In other words, it is a proof (including all assumptions) that can be written on a large enough sheet of paper.
In such a logic, one can regard the existential quantifier, for instance, as derived from an infinitary disjunction.
Logicians in the early 20th century aimed to solve the problem of foundations, such as, "What is the true base of mathematics?"
In the words of David Hilbert (referring to geometry), "it does not matter if we call the things chairs, tables and beer mugs or points, lines and planes."
), give a finite number of propositions expressed in those symbols, which were to be taken as "foundations" (the axioms), and some rules of inference which would model the way humans make conclusions.