Hi, I'm sorry for being lazy again and asking for help here instead of looking stuff for myself, but I'm being lazy merely because I've tons of other stuff to read. What I want to understand is: 1) Under what conditions does Gödel's incompleteness theorem 1 hold? (That for theories defined in a certain way, statements exists which are true but unprovable) 2a) Why does geometry not obey these conditions? 2b) What properties does geometry for Tarski to be able to prove geometry to be complete? 3a) Kant's point is that the way we experience the world, shapes our theories a priori. His idea was that math (and logic) fundamentally is a priori: there are certain basic assumptions in math and logic which are given before every conscious theory. So, math is not based on analysis of given things and experiences: rather, it starts with synthetic assumptions, something we add to the things that are given, by experiencing them. 3b) Gödel was very much influenced by Kant, and his incompleteness theorem was inspired by or at least backed up by the idea that math is a priori. It is because our theories about logic are incomplete, that it becomes necessary to say we need the way our thinking works to found this logic. This is the line of thought Gödel would use (I think). I'm not stating that he's right, but that this was his goal. So, I'm wondering, what does Gödel's idea and Tarski nuance imply for our possible knowledge? So, please, I encourage to take this to a broader level than certain regions of mathematics, or certain regions of logic, and see for what kind of knowledge in general, Tarski's and Gödel's theories hold. There's been a lot of discussion on this forum about Gödel, and Tarski - much of these discussions have been totally messed up because people were divided in two sides, the laymen who saw all kinds of strong implications in Gödel etc. , and the mathematicians who became increasingly frustrated by this (and understandably so). So, I'd like this thread to be about stating what implications are present, and nothing more than this. So as to increase understanding. Thank you.