Equivalence modulo certain theory

I've recently came across the following theorem:

Theorem 3.2.4 Let $\kappa$ be an infinite cardinal. The $DC_\kappa$ is equivalent to $FA_\kappa(\Gamma_\kappa)$ module the theory $ZF+\forall\lambda(\lambda<\kappa\rightarrow DC_\lambda).$

It can be found here:
http://www.logicatorino.altervista.org/matteo_viale/thesis-parente.pdf
on page 39.

Unfortunately I cannot find any other papers where this phrase that "something is equivalent to something modulo something" is used. What does that mean exactly? Is that this the same as saying:
Let $\kappa$ be an infinite cardinal. The $DC_\kappa$ is equivalent to $FA_\kappa(\Gamma_\kappa)$ assuming axioms $ZF+\forall\lambda(\lambda<\kappa\rightarrow DC_\lambda),$
or is it something else?

Solution 1:

Yes, "$A$ is equivalent to $B$ modulo $\Gamma$" means that $\Gamma\vdash A\leftrightarrow B$.

There are a number of - appropriately - equivalent ways to express this, including:

• $\Gamma\models A\leftrightarrow B$ (via the soundness and completeness theorems).

• $\Gamma\cup\{A\}\vdash B$ and $\Gamma\cup\{B\}\vdash A$ (via the deduction theorem).

Note that the "base theory" is often suppressed; e.g. the very common statement

the Axiom of Choice and Zorn's Lemma are equivalent

is really shorthand for

the Axiom of Choice and Zorn's Lemma are equivalent modulo $\mathsf{ZF}$.

Finally, it's worth noting that in place of "modulo" other words and phrases can appear, the most common in my experience being "relative to" and "over."