Sets Godel attempted to prove two things:
1) you can't prove that math is true and consistent
2) there are parts of math you can never prove from any group of axioms
This is the minimum of what he attempted to prove from everyone Ive asked and from all the sources I've seen
The comment I made about steel goes back to my reading of a book called LSD and the Search for God. It mentioned someone on acid who said he could taste the categorical imperative and that it tasted like steel (that is, as absolute). When I read this I thought "that is how I feel about mathematics!" and I went on to read a book about Post Modernism and Mathematics, and now I am in the middle of another book called Why is There a Philosophy of Mathematics at All. I am much more interested in the philosophy of mathematics than in the equations, I admit, and would rather read Heidegger of the foundations of mathematics than study equations. Having said that, I see that Russell's paradox refutes the very
idea of having sets UNLESS we give do credit to how language works. Post modernism has all along said that mathematics is based on relative language use, and although I somewhat agree with them on this, I do believe there could be a way to get to an objective mathematics that could show its truth and perhaps prove everything that is provable in its field. I have not misused what a set is because it's not complicated. Everyone says a set is a collection of members. 6 is 1, 1, 1, 1, 1, and 1 taken together as a whole (I said nature but I should have said whole to be clearer). It's a set of 6 1's. The first 10 whole numbers can be taken as a group and can be a set. For a set to include itself, I agreed, is weird. That is why I asked if it can be shown that a set can include itself since i find this very awkward but also interesting. The Liar paradox is not identical to Godel's theorems. I know that. But I sense that there is something in language, like in Russell's paradox, that is tripping people up and encouraging them to hold Godel's position. If someone did math an LSD, for example, the situation of language's use in mathematics might completely change. We have to be very careful with language. That's all I am basically saying