>>12240434I mean did you google your terms?
Sounds like
https://en.wikipedia.org/wiki/Complete_theoryanswers your last question right away.
See also
https://en.wikipedia.org/wiki/Categorical_theoryFor a short history, Frege came up with quantifiers etc. in 1879. Around that time, or shortly after, people like Dedekind used this fully formal language to write down math. Around that time, Cantor came up with transfinite ordinals (to deal with function theory) and this was all fully formalized in the decades to come. Logicans like Russel did programs to capture math via logic and all this history.
So regarding the first question, Peanos original formulation was second order, meaning you have not just quantifiers over one domain of discourse, but also over collections of those. See
https://en.wikipedia.org/wiki/Second-order_arithmeticBut if you say Peano axioms, I'd say it's not strictly tied to second-order languages.
If you interpret quantifiers to range over just all there is of their respective sort, then second order logic doesn't have the nice metamathematical properties that first order logic has. E.g. completeness (the syntax is in some sense faithful to the models for a theory - at least for those things that all models share. This is not necessarily related to incompletness theorems or decidability).
But second order PA is categorical.
Meanwhile, PA in first order logic doesn't capture any particular model. If you look at the syntax of PA and prove theorems, you actually never prove statements that are just true for "the first infinite amount" of numbers.
After all numbers "N" we know, one can attach a dense (like Q) collection of infinite elements (like Z). Roughly, there's models like "M=N+Q·Z" and whenever you use first-order logic to prove any statement, it can also be understood to be a statement about this weird model.
