Infinitely long formulas

Solution 1:

It's not that you can't, it's just that the standard definition of predicate logic requires formulas (and proofs) to have finite length.

There are infinitary logics where you can have infinite conjunctions and disjunctions, or infinitely many quantifiers. And that all works out fine. But once you move from the "finite realm", set theory starts playing a much more significant role. Which makes things far more complicated (for example, the compactness and completeness theorems no longer hold in general).

Solution 2:

In a comment to Asaf you write,

Take the inductive definition of formula-hood in plato.stanford.edu/entries/logic-classical. Where does it state or imply that a formula may only be finite? Nowhere seemingly. Or it must be implicit in something that I am not seeing.

Finiteness is implicit there - in the precise meaning of "inductive definition!"

The inductive definition basically says, "The set $Form$ is the smallest set containing [stuff] and closed under [operations]." (Another way this is phrased is in three clauses, with the last being "and nothing is in $Form$ if it is not required to be by the previous clauses, but this is less precise.) A formula is then an element of $Form$.

The point is that since these operations and starting formulas are finitary, we never get infinitely long expressions in $Form$. Specifically, what you need to convince yourself of is:

The class of finitely long formulas satisfies the closure properties in the definition.

This exactly says that nothing else is a formula. And the fact that this closure holds is because the "basic operations" of formula-forming are finitary.

It might help to consider a simpler inductive definition first: $\mathbb{N}$ is the smallest set containing $0$ and closed under $a\mapsto a+1$. Now there's nothing in this definition explicitly excluding $\pi$ from being a natural number ... except for that word "smallest." This is the essence of inductive definability.

Solution 3:

Just because you can indefinitely disjunct yet another term to your ever-growing disjunction does not mean that you ever reach an infinitely long disjunction, just as much as you never get to an infinite number just by adding $1$ to an ever-growing number.

Indeed, by induction over the recursive ddefinition of expressions you can easily prove that every statement has to be of finite length:

Base: every atomic statement like $A$ or $B$ is of finite length

Step: if $\varphi$ and $\psi$ are of finite length, then $\varphi \lor \psi$ is of finite length as well (and same for the other operators)