Basic subtoposes of the effective topos

Lifschitz Realizability as a Topological Construction

“…Instead of topos theory we work in the set theory CZF, which is regarded as a predicative theory for mathematics. Instead of Lawvere-Tierney topologies, we will use formal topologies and a predicative notion of topological model due to Gambino. Aside from this difference in presentation, our results are more general than Van Oosten's in two ways (although the first of these does relate to some more recent results by Lee and Van Oosten in [11]).…”

“…The following lemma will be key to showing later that certain choice axioms and existence properties hold. It appears to be related to the constructions developed by Lee and Van Oosten in [11,Sections 4 and 5]. We will return to this point in section 8.1.…”

“…Connections to topos theory. The L n considered in this paper appear to be strongly related to the local operators in the effective topos previously considered by Lee and Van Oosten in [11], specifically to the local operators corresponding to finitary sights. We expect that in fact these local operators can be obtained by carrying out the construction of L n in the effective topos.…”

“…Conjecture 8.1. Some of the local operators in Eff considered in [11] have counterparts in the toposes RT(K 2 ), RT(K REC 2 , K 2 ) and (Eff ↓ ∆).…”

A Kuroda-style j-translation

“…(The connection between Cohen's strong forcing and Kuroda's negative translation was pointed out by Jeremy Avigad [1].) Another definition which can be explained by the Kuroda-style j-translation is Definition 6.1 in [7]. I do not doubt that some readers are able to come up with other examples as well.…”