A note on Grzegorczyk's logic

Grzegorczyk’s modal logic (Grz ) corresponds to the class of upwards well-founded partially ordered Kripke frames, however all known proofs of this fact utilize some form of the Axiom of Choice; G. Boolos asked in [1], whether it is provable in plain ZF . We answer his question negatively: Grz corresponds (in ZF ) to a class of frames, which does not provably coincide with upwards well-founded posets in ZF alone. Definition 1 Grzegorczyk’s logic (Grz ) [2] is a normal modal logic axiomatized by the schema ( (φ→ φ)→ φ)→ φ. We denote by K1 the class of upwards well-founded posets, K3 the class of posets without any strictly increasing infinite chain, and K2 the class of posets 〈W,≤〉 satisfying ∀X ⊆W (X 6= ∅ → ∃x ∈ X ∀y ≥ x∀z ≥ y (z ∈ X → y ∈ X)). (1) Recall that the Principle of Dependent Choices (DC ) is the following weak version of the Axiom of Choice: let R be a binary relation on a nonempty set A such that ∀x ∈ A∃y ∈ A 〈x, y〉 ∈ R, then there is an infinite sequence {an; n ∈ ω} ∈ A such that 〈an, an+1〉 ∈ R for every n ∈ ω. Lemma 2 ZF proves K1 ⊆ K2 ⊆ K3. Proof: If 〈W,≤〉 ∈ K1, and X ⊆ W nonempty, then any <-maximal element x ∈ X witnesses that (1) holds, hence 〈W,≤〉 ∈ K2. Assume that there is 〈W,≤〉 ∈ K2 \ K3. Fix an infinite increasing chain x0 < x1 < x2 < · · · in W , and put X = {xn; n odd }. Then for any x ∈ X there are z ≥ y ≥ x such that z ∈ X and y 6∈ X, contradicting (1). Proposition 3 (ZF `:) A frame W = 〈W,≤〉 is a model of Grz under all valuations if and only if W ∈ K2.