I am learning set theory on my own at the moment, and I realised I can't avoid not to learn logics. There is a strong connection between these two. such that, proofs for sets are based on logics.
I don't understand why and how these two fields are connected. For me, I reviewed them as separated fields. and from my past experience, one field is kind of self-contained, like axioms of Euclidean geometry are contained in geometry. axioms of real analysis are contained in real numbers.
With my limited knowledge about logic, It is based on statements, and a statement is either true or false. The rules of negations, contradictions etc completely make sense within the logic land!
However, in set theory. Objects are undefined, whatever objects which meet some conditions become sets. Sometimes we know what objects are, maybe even the exact and complete list of objects. Most of time we don’t.
Therefore, How can we apply logic rules to a set of undefined objects? (did I miss something very important here?)
Hence I am not convinced that 'if the rule is true in logic, then it is also true for sets'? for example, let A,B,C be sets, $if A \subseteq B, and B \subseteq C, then A \subseteq C$, due to Transitive Inference.