Of course, one has mathematics which is pursued for its own sake. However, suppose one wished to justify the connection of a theory to the physical world (beyond the trivial bit about the physical brain that thinks it up). The problem is that measurements are all rational numbers, but we posit theories with uncountable models. (I am thinking of the class of measurements as the universe of my model.) Many theories which have uncountable models can be justified by the L÷wenheim-Skolem Theorem, whereby any first-order theory with an uncountable model also possesses a countable model, so we can say that in using the uncountable interpretation we are just making a shortcut, but in reality we are describing a countable model. However, there are second-order theories which are not reducible to first-order, and where the L-S Theorem fails. Hence the previous justification also fails. Perhaps because of this failure most papers concentrate on first-order theories, but nonetheless second-order theories prove useful in describing certain phenomena. But apart from the justification "it's useful, so use it", can one justify using such a theory which does not correspond to a model with the universe being restricted to the class of possible measurements?