# How do set theory, and formal logic fit in together?

Im at that stage in my mathematical understanding where I kinda understand what set theory is and what first order logic is but dont really understand how they fit together to create Mathematics. I assume that the ZF system uses first order logic to create the foundations of mathematics and in the grand scheme of things, set theory is dependent on logic for its existence whereas logic or any formal system can exist on its own. Is this the correct view?

• Neither mathematical logic nor set theory "create Mathematics" ... Math "was there" centuries before math log and set theory. The "foundational" point of view was linked to the historical "foundational schools" : Logicism, Formalism, Intuitionism and must be understood in that setting. A "simple" point of view is that math log and set theory provide the language and the "tools" to represent the existsing math theories in a "uniform" way: but this does not means that math "dependent on logic [and set th] for its existence". Aug 4, 2015 at 7:57

Not the best answer, but I'll try with a short answer. Some of the set theory books, you will find it says for example $A\subseteq A\cup B$, where $A$ and $B$ are both sets. If we want to check if it's true, we have to prove that the statement $(x\in A)\implies (x\in A\cup B)$ is true. Note that this is equivalent to $(x\in A)\implies [(x\in A)\vee (x\in B)]$, and it's actually true. The reason why it's true is because the statement $p\implies (p\vee q)$ is a tautology which can be read more in Mathematical logic. There is a nice table that shows different tautologies in Rules of Inference. The rule we used here is called "Addition".