We say that two structures are isomorphic if there is an isomorphism from one to the other.
There are, in addition, categorical axiomatizations of arithmetic, where a set of sentences is categorical if, and only if, all of its models are isomorphic, which means that there is only one model up to isomorphism.
In 1904 Oswald Veblen stole a term from Kant and described a theory as categorical if it has just one model up to isomorphism (i.e. it has a model and all its models are isomorphic to each other).
Hurley calls it the isomorphism hypothesis.
Then f is an isomorphism from M to a structure whose universe is the disjoint union of a set A and its power set P(A) and which assigns A to I, assigns P(A) to S, and assigns the membership relation ∈ to E. So roughly speaking, σ defines the power-set operation "to within isomorphism".
In the Tractatus, declarative sentences (Sätze) are said to be facts (arrangements of names), and states of affairs (Sachlagen, Sachverhalte, Tatsachen) are also said to be facts (arrangements of objects): a sentence is true just if there is an appropriate relation of correspondence (an isomorphism) between sentence and state of affairs.
Interpretations range from treating the form of isomorphism as direct and literal, so that, e.g., the eyes become speckled when viewing a robin's egg, to attenuated, where the isomorphism is more akin to that enjoyed between a house and its blue-print.
Also difficult is the notion of isomorphism appealed to in (iii).
There must be a determinate and expressible structural isomorphism, even though one could not say that the blueprint realizes the form of the house.
Second, objects are always characterized up to isomorphism (in the best cases, up to a unique isomorphism).
Bisimulation is a weaker notion than isomorphism (a bisimulation relation need not be 1-1), but it is sufficient to guarantee equivalence in processing.
