How does model theory handle limit of a set of formulas

Question

Suppose $f_n$ be $n^{th}$ formula of finite length, and $\bigwedge\limits_{k<n}f_k\to F(\omega)$ as $n\to\omega$, where $F(\omega)$ is an infinitary formula. I'd like to know how it is handled in model theory. In other words, how does (infinitary) model theory treat the limit of formulas of finite length?

Answer 1

I'm not entirely sure I understand your question, but I'll give it a go:

If $(\varphi_n)_\omega$ is a countable family of formulas, then (in certain logics, like $\mathcal{L}_{\omega_1 \omega}$) we can form the formula

$$ \bigwedge_{n \in \omega} \varphi_n $$

We say that a model $\mathfrak{M}$ satisfies a formula of the above type whenever it satisfies each of the $\varphi_n$ in turn. That is

$$ \mathfrak{M} \models \bigwedge_{n \in \omega} \varphi_n \iff \mathfrak{M} \models \varphi_n \text{ for every $n \in \omega$}$$

As a cultural note, in my experience logicians don't view these as being "limits" of finite conjunctions. Infinitary conjunctions and disjunctions are pieces of syntax just like any other.

If you want a good resource for working with these logics, and rigorously considering these kinds of conjunctions (as you've expressed in the comments), I recommend this pdf on Infinitary Logic by David Marker

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Edit: For larger logics yet, the standard reference is Dickmann's Larger Infinitary Languages, but as a quick summary of the semantics, say $\overline{v}$ is a family of less than $\lambda$ variables and $\varphi$ is a formula which has (possibly a subset of) the $\overline{v}$ free. Then

$$ \exists \overline{v}. \varphi(\overline{v}) $$

$$ \forall \overline{v}. \varphi(\overline{v}) $$

are both formulas, and

$$ \mathfrak{M} \models \exists \overline{v}. \varphi(\overline{v}) \iff \text{There are elements } \overline{x} \in \mathfrak{M}^\lambda \text{ so that } \mathfrak{M} \models \varphi(\overline{x}) $$

$$ \mathfrak{M} \models \forall \overline{v}. \varphi(\overline{v}) \iff \text{Every vector of elements } \overline{x} \in \mathfrak{M}^\lambda \text{ satisfies } \mathfrak{M} \models \varphi(\overline{x}) $$

Infinitary conjunctions and disjunctions are still defined as above, but with possibly more than $\omega$ many formulas.

As an example (paraphrased from Dickmann):

In $\mathcal{L}_{\kappa^+ \kappa}$ we can assert that a model has cardinality $< \kappa$:

$$ \bigvee_{\text{Cardinals } \lambda < \kappa} (\exists (v_\alpha)_{\alpha < \lambda} . \left [ \bigwedge_{\gamma \neq \delta < \lambda} (v_\gamma \neq v_\delta) \land \forall y . \bigvee_{\gamma < \lambda} (y = v_\gamma) \right ] $$

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I hope this helps ^_^