Model-Theoretic Connections

Ultraproducts and Los's Theorem

Łoś's theorem is the transfer principle: a sentence holds in an ultraproduct exactly when it holds on a large set of coordinates. Everything model-theoretic in this stream follows from it.

Engineering · Mathematics5 min readKV-MATH-0251

Learning objectives

Define the ultraproduct and identify the role of maximality.
State Łoś's theorem precisely.
Follow the induction, especially the negation and existential cases.
Construct the canonical elementary embedding into an ultrapower.
Apply Łoś to derive compactness.
Recognise the non-standard models the construction produces.

01Ultraproducts

An ultraproduct is a reduced product over an ultrafilter. Maximality of the filter is the entire difference, and it changes the preservation behaviour completely.

Reduced product

Filter decides some subsets

For a subset S the filter may contain neither S nor its complement. Disjunctions can then fail to transfer, so only Horn sentences are preserved.

Ultraproduct

Ultrafilter decides every subset

Exactly one of S and its complement lies in U. Every disjunction is resolved, and every first-order sentence transfers.

Recall from the Boolean stream that an ultrafilter is a maximal proper filter, characterised by deciding every element. That decisiveness is precisely what the induction in Łoś's theorem needs.

02Łoś's theorem

Key resultŁoś's theorem

For structures A_i indexed by I and an ultrafilter U on I, and for any formula φ and elements of the ultraproduct:
∏A_i/U ⊨ φ[f⃗/U] ⟺ { i ∈ I : A_i ⊨ φ[f⃗(i)] } ∈ U.

In words: a formula holds in the ultraproduct exactly when it holds in 'almost every' factor, where U decides what almost every means. This is a complete transfer principle — not merely for a restricted class of formulas but for all of them.

03The proof

ProcedureŁoś by induction on formula structure

in: formula φ → out: the transfer equivalence for φ

atomic: holds by the definition of θ_U and the relation clause
conjunction: ⟦φ & ψ⟧ = ⟦φ⟧ ∩ ⟦ψ⟧; U is closed under intersection ✓
negation: ⟦¬φ⟧ = I ∖ ⟦φ⟧
U is an ULTRAfilter, so exactly one of ⟦φ⟧, I∖⟦φ⟧ lies in U
hence ⟦¬φ⟧ ∈ U iff ⟦φ⟧ ∉ U — exactly what is needed
existential: if ⟦∃x φ⟧ ∈ U, choose a witness in each such coordinate
(AXIOM OF CHOICE) and let g be the resulting element of the product
then ⟦φ(g)⟧ ⊇ ⟦∃x φ⟧ ∈ U
conversely a witness in the ultraproduct gives witnesses on a large set
universal: rewrite ∀ as ¬∃¬ and apply the previous two cases

Two steps are load-bearing: negation needs maximality of U, and the existential case needs choice to assemble the witness function. Caveat: dropping maximality breaks the negation case, which is exactly why reduced products only preserve Horn sentences.

CautionThe negation case is where ultrafilters are essential

For a non-maximal filter it is possible that neither ⟦φ⟧ nor its complement lies in F, so neither φ nor ¬φ holds in the reduced product though every factor decides. Maximality is not a convenience — the theorem is false without it.

04Ultrapowers and the diagonal embedding

An ultrapower is an ultraproduct of a constant family. It comes with a canonical elementary embedding of the original structure.

d : A → A^I/U d(a) := the class of the constant function with value a

By Łoś, A ⊨ φ[a⃗] iff the constant function satisfies φ in the ultrapower, since the agreement set is all of I. So d is an elementary embedding.

Principal U gives nothing
The ultrapower collapses to A itself and d is an isomorphism.
Free U gives a proper extension
For infinite I and free U the ultrapower is strictly larger, and d is a proper elementary embedding.
Consequence: elementary extensions on demand
Every infinite structure has proper elementary extensions, constructed without compactness.
Consequence: non-standard models
The ultrapower of the natural numbers over a free ultrafilter contains elements exceeding every standard natural — an infinite number, in a structure elementarily equivalent to the standard one.

05Applications

Compactness

The clean proof

Index finite subsets of a theory, take a model of each, and form the ultraproduct over an ultrafilter containing the sets of indices above each finite piece. Łoś gives a model of the whole theory.

Non-standard analysis

Infinitesimals

The ultrapower of the reals contains elements smaller than every positive real. Robinson's framework is built on exactly this construction.

Jónsson's lemma

Locating subdirect irreducibles

The P_U operator in the statement is the ultraproduct operator, and the proof constructs an ultrafilter from the congruence analysis.

Keisler–Shelah

Elementary equivalence characterised

Two structures are elementarily equivalent if and only if they have isomorphic ultrapowers — an algebraic characterisation of a logical notion.

Preservation theorems

The standard technique

Most preservation results are proved by constructing ultraproducts with prescribed properties.

Decidability transfer

Feferman–Vaught

The theory of a product reduces to the theories of factors plus the index Boolean algebra, which underlies the Chapter IV decidability results.

06What ultraproducts cost

The construction is powerful and entirely non-constructive.

The dependencies
Ingredient	Requires
Existence of a free ultrafilter	BPI — not provable in ZF
The existential case of Łoś	some choice to select witnesses
Compactness via ultraproducts	BPI, and compactness is equivalent to it
Ultraproducts over principal ultrafilters	nothing — but they give nothing
Ultraproducts of finitely many finite structures	nothing — they collapse

NoteWhy the finite collapse matters

An ultraproduct of finitely many structures is isomorphic to one of them, because the ultrafilter on a finite index set is principal. This is what makes Jónsson's lemma computationally usable for finitely generated varieties: P_U disappears and the conclusion becomes a finite, choice-free statement.

Frequently asked

Is the ultraproduct construction canonical?

No — it depends on the choice of ultrafilter, and different free ultrafilters can give non-isomorphic ultraproducts. Whether all free ultrafilters on the naturals give isomorphic ultrapowers of a fixed structure is independent of ZFC, being related to the continuum hypothesis.

Why does the ultrapower of the naturals have infinite elements?

Take the identity function on I = ω. For each standard n, the set of coordinates where the identity exceeds n is cofinite, hence in any free ultrafilter. So by Łoś the class of the identity exceeds the image of every standard n. It is an element of a structure elementarily equivalent to the naturals, yet larger than all of them.

Does Łoś's theorem hold for infinitary logic?

No. The induction relies on formulas being finite, so that agreement sets can be combined by finite intersection. Infinitary conjunctions would need closure under infinite intersections, which filters do not have. This is one of several places where finiteness of syntax is load-bearing.

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.