 Perhaps the philosophy of second-order logic is that if a structure can be characterized, it exists, echoing Hilbert’s formalist philosophy. However, even in that sense we need to know that the characterizing sentence \(\theta_\ma\) is consistent. A trivial approach is to observe that \(\ma\) itself certainly satisfies \(\theta_\ma\), but this clearly begs the question. Still, this is how Hilbert argues for the consistency of his axioms for the real numbers (1900). If \(\ma=(\oN,+,\cdot)\), an Axiom of Infinity saying that that a proper subset of the domain has the same cardinality as the entire domain, is what is needed. In combination with the Comprehension Axiom Schema the existence of \(\ma\) follows. For \((\oR,+,\cdot,0,1)\) this is not enough, we need a stronger Axiom of Infinity, for example the existence of a dense order which satisfies the Completeness Axiom: Every bounded non-empty set has a supremum. It seems inevitable that when we move from a structure to a bigger structure we need to make a largeness assumption. Such assumptions are called Large Domain Assumptions in Väänänen (2012). In set theory this aspect is taken care of at the outset by assuming the Power Set Axiom and the Replacement Axioms which together make sure that there are large enough sets. In second-order logic we have to assume new and new Large Domain Assumptions, as we move along. This can be seen as a drawback, having to make new assumptions all the time. On the other hand it can be seen as an asset, not having to make too ambitious assumptions before they are really needed.