  • @NoahSchweber: Thank you. – rajashekar Oct 07 '21 at 03:09

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