Realizing Realizability Results With Classical Constructions

Abstract

J.L. Krivine developed a new method based on realizability to construct models of set theory where the axiom of choice fails. We attempt to recreate his results in classical settings, i.e. symmetric extensions. We also provide a new condition for preserving well-ordered, and other particular type of choice, in the general settings of symmetric extensions.

A program for the full axiom of choice

“…Thanks to Asaf Karagila for several fruitful discussions [Kar19]. He observed that, by exactly the same method, we obtain a program, not only for AC, but for every axiom which can be shown compatible with ZFC by forcing (Theorem 5.7); for instance CH, 2 ℵ 0 = ℵ 2 + Martin's axiom, etc.…”

Forcing revisited

“…If we then take the direct limit of this sequence, we end up with an ill-founded structure. 13 To avoid this, we want to make it impossible for such ill-founded structures to result from taking these natural limits. To do this, we would like to be able to track all of the possible evolutions and ensure that we will always evolve a wellfounded structure.…”

“…12 I.e., the submodel of 𝐹(𝑞) generated by the pointwise image of 𝐹(𝑝) through 𝐹(≤ 𝑝,𝑞 ) is isomorphic to 𝐹(𝑝). 13 More specifically, if lim → 𝐹 = lim → ⟨⟨𝐹(𝑝 𝑛 )⟩, ⟨𝐹(≤ 𝑝 𝑛 ,𝑝 𝑚 )⟩ 𝑛≤𝑚∈𝜔 ⟩ = ⟨𝑀 ∞ , 𝐸 ∞ ⟩, then 𝐸 is ill-founded on 𝑀. To see this let 𝑔 𝑛,∞ ∶ 𝐹(𝑝 𝑛 ) → 𝑀 ∞ be the canonical map from 𝑀 into the limit.…”

“…Traditional forcing can be understood as a specific example of this technique and it has been deployed to generate models of theories that had not previously been considered using symmetric extension. Thus far, these construction have been matched using symmetric extensions by [13], however, the more general question of the relationship between realizability models, symmetric extensions and ultrapowers from restricted targeted presheaves remains open.…”