Questions: Isomorphisms and Structural Equivalence

Question 1

The structures (ℤ, <) and (ℚ, <) — the integers and rationals under their usual orderings — satisfy different first-order sentences (for example, ℚ satisfies 'between any two elements there is another,' while ℤ does not). What can we conclude?

• A. They are isomorphic, because both are infinite linear orders
• B. They are not isomorphic, and since they differ on a first-order sentence, they are not even elementarily equivalent
• C. They are elementarily equivalent but not isomorphic, because they have the same cardinality
• D. They cannot be compared because one has a different language than the other

Answer: (B)

If two structures disagree on any first-order sentence, they cannot be isomorphic (isomorphic structures satisfy exactly the same sentences) and they are not elementarily equivalent. The density property 'between any two elements there is a third' is a first-order sentence true of ℚ but false of ℤ (there is nothing between 1 and 2 in ℤ). So ℤ and ℚ are neither isomorphic nor elementarily equivalent as ordered structures.

Question 2

An isomorphism f: M → N differs from a mere embedding g: M → N in which crucial respect?

• A. An isomorphism preserves all first-order formulas; an embedding preserves only universal sentences
• B. An isomorphism must be surjective — its image is all of N; an embedding's image may be a proper substructure
• C. An isomorphism is always elementary; an embedding may fail to preserve existential formulas
• D. An embedding requires a bijection; an isomorphism only requires injectivity

Answer: (B)

Both an embedding and an isomorphism are injective maps that preserve and reflect atomic formulas. The difference is surjectivity: an embedding's image may be a proper substructure of N (it identifies M with a copy of itself inside N), while an isomorphism is bijective — M is put in perfect one-to-one correspondence with all of N. Isomorphisms are 'perfect translations' of one structure into another; embeddings are 'inclusions into a larger structure.'

Question 3

If M and N are isomorphic, then every first-order sentence true in M is also true in N.

• T. True
• F. False

Answer: True

This is the fundamental theorem of isomorphisms: an isomorphism f: M → N is a bijection that preserves all atomic formulas, and by induction on formula complexity, it preserves the truth of every first-order formula. Intuitively, renaming elements by f produces an indistinguishable copy — no formula can detect whether you're evaluating it on M or on its f-image in N. Isomorphism is the strongest notion of structural equivalence precisely because it preserves all logical properties, not just atomic ones.

Question 4

Two structures that satisfy exactly the same first-order sentences is expected to be isomorphic.

• T. True
• F. False

Answer: False

This is false, and it is one of model theory's most important lessons. Elementary equivalence (satisfying the same sentences) is strictly weaker than isomorphism. A classic example: (ℚ, <) and any countable dense linear order without endpoints are elementarily equivalent to each other, but a countable dense linear order and an uncountable one (like ℝ) also satisfy the same first-order sentences — yet they cannot be isomorphic since they have different cardinalities. Compactness and Löwenheim-Skolem theorems guarantee the existence of models of different sizes that are elementarily equivalent.

Question 5

What is the difference between isomorphism and elementary equivalence, and why does model theory need both concepts?

Model answer: Two structures are isomorphic if there exists a bijection between them that preserves all atomic formulas — they are literally the same structure with renamed elements, and they agree on every first-order sentence. Two structures are elementarily equivalent if they satisfy exactly the same first-order sentences, but there may be no bijection between them (they might have different cardinalities). Isomorphism implies elementary equivalence, but not conversely. Model theory needs both because isomorphism is too strong for classifying models of most theories: by Löwenheim-Skolem, a theory with an infinite model has models of every infinite cardinality, which cannot all be isomorphic. Elementary equivalence is the coarser tool for identifying models that are logically indistinguishable even when structurally different in size.

This distinction drives much of model theory. A theory is categorical in cardinality κ if all its models of size κ are isomorphic — this is a very strong condition. Most theories have non-isomorphic models of the same cardinality. Elementary equivalence classes partition models by logical indistinguishability, providing a finer analysis than 'same theory' but coarser than isomorphism. The interplay between these equivalence relations is studied through tools like back-and-forth systems, Ehrenfeucht-Fraïssé games, and Scott sentences.