@Knight Hello! FOL (first-order logic) was not designed all at one go. Russell and Wittgenstein never understood logic, so they are irrelevant. The only thing Russell did was to find an inconsistency in Frege's system which we now know as Russell's paradox. But Peano was one of the early people who contributed significantly to logic, though he was not logical enough to see that his axiomatization of N failed to be useful. That is why all logicians now define PA differently from Peano.
Some history digging suggests that true FOL roughly began with Boole's students De Morgan and Pierce. Pierce in particular first got something like FOL without quantifiers in 1870, and then later in 1885 he introduced the quantifiers, though he used Π,Σ instead of ∀,∃. It seems that FOL was used in parallel by mathematicians working in analysis (where it is crucial to manipulate quantifiers correctly), including Dirichlet, Riemann, and Weierstrass, but they never extracted out the logic.
In other words, the mathematicians working in analysis used FOL reasoning correctly without knowing what they were using.
Nevertheless, it was not until Godel that even basic facts about FOL became known, including the semantic completeness theorem. Prior to that FOL was just used for mathematics but not studied as a mathematical object. But it's impossible to think of, much less grasp, this theorem without having a 100% grasp of true FOL, which includes its semantics and a 100% precise deductive system.
That is why one could argue that the field of mathematical logic only truly began with Godel.
By the way, it seems that someone (I think you can guess who) flagged your chat message in another room after you had left it. I do not think that flag was reasonable. If that happens again, I recommend that you really contact the moderators. =)
@Knight Originally, before Godel, people were just using FOL. FOL itself does indeed refer to the logic itself. After Godel, people started studying FOL mathematically, not just use it for mathematics like in the past.
@Knight: If you understand FOL semantics and deduction already, then it's easy to understand at least the statement of Godel's semantic completeness theorem. I can state it here for you:
> Given any set A of axioms over a language L, the set of sentences that we can deduce from A is exactly the set of sentences that is true in all models of A.
This is not a trivial theorem to prove, though it is now in every introductory undergraduate course on logic. In Godel's time, it was groundbreaking.
