Lets say that I tell to a person "A" that I believe that the Got exists. For the person "A" it seems therefore obvious to imagine that I also believe in a lot of things that flow from such a statement. But lets say, that me and the person "A" made a following agreement between the two of us: "If I say that I believe in something to be true, I have to say it explicitly. We cannot make assumptions." Therefore the easiest way to tell to the person "A" explicitly all the things that I believe in when I'm saying that I believe in Got, I have two options:
1) To list all the things that I believe are true. The disadvantage would be that such a list would be almost infinitely long.
2) To make a formal system so that person "A" could deduce all the theorems/truths that flow from that system. Therefore this approach would be way more easier for me to say all the things that I believe in without relying on person's "A" assuming.
Is it possible to view a formal system as a short way of saying all the things that I believe in?
It is highly unlikely that you actually believe all the things that logically follow from some set of axioms, no matter what that set of axioms is: sometimes the connection between the axioms and its theorems is just too hard to grasp.
Take some basic axioms in mathematics, for example. You may well believe in all those axioms, as they are expressing elementary propositions about numbers or sets ... but do you believe all its consequences?
Professional mathematicians discover new mathematical theorems every day, meaning that they weren't sure whether something was a theorem before, and so they may not have believed it beforehand, even though all along it was a consequences of axioms they did believe in.