No exhaustive chain of safe words for the two's-complement order

There is no sequence $P_0,P_1,P_2,\dots$ of finite lists of integers such that

1. each $P_n$ is safe for the two's-complement order $\prec_{\mathrm{true}}$,
2. each $P_n$ is a prefix of $P_{n+1}$, and
3. every integer occurs in some $P_n$.

So Ho's construction cannot be carried out on $\mathbb{Z}$ with the fixed completion order $\prec_{\mathrm{true}}$. A construction for the goal must change the completion order between stages or use different ideas.

Formalization Note compLt R P x y holds when both lie in P with x at a smaller index, or x ∈ P and y ∉ P, or neither lies in P and R x y. Safe R P says there are no a and r ≠ 0 with compLt R P a (a+r), compLt R P (a+r) (a+2r) and compLt R P (a+2r) (a+3r). epsLt ε x y is x ≠ y ∧ (x / 2^k % 2 = 1 ↔ ε k = true) with k = padicValInt 2 (y - x), i.e. $k=\nu_2(y-x)$; integer division is floor division, so x / 2^k % 2 is the $k$-th two's-complement digit of $x$.