does this intuition regarding Godel incompleteness theorem makes sense?
Without diving into the details of the formal proof, I had the following intuition why math would be incomplete:
Take any known normal number, or generate one (if that's allowed?) and convert the "random" sequence of integers of this number into a 3d random walk, for example by writing it base 6 and mappinbg that to the 6 possible directions in the walk. 3d random walks return with a probability less than 1, so for some finite number n, the walk returns exactly n times. In other words, it's almost surely, that "the walk returns less than n+1 times" is true, for some number n. But to prove this would be impossible, since you would have to check infinitely many positions. It's intuitive to assume that such a statement is true for no real reason, for me at least. Some numbers can be expressed in a finite expression, like pi if we assume it's normal,but contain infinite information. it's therefore not surprising or counter intuitive that we can also find true statements that can't be proven, to me at least.
Does this intuition make sense or am I way off? Similarly, this intuition matches my intuition about the Halting problem, a programma checking whether such walk returns n +1 times would run forever, thus we could never know whether this statement is true,even though it's a fact.
Does this make sense or am I way off? I know i haven't mentioned self reference, which seems an important factor in these kind of theorems, which makes me less sure about my intuition of these concepts. Is self reference only a trick to prove these theorems, or is it fundamental to incompleteness and unproveable statements.
Yes and no.
Yes, incompleteness definitely involves infinity of some kind: as long as everything is finite, you can in principle simply have one big, but still finite, list of answers/results/truths/theorems.
However, I believe your reasoning/intuition is a little bit off in that you seem to be suggesting that any time we're dealing with infinity you automatically get incompleteness, since we cannot check all of those infinite cases ... or run a machine infinitely long. However, note that I can prove all kinds of things about all infinite numbers without having to go through all numbers one by one. And for simple enough machines I can prove that they will run forever, without me having to run them forever.
In sum: you're on the right track: yes, infinity is necessary for incompleteness. However, infinity is not sufficient for incompleteness.
To get incompleteness there needs to be something else going on: some kind of information complexity regarding the domain that prevents completeness. We see this reflected in Godel's incompleteness theorem in that it states that any 'sufficiently strong' theory is incomplete. That is, 'simple enough' domains are perfectly axiomatizable even if they are of an infinite nature. e.g. Presburger Arithmetic.