ZFC with 1st order logic is known.
My question may be formulated either way
- What is the definition of ZFC2 (ZFC+second order logic)? (I assume that formulas also have quantifications of two kind over object and over predicates. Then what are some examples of statements in ZFC2 that cannot be proved in ZFC1?)
- What are the sources with precise definitions of second-order logic?
- Are there exist a "standard" deductive system of second-order logic? (what exactly reasearchers mean talking about second order logic)
- What are the reasons to use the second order logic while one may have a type theory?
Question is motivated by https://mathoverflow.net/questions/409521/does-there-always-exist-a-categorical-extension-of-zfc-2-with-no-set-models and What's a good introduction to Second Order Logic (has no accepted answer)
