# Questions tagged [infinitary-logic]

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### Must models of the following theory satisfying opposing infinitary sentences, satisfy opposing finitary sentences?

This is a follow-up to posting titled "Is this theory finitary first order complete?"
Recall the theory presented at that posting. Replace the size axiom by the following:
$\textbf{...

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### Are there strong set theories written in infinitary language, that are finitary FOL complete?

Are there set theories that extend some complete infinitary language $\mathcal L_{\kappa, \lambda}$, prove all axioms of $\sf ZFC$, and are finitary $\textbf{FOL}$ complete? That is, every sentence in ...

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### Is this theory finitary first order complete?

If we coin a theory in $\mathcal L_{\omega_1, \omega}$ that begins with constructing pure true well founded finite sets, then the set of all true well founded hereditarily finite sets, then builds up ...

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### Is ZF + Def a conservative extension of ZFC+HOD? If not, what are counter-examples?

$\sf ZF + Def$ is the theory that extends $\mathcal L(=,\in)_{\omega_1,\omega}$ with axioms of $\sf ZF$ (written in $\mathcal L(=,\in)_{\omega, \omega}$) and the axiom of definability:-
$\textbf{...

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### Can generalization of non-redundant construction in $V_\omega$ be consistent?

The following posting is along the general line of thought presented in this earlier posting titled "Is it possible to derive the rules of set theory as transfers from the pure finite set world, ...

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### Are there second (or higher) order infinitary logic languages? References?

Reading on Infinitary languages I'm only seeing first order infinitary languages $\mathcal L_{\kappa, \lambda}$, i.e. in all of these languages no quantification over predicate and function symbols is ...

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### Is bounded-$\mathcal L_{\omega_1,\omega_1}$ significantly different from $\mathcal L_{\omega_1,\omega_1}$?

Take the language $\mathcal L(=,\in)_{\omega_1,\omega_1}$, if we restrict infinite quantification strings, i.e. of the forms $``(\forall v_i)_{i \in \omega}"; ``(\exists v_i)_{i \in \omega}"$, to ...

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### How do these two principles of Foundation written in $\mathcal L_{\omega_1,\omega}$ compare?

This comes in continuation to posting titled "Can Foundation be captured for some $\mathcal L_{\omega_1, \omega}$ theories?".
Here, an attempt at a stronger notion of Foundation, yet ...

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### Can we have ZF + Definability + True Foundation + True Finiteness? Can it be Categorical?

Lets extend $\mathcal L_{\omega_1, \omega_1}$ with axioms of equality and of:
$\sf ZF + Definability+Ture$-$\sf Foundation+True$-$\sf Finiteness $
Where $\sf ZF$ is written, as usual, in $\mathcal L_{...

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### Can Foundation be captured for some $\mathcal L_{\omega_1, \omega}$ theories?

This posting is a continuation to an earlier one titled "Can Foundation be captured in $\mathcal L_{\omega_1, \omega}$ ?"
It appears that capturing foundation is problematic at every $\...

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### Can Foundation be captured in $\mathcal L(\omega_1,\omega)$?

Working in $\mathcal L_{\omega_1, \omega}$, can Foundation be captured?
My idea is to formalize a theory where all of its models are the well founded pointwise definable models of $\sf ZFC$. I attempt ...

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### Is this theory of well founded countable sets formalized in infinitary logic, complete and categorical?

Working in $\mathcal L_{\omega_1, \omega_1}$, add symbol $=$ with its axioms; add symbol $\in$ and axiomatize:
$\textbf{Extensionality: } \forall x \forall y : \forall z (z \in x \leftrightarrow z \in ...

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### Does adding definability axiom expressed in infinitary language to ZF, let all models be pointwise definable?

This posting is generally related to a prior posting titled "Are all constructible from below sets parameter free definable?"
If we work in infinitary language $\mathcal L_{\omega_1, \omega}$...

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### End elementary extension in infinitary logic of some $L_\alpha$ producing a $L_\beta$

Let $L_\alpha$ be some admissible level of the constructible hierarchy and $M \supseteq L_\alpha$ an extension of $L_\alpha$. I am looking for conditions under which $M \simeq L_\beta$. It is not ...

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### Complexity of infinitary satisfiability, part 2

This question is a follow-up to this one, which was almost entirely answered by Farmer S. Throughout, we work in $\mathsf{ZFC+V=L}$.
Given a "pre-admissible" (= admissible or limit of ...

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### What does the Ehrenfeucht-Fraïssé game on structures with infinitely many relations tell us?

EF-games are typically presented for structures with finitely many relations, and if you want to extend them to structures with functions, you can relationalize the functions. This seems to be to ...

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### Can having no more than countably many classes, be inferred from, having every class being countable?

In a theory having proper classes it won't be that easy to phrase how many classes we have in comparison to the number of elements of some class. Here, I'll adopt the following method:
We'd say that: ...

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### Can ZFC + Classes + definability rule manage to prove all classes countable?

Working in bi-sorted $L_{\omega_1,\omega} (=, \in)$, if we write $\sf ZFC + Classes$ as it is; i.e., in bi-sorted $L_{\omega, \omega} (=,\in)$, and add the following definability rule written in bi-...

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### Sharp Craig interpolation theorem for $L_{\omega_1 \omega}$

I’d like to know if a sharp version of Craig’s interpolation theorem for $L_{\omega_1 \omega}$ is already known or exists in the literature. By a “sharp” version of this theorem, I mean something like ...

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### When do infinitary compactness numbers exist?

For a logic $\mathcal{L}$, let the compactness number of $\mathcal{L}$ (if it exists) be the least $\kappa$ such that every $(<\kappa)$-satisfiable $\mathcal{L}$-theory is satisfiable. Note that ...

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### On undefinability of well-orderings in $L_{\omega_1,\omega}$

It is a well-known theorem that well-orderings can not be characterized in $L_{\omega_1,\omega}$.
In particular, if $\psi$ is an $L_{\omega_1,\omega}$-sentence in a vocabulary $\tau$ that contains a ...

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### The Strong Compactness Cardinal of $n$-th Order Logic

I was reading Kanamori's The Higher Infinite, when I came across the fact that for any extendible cardinal $\kappa$ and any $\mathcal{L}_{\kappa,\kappa}^n$-theory $T$, $Sat(T)\Leftrightarrow \forall t\...

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### How expressive can $\mathcal{L}_{\kappa,\kappa}$ be?

(Note that the logic systems described in this question only refer to logic systems restricted to the language $\in$)
1. Can $\mathcal{L}_{\kappa,\kappa}$ express $n$-th order finitary logic? It is ...

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### Elementary extensions of infinitary languages

Let $R_\alpha$ be the $\alpha$-th infinite regular ordinal. This question assumes AC, so the following is true:
$$R_\alpha=\left\{
\begin{array}{ll}
\omega_\alpha & \alpha=\kappa+n\;\mathrm{...

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### Models with few types in infinitary logics

Let $\mathcal{L}_{\kappa \lambda}$ denote the infinitary logic that allows conjunction of less than $\kappa$-many formulas and simultaneous quantification of less than $\lambda$-many variables. It is ...