It is a theorem of second order Peano arithmetic that all models are uniquely isomorphic. Note that Eliezer does not say that he believes in second order logic, but only makes a conditional statement. The problem with second order logic is that it refers to the undefined term "property." Properties are pretty close to sets, so if one believes that this term is sensible, one seems to believe in a preferred model of set theory. One could talk about second-order logic only relative to a first order theory of set theory, but then one only has a relative uniqueness statement.

Eliezer seems to have brought up second-order arithmetic not because he thinks it's a good idea, but because he thinks Nelson is using it. In fact, Nelson is nervous about set theory, so he interprets induction not for arbitrary properties, but only for formulas of the language. Then induction becomes a first-order axiom scheme and Gödel says that there are many models.

Platonism refers to the inference system component of the human decision problem, while the notion of things being real refers to the outcome (reality) concept (defined with respect to this inference system). People could turn out being able to reason about infinite, but without the infinite being real (i.e. reality being infinite). Infinite could, for example, help to model uncertainty about the world.