Question: Does "first-order logic" (i.e. any elementary first-order theory, see below) permit infinite conjunctions? (Of only countable cardinality or of arbitrary infinite cardinality?)

It seems like any infinite axiom schema is implicitly an infinite conjunction (cf. this related question), and there are numerous examples of elementary first-order theories with infinite axiom schemas. So probably the answer is yes, but if so this is rarely stated explicitly, so I want to confirm. Many sources actively give the impression that only finite conjunctions are allowed.

Following Knight and Keisler's introduction to infinitary logic, $\omega$ denotes the first infinite ordinal (countable cardinality), and $\omega_1$ denotes the first uncountable ordinal, for a given vocabulary $L$ and infinite cardinals $\mu \le \kappa$, $L_{\kappa \mu}$ is the infinitary logic with:

  • no greater than $\kappa$ variables,
  • conjunctions and disjunctions over sets of formulas of size strictly less than $\kappa$, and
  • existential and universal quantifiers over sets of variables of size strictly less than $\mu$.

Based on these definitions, Keisler and Knight claim that $L_{\omega \omega}$ is ordinary elementary first order logic with (countably many variables,) finite conjunctions, finite disjunctions, and finite quantifiers.

They define an elementary first-order theory (which is basically what I mean by "first-order logic" in the question title) as any set (of unspecified cardinality?) of sentences in $L_{\omega \omega}$ closed under logical consequence.

Infinite disjunctions definitely are NOT allowed in first-order logic, see either [this related question] or [this other related question]. It is also probably not a coincidence that all of the examples of $L_{\omega_1 \omega}$ theories given in section 1.1 of Knight and Keisler's introduction to infinitary logic (i.e. "simplest possible" theories expressible in $L_{\omega_1 \omega}$ and not $L_{\omega \omega}$) use countable disjunctions but do not use infinite conjunctions.

Another related question is about how quantifiers in the elementary first-order object theory correspond to infinitary conjunctions/disjunctions in the metatheory for infinite models. However I am asking about infinitary conjunctions/disjunctions in the object theory, not the metatheory.


1 Answer 1


You have to distinguish between what individual sentences can do and what theories can do. Theories, being arbitrary sets of sentences, are able to perform arbitrarily large conjunctions in the following sense:

For any set of theories $\{T_i: i\in I\}$, the theory $T:=\bigcup_{i\in I}T_i$ "behaves like the conjunction of the $T_i$s" in the following sense: the models of $T$ are exactly those structures satisfying each $T_i$.

On the other hand, even if each $T_i$ consists just of a single sentence $T_i=\{\varphi_i\}$, there may be no single sentence $\psi$ such that the models of $\psi$ are exactly the models of each $\varphi_i$. For a concrete example, there is no single sentence true in exactly the infinite structures, even though for each $n$ there is a sentence true in exactly the structures of size $\ge n$.

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    $\begingroup$ Is a union of theories always a theory? I thought that wasn't the case. Also, how can a theory consist of a single sentence. Doesn't a theory need to contain every tautology? $\endgroup$ Jan 6 at 5:58
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    $\begingroup$ @JasonSwanson Some texts define "theory" as "deductively closed set of sentences," others define it as simply "set of sentences." I'm using the latter definition here, but it sounds like you're used to the former. $\endgroup$ Jan 6 at 14:03
  • $\begingroup$ Sorry for slow reply. This makes sense -- it seems I was conflating syntax and semantics (again). The last example is nice, and seems to explain the claim I once heard that "first order logic can't distinguish between the finite and infinite". To confirm understanding, 1) syntactically individual sentences in FOL can not have infinite conjunctions,and 2) FOL theories with arbitrarily many individual sentences are syntactically valid, even when not finitely axiomatizable, even though such theories could be "semantically equivalent" to a single sentence in FOL + an infinitary conjunction symbol? $\endgroup$ Jan 8 at 4:23
  • $\begingroup$ Anyway regarding semantics of non-finitely-axiomatizable theories, and/or how one would specifically define the semantics of "FOL + an infinitary conjunction symbol", I and others can see your answers to previous questions: math.stackexchange.com/a/3287026/606791 , math.stackexchange.com/questions/2611753/… Thank you for all of your help with this! (@JasonSwanson Regarding the union of theories thing, I just interpreted it as the "theory generated by the union", e.g. similar to what you have to do with $\sigma$-algebras.) $\endgroup$ Jan 8 at 4:32

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