11 votes
Accepted

When do infinitary compactness numbers exist?

The compactness number for $\mathcal L_{\kappa,\kappa}$ is equal to the least $(\kappa,\infty)$-strongly compact cardinal. A cardinal is $(\kappa,\infty)$-strongly compact if for every set $X$, there ...
Gabe Goldberg's user avatar
11 votes
Accepted

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

The answer is no. Suppose that $\kappa$ is a Mahlo cardinal. This is $\Pi^1_1$ expressible in $V_\kappa$, since it amounts to the assertion that every closed unbounded subset of $\kappa$ contains a ...
Joel David Hamkins's user avatar
8 votes
Accepted

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

Yes, this is categorical and hence complete (with respect to any satisfactory notion of proof, that is). Specifically, I claim that any model $M$ of your theory is isomorphic to $\mathsf{HC}$, the set ...
Noah Schweber's user avatar
7 votes
Accepted

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

The models of your theory are exactly the pointwise-definable (with respect to $\mathcal{L}_{\omega,\omega}$ as usual) well-founded models of $\mathsf{ZF}$; incidentally, your true finiteness axiom is ...
Noah Schweber's user avatar
6 votes

Complexity of infinitary satisfiability, part 2

Here is a partial answer; it deals with those admissibles $\kappa$ large enough to see that $\mathrm{cof}(|\kappa|)=\omega$ and such that $L_\kappa$ has largest cardinal $\theta$. That is, let $\kappa$...
Farmer S's user avatar
  • 8,752
6 votes
Accepted

Models with few types in infinitary logics

Here is a partial answer: consistently, the generalization can fail for all uncountable $\kappa$. Namely: Suppose $\mathbb{V} = \mathbb{L}$ and let $\kappa$ be any uncountable cardinal. Let $\...
Danielle Ulrich's user avatar
6 votes
Accepted

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

Any model of ${\sf ZFC}+V=\sf HOD$ has an elementary equivalent pointwise definable model. If $M$ models $V=\sf HOD$, it has a parameter free definable well ordering, for each formula $φ$ consider the ...
Holo's user avatar
  • 1,645
5 votes
Accepted

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

Your Foundation axiom does not assert that there is no infinite descending sequence, but rather merely rules out sets at infinite set-theoretic rank. For example, if $x=\omega$, then we can find for ...
Joel David Hamkins's user avatar
5 votes
Accepted

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

Your new formulation of foundation is equivalent to the corresponding unbounded form. The reason is that every failing instance of the unbounded formulation will lead to a failing instance of the ...
Joel David Hamkins's user avatar
5 votes
Accepted

Elementary extensions of infinitary languages

$\alpha$-supercorrect cardinals are consistent, for any $\alpha$. Work in a mild second-order set theory, namely Gödel–Bernays set theory along with the assertion that there is a $\mathcal L_{R_\...
Kameryn Williams's user avatar
5 votes
Accepted

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

It actually isn't true that the complexity of $\theta$ could be meaningfully bounded in the term of complexities of $\varphi$ and $\psi$. Let us fix an arbitrary recursive ordinal $\alpha$. Below I ...
Fedor Pakhomov's user avatar
4 votes

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

Duplicator wins in E-F game for structures $A,B$ in position given by tuples $\vec{a}\in A$, $\vec{b}\in B$ and $2n$ turns left $\iff$ the same $L_{\infty,\omega}$ formulas of the quantifier rank $n$ ...
Fedor Pakhomov's user avatar
4 votes
Accepted

Is this theory finitary first order complete?

Let $\mathbb{K}$ be the class of well-founded models of $\mathsf{ZFC+V=L}$ + "There is no inaccessible cardinal." This is a subclass of the model class of your theory, but under mild ...
Noah Schweber's user avatar
3 votes
Accepted

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

To avoid triviality (e.g. "The true $\mathcal{L}_{\kappa,\lambda}$-theory of $V$") let's look specifically for theories which $(1)$ consist of adding to $\mathsf{ZFC}$ a single infinitary ...
Noah Schweber's user avatar
3 votes
Accepted

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

Yes, the theory is consistent, if ZF is consistent, because there are pointwise definable models of ZF. Any such model is a model of your theory, which is therefore satisfiable and hence consistent. ...
Joel David Hamkins's user avatar
2 votes

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

Since I don't have a good understanding of your Classes axiom/theory, let me answer instead for Gödel-Bernays set theory GBC, for which the answer is negative. We know that there are pointwise ...
Joel David Hamkins's user avatar
2 votes

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

Maybe this question was already resolved, but here is a solution in the literature. Magidor showed (Theorem 4) that $\kappa$ is extendible iff $L^2_{\kappa\kappa}$ is $\kappa$-compact, and that $\...
Tim Campion's user avatar
  • 61.6k
2 votes
Accepted

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

Sure there are. They even come up in practice from time to time; e.g. to show that assuming Vopenka's Principle the modal analogue of second-order logic (a la Hamkins/Woloszyn) has definable-in-$V$ ...
Noah Schweber's user avatar
2 votes

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

Your axiom schema is equivalent to being an $\omega$-model. Working inside an $\omega$-model, any counter example to your schema will be counter example of the axiom of foundation of ZFC, as you can ...
Holo's user avatar
  • 1,645
1 vote
Accepted

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

Lemma. A model of ZF satisfies the “Foundation” axiom if and only if it is an $\omega$-model, i.e., iff it satisfies the simpler $\mathcal L_{\omega_1,\omega}$-sentence $\forall x\in\omega\,\bigvee_{n\...
Emil Jeřábek's user avatar
1 vote
Accepted

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

(Turning some comments into an answer) The definition of $L(x,\alpha+1)$ was wrong, instead it should have been $$L(x,\alpha+1)\leftrightarrow\bigvee_{n\in\omega}\bigvee_{\varphi}\exists p_1,\ldots,...

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