# Why doesn't one assert in metamathematics that a sentence S is a logical consequence of the conjunction of a set of sentences?

In other words, why isn't there -- at least in standard textbook presentations of things like the deduction theorem and the compactness theorem -- a conjunction connective that is applied to sets of sentences?

It seems that the conjunction connective is simply understood or somehow built into the concept of logical consequence when we assert that a sentence is a logical consequence of a set of sentences.

We can also consider disjunctions. In ZF, we speak of the binary union of two sets, and we have the convenient tool of the union of an arbitrary infinity of sets, but the claim that for every family of sets there exists a union of that family is a premise (that is given the label "axiom"). More basic than the issue of the existence of an arbitrary union set is the formula involving disjunctions that we formulate in the hope that such a union set exists.

Returning to conjunctions, to formulate the intersection of an infinity of sets, we can begin with some family (i.e. set) m of sets. We use quantifiers to formulate what is conceptually simply the conjunction of sentences of the form (x ∈ y), as we allow the variable y to take on every value that is an element of m.

Although two formulations may seem to be identical in meaning, the choice of one formulation rather than another can have substantive mathematical effects. For example, using standard ways of formulating things, we would say that if S is a logical consequence of an infinite set of sentences then S is a logical consequence of some finite subset of the set of sentences. If we can use quantifiers to reformulate the conjunction of the infinite set of sentences as a single sentence, then that result isn't obtained.

The following from Wikipedia's sequent calculus article may be similar to what you are looking for:

The standard semantics of a judgment in natural deduction is that it asserts that whenever A1, A2, etc., are all true, B will also be true. The judgments

A1,...,AnB

and

⊢ (A1 ∧ ⋯ ∧ An) → B

are equivalent in the strong sense that a proof of either one may be extended to a proof of the other.

However, the sets involved here are finite.

Wikipedia contributors. (2019, May 10). Sequent calculus. In Wikipedia, The Free Encyclopedia. Retrieved 16:23, October 5, 2019, from https://en.wikipedia.org/w/index.php?title=Sequent_calculus&oldid=896391377