Help, Foundations of Mathematics: The Hilbert/Epsilon Operator, Bourbaki and Godel's Theorems

Question

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.

For example, Bourbaki defines:

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!

Answer 1

I won't try to explain how Bourbaki, Church, and Frege fit together, because they are not only giving different foundational systems but foundational systems for different parts of mathematics. For general information about foundations, you might look at W. S. Hatcher's book, "Foundations of Mathematics".

Your second question, "Perhaps through constructive means?", concerns Hilbert's motivation for introducing the epsilon operator. Hilbert wanted consistency proofs that are not merely constructive but finitistic, i.e., using only finite combinatorial arguments.

Bourbaki's motivation for using epsilon (or tau) had, as far as I know, nothing to do with consistency proofs. It was a convenient device for getting the quantifiers and the axiom of choice from a single symbolism.

Your third question concerns something you read to the effect that Gödel's incompleteness [not "inconsistency"] theorems depend on accepting that the human brain will always surpass a Turing machine .... Nonsense. Gödel may well have believed that brains will always surpass Turing machines, but his theorems do not in any way depend on such a belief.

I believe that covers all the question marks in your question, but I should also mention that I disagree with "Keeping the ordered pair primitive is confusing, perhaps related to Russel[l]'s point of view." It seems to me that the only reason for not keeping the ordered pair primitive is a desire to minimize the number of primitive concepts, reducing everything to sets. Apart from this reductionism, a primitive ordered pair seems less confusing than a set-theoretic encoding of ordered pairs. With a primitive ordered pair, one simply assumes as an axiom that two ordered pairs are equal iff they agree in both components; with a set-theoretic encoding, one has to prove that as a theorem, and I've seen students get the proof wrong (usually by overlooking the possibility that the two components of an ordered pair might be equal). Also, if I remember correctly and contrary to what you wrote, Russell was not particularly fond of primitive ordered pairs; when a set-theoretic encoding became available, Russell used it in the next edition of Principia.

Finally, Gödel's incompleteness theorems apply not only to arithmetic but to any theory in which some basic facts of arithmetic can be proved. The theory may be much richer, for example the Zermelo-Fraenkel set theory that is nowadays generally considered the standard foundation for mathematics.