Can a sentence go on and on forever? Logic’s strange answer
The Sentence That Never Ends
Imagine you want to say exactly how many things there are. “One, or two, or three, or four, or …” But you can never finish — the sentence just keeps going. Ordinary logic, the kind you learn in puzzles, cannot write that down. Its sentences must be finite, so ”…” is not allowed. Yet sometimes we need a logic that can talk about infinity directly: a logic that lets a sentence be infinitely long. Mathematicians and philosophers in the 20th century built such languages. They are called infinitary languages, and they reveal surprising truths about what logic can and cannot do.
The simplest infinitary language is called L(ω₁,ω). The “ω₁” means you can join together countably many smaller sentences with ∧ (and) or ∨ (or). The “ω” means your quantifiers, like “there exists” or “for all,” still range over only finitely many variables at a time. So in L(ω₁,ω) you could finally write: “x=0 ∨ x=1 ∨ x=2 ∨ …” That single infinite sentence captures that there are infinitely many counting numbers.
But why should you care? Because allowing infinity into logic makes some tasks possible and others impossible. Some of the nicest features of ordinary logic — like the compactness theorem — fall apart. Understanding why leads us deep into the nature of infinity itself.
What Can an Infinitary Sentence Say?
Ordinary first‑order logic cannot say “the only things that exist are the natural numbers 0, 1, 2, ….” Any set of axioms you write down will also have weird non‑standard models with extra, fake numbers. But in L(ω₁,ω), you can grab the natural numbers with a single sentence. You do it by listing every addition and multiplication fact: “0+0=0, 0+1=1,…” and “0·0=0, 0·1=0,…” joined together with an infinite ∧. Then you add that every number is built from 0 by the successor operation: for any x, x must equal “0”, or “the successor of 0”, or “the successor of that”, and so on. This infinite sentence has exactly one kind of model: the standard natural numbers.
A similar trick expresses that a set is finite. In L(ω₁,ω) you can write: “Either there is exactly 1 thing, or exactly 2 things, or exactly 3 things, …” — an infinite ∨ of finite descriptions. No finite sentence can do that.
If you allow even more powerful quantifiers — L(ω₁,ω₁) — you can also say that a relation is a well‑ordering. A well‑ordering is a linear order where every subset has a least element. The sentence that captures this uses an infinite quantifier: for any countably many elements x₀, x₁, x₂, …, one of them is the smallest. That tiny extra “for any infinite sequence” quantifier pushes the language into a new realm of strength.
The cost of this strength is that many comfortable facts about logic stop being true.
The Well‑Behaved Language: L(ω₁,ω)
Despite its infinite sentences, L(ω₁,ω) still behaves a bit like ordinary logic. Carol Karp (1922–1972) proved that it has a completeness theorem. That means if a sentence is true in every possible structure, there is a proof of it — if you allow proofs to have countably many steps. You add a new rule: from φ₀, φ₁, φ₂,… you can infer their infinite conjunction ∧φₙ. Deductions of countable length then turn out to be enough.
But the compactness theorem is gone. In ordinary first‑order logic, if every finite subset of a set of sentences has a model, the whole set has a model. For L(ω₁,ω) this fails spectacularly. Consider the sentence σ that captures the standard model of arithmetic. Add uncountably many new constant symbols c₀, c₁, c₂,… and demand that they are all different: cᵢ ≠ cⱼ for i≠j. Any countable chunk of these demands can be satisfied — you can fit those constants into a countable model — but the entire uncountable set cannot, because the model can only hold countably many distinct elements while satisfying σ. So compactness, a cornerstone of first‑order logic, breaks.
Scott’s Isomorphism Theorem offers a silver lining. If you have a countable structure with countably many relations, a single L(ω₁,ω) sentence can describe it exactly up to isomorphism among countable models. This is impossible in ordinary logic. Hanf numbers, which measure how large models a sentence can have, also become enormous for L(ω₁,ω): the least cardinal that forces arbitrarily large models is larger than many infinities you can easily name.
When Infinity Gets Out of Control: L(ω₁,ω₁)
Once you allow infinite quantifiers, logic begins to resemble second‑order logic. You can quantify over infinite sets of individuals, not just individuals one by one. In L(ω₁,ω₁), you can say “every countable set has a least element” — that’s how well‑orderings are captured. But this power comes with a devastating limitation: undefinability of truth.
