# Tag Info

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 ...
• 7,684
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 ...
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 ...
• 19.3k
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 ...
• 19.3k

### 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 $\kappa$...
• 8,752
Accepted

• 2,087
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 ...
• 5,461

### 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 $n$ ...
• 5,461
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 ...
• 19.3k
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 ...
• 19.3k
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. ...

### 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 ...
### 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,...