Model-Theoretic Connections

Elementary Substructures and the Lowenheim-Skolem Theorems

A substructure that agrees with its parent on every formula, not just on the atomic ones. The Tarski–Vaught test makes the condition checkable, and Löwenheim–Skolem makes such substructures plentiful.

Engineering · Mathematics6 min readKV-MATH-0249

Learning objectives

Distinguish a substructure from an elementary substructure.
State and apply the Tarski–Vaught test.
State the downward and upward Löwenheim–Skolem theorems.
Construct elementary substructures of prescribed cardinality.
Explain the Skolem paradox and its resolution.
Recognise elementary chains and their unions.

01Elementary substructure

B is a substructure of A when B ⊆ A is closed under the operations with relations restricted. B is an elementary substructure when A and B agree on every formula evaluated at elements of B.

B ≺ A ⟺ B ⊆ A and for every formula φ and every b⃗ ∈ B:
B ⊨ φ[b⃗] ⟺ A ⊨ φ[b⃗]

Substructure requires agreement on atomic formulas only. Elementary substructure requires it for all formulas, including quantified ones.

CautionSubstructure and elementary substructure are far apart

The even integers form a substructure of the integers under addition. They are not an elementary substructure: the formula ∃y (y + y ≈ x) is satisfied by every element of the even integers within the even integers only when the witness is also even, and the two structures disagree on which elements are doubles. Quantifiers see the difference; atomic formulas do not.

The difficulty is that quantifiers range over the whole structure. A quantified formula true in A may need a witness lying outside B, so B can fail to reproduce A's verdict even while agreeing on everything atomic.

02The Tarski–Vaught test

Checking every formula is impossible directly. The test reduces the condition to existential formulas with a single quantifier.

ProcedureThe Tarski–Vaught test

in: A, B ⊆ A → out: whether B ≺ A

input: structure A and a substructure B ⊆ A
condition to check:
for every formula φ(x, y⃗) and every b⃗ ∈ B,
if A ⊨ ∃x φ(x, b⃗) then there is some b ∈ B with A ⊨ φ(b, b⃗)
in words: every existential statement about parameters from B that holds in A
already has a witness inside B
if the condition holds, then B ≺ A
proof: induction on formula complexity; the only hard case is ∃, handled
exactly by the hypothesis

Correctness: induction on formulas, with negation and conjunction immediate and the existential case supplied by the hypothesis. Universal quantifiers are handled by rewriting ∀ as ¬∃¬. Caveat: B must already be a substructure — closure under the operations is assumed, not implied.

The test is what makes elementary substructures constructible. It says one need only close a set under witnesses for existential formulas, which is a finitary closure operation and therefore terminates in a set of controlled size.

03Downward Löwenheim–Skolem

Key resultThe downward theorem

If A is a structure for a language L and X ⊆ A, then A has an elementary substructure B with X ⊆ B and |B| ≤ max(|X|, |L|, ℵ₀). In particular every structure in a countable language has a countable elementary substructure.

Start with the seed set
Take X, the elements one wishes to retain.
Close under Skolem witnesses
For each formula ∃x φ(x, b⃗) with parameters from the current set that holds in A, add one witness. This requires choice.
Iterate ω times
Each round adds at most max(|current|, |L|, ℵ₀) elements, and ω rounds keep the bound.
Apply Tarski–Vaught
The limit set satisfies the test by construction, so it is an elementary substructure.

The construction is the finitary closure argument from Chapter I applied to formulas rather than operations, which is why the cardinality bound comes out as it does — formulas are finite, so each round adds boundedly many elements.

04Upward Löwenheim–Skolem

The upward direction produces larger models rather than smaller substructures, and it is a corollary of compactness.

ProcedureBuilding arbitrarily large elementary extensions

in: infinite A, cardinal κ → out: elementary extension of size ≥ κ

input: infinite structure A, cardinal κ ≥ max(|A|, |L|)
expand the language by κ new constant symbols c_i
form the theory T := Th(A with names for all its elements)
∪ { c_i ≉ c_j : i ≠ j }
every FINITE subset of T is satisfiable in A itself
(finitely many new constants can be interpreted as distinct elements,
since A is infinite)
by compactness T has a model B of size at least κ
the interpretation of the original names embeds A elementarily into B
output: B ≻ A with |B| ≥ κ

Compactness is doing all the work, and compactness rests on BPI. Caveat: the hypothesis that A is infinite is essential — a finite structure has no proper elementary extension.

Combining the two directions: any theory with an infinite model has models of every infinite cardinality at least the size of the language. Cardinality is entirely invisible to first-order logic above that threshold.

05The Skolem paradox

Zermelo–Fraenkel set theory, if consistent, has a countable model — yet ZF proves the existence of uncountable sets. The apparent contradiction is the Skolem paradox.

Inside the model

The set is uncountable

The model contains a set x and satisfies 'x is uncountable', meaning no bijection between x and ω exists within the model.

Outside the model

Everything is countable

From the ambient metatheory the model is countable, so x has countably many elements and a bijection with ω exists — but that bijection is not an element of the model.

Key resultThe resolution

Uncountability is not absolute. It is a statement about the non-existence of a bijection, and whether such a bijection exists depends on which sets the model contains. The paradox dissolves once one distinguishes satisfaction within a model from truth in the metatheory — the same distinction Tarski's definition makes precise.

06Elementary chains

A chain of structures each an elementary substructure of the next has a union that is an elementary extension of every member.

A₀ ≺ A₁ ≺ A₂ ≺ ⋯ ⟹ A_i ≺ ⋃_n A_n for every i

The Tarski elementary chain theorem. Proved by induction on formulas, using that every formula mentions finitely many elements, all of which appear at some finite stage.

This is the standard device for constructing models with prescribed properties: build a chain handling one requirement at a time, then take the union. It is the model-theoretic counterpart of the ascending-union argument used for generated subuniverses in Chapter II, and it works for the same reason — formulas, like terms, are finite.

Frequently asked

Does the downward theorem need the axiom of choice?

Yes — selecting a witness for each existential formula is a choice. The theorem is not provable in ZF alone. The upward direction needs compactness, hence BPI, which is weaker.

Can a finite structure have a proper elementary extension?

No. A finite structure satisfies a sentence asserting exactly n elements exist, and any elementary extension must satisfy the same sentence, so it has exactly n elements too. Elementary extensions of finite structures are isomorphic to them.

Is every substructure of a countable elementary substructure elementary?

No — elementarity is not inherited by further substructures. B ≺ A and C ⊆ B gives no information about whether C ≺ B. The Tarski–Vaught test must be verified at each level, which is why elementary chains are constructed deliberately rather than found.

Sources and further reading

S. Burris and H. P. Sankappanavar, A Course in Universal Algebra, Millennium Edition (a corrected re-typesetting of Springer GTM 78, 1981).
G. Grätzer, Universal Algebra, 2nd edition, Springer.
R. McKenzie, G. McNulty and W. Taylor, Algebras, Lattices, Varieties, Volume I.

Original KEVOS® explanatory article. Written from the topic map of the cited works; no text is reproduced from them.