Dana Scott (b. 1932) showed that the set of all logically valid sentences of L(ω₁,ω₁) cannot be defined within the structure that codes those sentences. The coding structure is the collection of hereditarily countable sets, written H(ω₁). You can assign a code to every formula, just as Gödel did for ordinary arithmetic. Scott proved that there is no formula of L(ω₁,ω₁) that says “code number x is a valid sentence.” Why? Because you can first characterize H(ω₁) inside the language (a single sentence says “this structure is H(ω₁)”), and then you could use it to define truth for the language itself — a contradiction akin to Tarski’s theorem on the undefinability of truth. So L(ω₁,ω₁) is incomplete in a deep sense: no complete proof system that stays inside the language is possible.
This shows that moving beyond finite quantifiers breaks the neat connection between syntax and semantics that first‑order logic and even L(ω₁,ω) enjoy.
Large Cardinals and the Compactness Problem
The failure of compactness for most infinitary languages raises a natural question: can any infinitary language be compact? The answer links logic to the theory of large cardinals — gigantic infinite numbers whose existence cannot be proved from the usual axioms of set theory. A cardinal κ is called weakly compact if the language L(κ,κ) is weakly κ‑compact (every set of sentences of size ≤κ that is consistent in chunks of size <κ has a model). It turns out that weakly compact cardinals must be inaccessible (unreachable by the usual operations of power set and union from smaller cardinals). In fact, the first inaccessible cardinal is not weakly compact, nor the second, nor any finite number of them — you need inaccessibly many inaccessibles below. These results are due to William Hanf and others in the 1960s.
Moreover, if you assume Gödel’s axiom of constructibility, there are no compact cardinals at all. So the hope for a simple compact infinitary logic collapses. Still, exploring compactness in this setting opened the door to understanding measurable cardinals and other large‑cardinal notions that now play a central role in set theory.
Barwise’s Fix: Admissible Sets
The failure of compactness seems total, but Jon Barwise (1942–2000) discovered a way to recover it for carefully chosen fragments. Instead of looking at the full language L(ω₁,ω), you restrict your formulas to those whose codes live inside a countable admissible set A. An admissible set is a collection of sets that is closed under basic operations (pairing, union) and satisfies weak separation and replacement schemes limited to Δ₀ formulas — roughly, formulas whose quantifiers are bounded by membership. These sets allow recursion theory and proof theory to work smoothly.
Barwise showed that if Δ is a set of sentences that is Σ₁‑definable over A (the set‑theoretic analogue of recursively enumerable), and every Δ′∈A that is a subset of Δ has a model, then Δ itself has a model. This Barwise Compactness Theorem gives back a usable compactness for an important class of sentences. A striking application: every countable transitive model of ZFC set theory has a proper end‑extension — a larger model in which no new elements appear below any old element. First‑order compactness cannot prove this; you need the infinitary version.
Thus, by tuning the language to the structure of definability inside admissible sets, logic regains some of its neatness. It shows that the boundary between logic and set theory is porous and deeply interconnected.
Why Should You Care?
You probably will never write an infinite sentence. But the story of infinitary logic matters because it shows that even the simplest‑seeming rules have sharp limits. The compactness theorem — that if every finite piece is possible then the whole is possible — feels obvious until you try it with infinite conjunctions and it fails. The completeness theorem — that valid sentences have proofs — works for L(ω₁,ω) but collapses for L(ω₁,ω₁). And the attempt to salvage compactness drags us into the highest reaches of infinity, where we must ask: do those cardinals even exist?
These results teach us that logic is not a finished tool. It is a landscape where power and control are in tension. Every time we stretch the rules, we pay a price in certainty or elegance. Seeing that landscape helps you appreciate why mathematicians and philosophers still argue about what counts as a legitimate language, a valid proof, or a real infinity. It is the same kind of curiosity that makes you ask whether there is a biggest number — and what happens if you try to count past it.
Think about it
- If you could write an infinite sentence, would you trust a proof that uses infinitely many steps? Why or why not?
- The compactness theorem fails for infinitary logic. Does that mean our gut feeling — “if every small part works, the whole thing works” — is wrong for infinity, or just that our logical tools need to be more careful?
- Imagine a language powerful enough to describe all truths about numbers. If the system itself cannot define its own notion of truth, is it still a complete logic? What would “complete” even mean in that world?
