Isomorphisms and elementary equivalence

Isomorphic structures exhibit identical truths—they have the same theory. But conversely, not all structures with the same theory are isomorphic.

Enjoy this installment from A Panorama of Logic, an introduction to topics in logic for philosophers, mathematicians, and computer scientists. Fresh content each week.

Please consider subscribing as a free or paid subscriber.

Isomorphism and elementary equivalence

An isomorphism of one structure M with another N in the same language is a way of realizing the two structures as copies of one another—it is a one-to-one correspondence of the individuals in the first structure with those in the second structure that respects all the atomic relations and operations between the two structures. An isomorphism of one order ⟨A, ◁^A⟩ with another ⟨B, ◁^B⟩, for example, is a bijective map π : A → B such that

\( x ◁ ^A y\quad\text{ if and only if }\quad π( x ) ◁ ^Bπ( y ). \)

More generally, an isomorphism of structure M with structure N is a bijective function π : M → N of the domains of these structures such that:

1. Every atomic relation instance holding in the structure M also holds for the corresponding instance in the structure N and vice versa:

   \( R^M( a_1,..., a_n )\quad\textup{ if and only if }\quad R^N\bigl( π( a_1 ),..., π( a_n )\bigr ). \)

2. The function operations of the language, if any, are respected by the isomorphism:

   \( π\bigl( f^M( a_1, ...,a_n )\bigr ) = f^N\bigl( π( a_1 ), ...,π( a_n )\bigr ). \)

3. And finally, the interpreted constants of the language, if any, are respected by the isomorphism:

   \( π( c^M ) = c^N. \)

Two structures are isomorphic, written M ≅ N, if there is such an isomorphism. Isomorphic structures are thus copies of each other with respect to all the fundamental structure that has been deemed important enough to include in the signature of the structure.

Mathematicians typically give enormous importance to their isomorphism concepts. According to the philosophy of structuralism, the genuinely mathematical ideas and properties are precisely those that are preserved by isomorphism. According to this view, properties of a mathematical structure that are not preserved by isomorphism are deemed inessential—if we had thought them to be essential we would have incorporated those features into the fundamental structure; we would have expanded the language to include suitable new relations or operations capable of expressing those features, and in this case they would have been preserved by isomorphisms in the expanded context.

This perspective can be seen as an abstract analogue of the Erlangen program in geometry, originating with Felix Klein, by which one specifies a geometry by providing a group of transformations, regarding a concept as geometrical in that geometry exactly when it is preserved by all those maps. The circumference of a polygon is a geometrical notion with respect to the group of all isometries of the plane, but not if we allow dilation. Chirality in three-space is a geometric notion when we allow only orientation-preserving isometries—such as the manner that a molecule might move about in a vacuum or fluid suspension—but not when we also allow reflections.

A key feature of the isomorphism concept is that