Student, you are using truth-functional logic to come to a Goedelian conclusion. I too have considered this seemingly undecidable concept of Nothing. I have found mathematically, what you will arrive at is a contradiction, albeit one that is inconclusive.
But, lets say we use a modal logic standpoint.
Lets further specify that ‘nothing’ is a mood of sorts. So, as an example “It is possible that ‘nothing’ might exist” would mean that non-existence is just a possibility, and thus DOES have a strange existence, as possibility. We need not subject the possibility to the rigid rules of T-F logic. In fact, we can’t because modal logic rejects the T-F logic schema. Nothingness then becomes just a possible thing, amongst many other states. As long as it remains possible it can be in a sort of quantum physics state of being and non-being.
If this seems dubious to you, I’ll be the first to agree. Modal logic is a weak form of reasoning. It can’t establish conclusions like T-F logic. But, it opens a pathway to a different view of existence. It is also how our minds seem to work. Being Dutch, your great mathematician Brouwer questioned an idea in math logic which I love him for doing. He took out the negative X negative= positive convention in algebra and found so many conclusions in abstract algebra would have to change. But nobody seemed to care. Well that’s an overstatement. Nobody seemed to pick up on it. I’m writing an article now about this great man and his intuitionist philosophy. But I am digressing with this comment. Back to your question…
So, as an answer to your opening query: Does Nothing exist? Yes, but only as a possibility. Of course you could then ask…well does impossibility exists? I would have then say: impossibility is the set in which ‘nothing’ is contained. What? Yeah it gets me too. If I could embed math symbols it would be clearer. I seem to be going back to T-F logic with that comment and I’m not. I’m saying of all the possible states, impossibility is the possible state of being ‘nothing’. I know this seems like semantics.
Well I guess I failed to convince you. No big deal, I’m still trying to convince my daughter to stopping believing in that foolish Christianity stuff with the Jesus and the God and all that rot!
Later man.