Can anyone explain how this fits together or provide direction to someone learning meta-mathmatics from the ground up?
I started reading Bourbaki's Theory on Sets, Alonzo Church's Introduction to Mathematical Logic and Frege Begriffsschrift because they appear to provide the lowest level insight into the foundation, if there is any. I've constructed logic trees using the Hilbert / Epsilon operator as Bourbaki does and learned the way that Bourbaki uses the operator generates an extreme amount of signs to represent a simple ordinal number 1. There are some that claim this is due to the way the operator is used for quantification. The Hilbert operator's usage generates many signs and it's my understanding may be compared to Skolem which uses less.
Where $T_x(R)$ represents a distinguished object where some value is substituted for x and the relation R holds true.
Another interesting item in Bourbaki, is that they held the ordered pair as primitive only until many years later which Kuratowski's definition was used (generating even more signs). Keeping the ordered pair primitive is confusing, perhaps related to Russel's point of view.
Hilbert believed the operator would be helpful in providing consistency proofs. Perhaps through constructive means? I'm assuming this is why the operator was important. It's also my understanding that Godel's inconsistency theorems, if accepted, indicate that Hilbert's goal of proving consistency is impossible, at least for arithmetic. I'm unsure how it applies outside of arithmetic. I have also read that Godel's theorem depends on accepting that the human brain will always surpass a Turing machine in it's intelligence or decision making (therefore the brain can imagine numbers that a Turing machine cannot?).
It's also odd that Bourbaki choose to leave the Hilbert / Epsilon included in their versions even after Godel's announcement occurred. Maybe the group was just firm believers in Hilbert's work, or perhaps there is some other underlying reasons for it. I'm also surprised that such a smart man, Hilbert, had so little to say about the impact of Godel. Perhaps at that point of his career he had other responsibilities.
I appreciate anyone's point of view here and suggestions on paths to explore. I suspect there may be philosophical stances that influence all of this.
Thanks so much!