Disclaimer: Please excuse any lack of precise mathematical expression, I am just trying to wrap my head around the concepts right now.
I often read that it is possible to express and/or deduce all other mathematical theories, theorems and axioms (is this correct?) from the axioms and/or notation of set theory. Based on this, I ask myself the following questions:
- To what extent is this true? What is there that cannot be expressed in terms of set-theoretical notation, and is there anything that cannot be deduced using the axioms of ZFC?
- Who exactly came to the conclusion that all this is possible? I have read that Bertrand Russell tried to formalize and break down all of mathematics so that it can be expressed in terms of formal logic (Principia Mathematica), but I haven't found anything about a mathematician who has managed to break down all of mathematics so that it can be expressed in terms of set theory. Can someone provide me with some historical background - how did mathematicians find out that all of mathematics can be expressed in terms of ZFC?
Thank you very much in advance! You are helping out a high schooler in great despair.