# A weakening of cardinal compactness - is it equivalent?

I was messing around with the intuition behind the size of weakly compact cardinals (in their usual characterization). I found an interesting, seemingly weaker LCA which still implies weak inaccessibility.

I started by making an intuitively powerful property that can be stated as an $\mathcal{L}_{\kappa,\kappa}$ scheme. I ended up with three properties:

• $\text{LZ}_\lambda$ for $\lambda<\kappa$ is the $\mathcal{L}_{\kappa,\kappa}$-sentence "$\exists_{\alpha<\lambda}x_\alpha(\bigwedge_{\alpha<\lambda}\bigwedge_{\beta\neq\alpha}x_\alpha\neq x_\beta)$" which is actually true in a structure $\mathcal{M}$ iff it's universe has size at least $\lambda$ (it's not hard to see why).
• $\text{LA}_\lambda$ for $\lambda<\kappa$ is the $\mathcal{L}_{\kappa,\kappa}$-sentence "$\exists_{\alpha<\lambda}x_\alpha\forall X(\bigvee_{\alpha<\lambda}x_\alpha=X)$" which is true in a structure $\mathcal{M}$ iff it's universe has size at most $\lambda$.