Just a small note: I’m taking High School Geometry, so in no way am I advanced in mathematics.
Recently I’ve started a project attempting to explain Gödel’s incompleteness theorems. However I can’t seem to comprehend the difference between the 2 theory’s.
From what I’ve gathered the first incompleteness theorem states:
- Any axiomatic system will always be incomplete. The only way for the system to be complete is to have it be inconsistent (it can only be complete if there is an axiom that is different or doesn’t fit in with the rest of the system)
- In simplest terms: Any consistent axiomatic system will always be incomplete.
- There are statements within the system which are unprovable by the systems axioms.
I’ve also noted that the second incompleteness theorem states:
- If your system is consistent (no flaws; doesn’t have anything that contradicts the rest of the system), then there is no way to make it complete. There is always some formula that you can neither prove, nor prove it’s negation (that it’s not true)(there will always be a formula that can be found, but it’ll be unknown it’s it true, or false).
Can someone please help my clarify if I’m correct (or somewhat correct) with my notes. If I’m not an explanation would be very much appreciated.
Some elements of what you say are good informal approximations to what Gödel's theorems say, while others could be improved.
Firstly, incompleteness does not mean that there are sentences within the system which are unprovable. For example, in any axiomatic system which attempts to capture elementary arithmetic, we do not want $0 = 1$ to be provable. Rather, it means that there is a sentence $A$ such that neither $A$ nor its negation $\neg A$ is provable in the system. Informally speaking, incompleteness means that there is some sentence $A$ such that the axiomatic system does not settle the question of whether $A$ holds or not.
Secondly, Gödel's theorems do not apply to all axiomatic systems, but only those which are "rich enough" in some technical sense. Roughly, Gödel’s theorems apply to axiomatic systems where one can talk about multiplication and addition of natural nuumbers and which are strong enough to prove some elementary facts about multiplication and addition. Gödel’s theorems do not apply, for example, to so-called Presburger arithmetic, which is a complete and consistent system, but it is not very interesting mathematically in that it only allows one to talk about addition. The details of the system are not terribly important, the upshot is that Gödel's theorems do not apply to every possibly axiomatic system, but the systems that they do not apply to are "missing" some important mathematical notion, such as multiplication.
Thirdly, what you state as Gödel's second theorem is in fact an informal rendering of Gödel's first theorem. Gödel's second theorem is more difficult to understand at the high school level because it talks about an axiomatic system being able to "prove its own consistency". The theorem states that no "rich enough" (see the previous paragraph) consistent system can "prove its own consistency".
This is nice enough as a slogan, but the devil is in the details. Making it precise what it means for a system to "prove its own consistency" is more complicated than simply defing what an incomplete theory is. I will not attempt to explain what this means at a high school level. Instead, let me suggest to focus first on properly understanding and explaining Gödel's first theorem, which is on its own no mean feat at the high school level.
In particular, one thing that I notice you do in your explanation is that you offer in different places several slightly different and somewhat conflicting definitions of a single notion. You give no less than three or four different meanings to the notion of an (in)consistent system:
The key thing about understanding advanced mathematics (as opposed to high school mathematics) is getting used to the fact that mathematical notions, such as the notion of a consistent system, are not just given informal and intuitive explanation, but they have precise definitions. In this case, a system is consistent if one cannot deduce every sentence using the axioms and rules of the system. Equivalently, a system is consistent if one cannot deduce a contradictory sentence using the axioms and rules of the system, i.e. a sentence of the form $A \wedge \neg A$ (informally, "$A$ and not $A$").
If you want to properly explain Gödel's theorems, I would say the most important thing is carefully explaining the definitions involved in the theorems and making sure that your intended audience (which might very well be just you yourself, I don't know) understands that the theorems are not just some vague, informal statements but that they have a very precise meaning which the audience can decypher through understanding these definitions.
Good luck!