#paper # Abstract We show that, assuming the consistency of a supercompact cardinal, the first (weakly) inaccessible cardinal can satisfy a strong form of a Löwenheim-Skolem-Tarski theorem for the equicardinality logic L(I), a logic introduced in [5] strictly between first order logic and second order logic. On the other hand we show that in the light of present day inner model technology, nothing short of a supercompact cardinal suffices for this result. In particular, we show that the Löwenheim- Skolem-Tarski theorem for the equicardinality logic at κ implies the Singular Cardinals Hypothesis above κ as well as Projective Determinacy. # Introduction **Def** $I$ is the equicardinality quantifier. $Q^{eq}$ is the equicofinality quantifier **Definition 4** The Löwenheim-Skolem number LS(L) of L is the smallest cardinal κ such that if a theory $T ⊂ L$ has a model, it has a model of cardinality lt; max(κ, |T|)$. The Löwenheim-Skolem-Tarski number LST(L) of L is the smallest cardinal $κ$ such that if $A$ is any $τ$ -structure, then there is a substructure $A'$ of $A$ of cardinality lt; κ$ such that $A' ≺_{L} A$. **Basic facts** - LS(L) always exists, if LST(L) exists then it is $\geq$ LS(L). - LST of first order logic is $\aleph_{1}$ - LST of $L(Q_{1})$ and $L(Q_{1}^{2})$ is $\aleph_{2}$ - For second order logic, $LS(L^2)$ is the supremum of $Π_2$ -definable ordinals (Väänänen), which means that it exceeds the first measurable, the first κ+ -supercompact κ, and the first huge cardinal if they exist. Theorem 5 (Magidor) 1. Suppose $κ$ is a [[Strong]] cardinal , then $LS(L^2 ) < κ$. 2. $LST(L^{2})$ exists if and only if [[Supercompact]] cardinals exist, and then $LST(L^2)$ is the first of them. Theorem 6. [[Vopenka|Vopěnka’s Principle ]] holds if and only if every logic has a Löwenheim-Skolem-Tarski number. Theorem 7 ([14]) 1. $LST(L(I))$ exists only if inaccessible cardinals exist, and then $LST(L(I))$ is at least as large as the first of them. 2. $LST(L(I, Q^{ec} ))$ exists only if Mahlo cardinals exist, and then the cardinal $LST(L(I, Q^{ec} ))$ is at least as large as the first of them.