2026/07/18 by José de Jesús Pelayo Gómez
Mathematics · #math.LO
We develop a machinery for deriving ideals on a countable set from a partition of ω indexed by ω<ω. A derivative operator on trace trees, parametrized by an auxiliary ideal \mathcal J, yields a strict transfinite hierarchy \mathcal H\mathcal Jα of proper ideals, tall from level one onward and independent of the chosen partition. Its finite levels are exactly the Fubini powers, \mathcal Hn\congFin⊗(n+1), while \mathcal Hω=\bigcupn\mathcal Hn amalgamates all finite powers. We establish presentation independence, local homogeneity, Fubini recursion, and Π11-completeness of the full hierarchy. Let \mathcal Fω be Kwela's canonical inductive limit and \mathcal F'ω his independent-partitions limit. We prove \mathcal Fω\not≤K\mathcal Hω, although \mathcal F'ω\sqsubseteq\mathcal Hω and every finite coherent fragment of a putative reduction is realizable over \mathcal Hω. The proof introduces essential depth, an invariant monotone along Katetov reductions of Fin\otimesFin, and gives the sharp non-extension bound N(m)=m+2. Thus \mathcal Hω contains no isomorphic copy of \mathcal Fω, and \mathcal Fω\not≤K\mathcal F'ω. Applications include chromatic ideals \mathcal Gk∈\mathcal H2∖\mathcal H1 whose inclusion order records divisibility and whose Katetov order records arithmetic. We also prove the orthogonality of Cantor--Bendixson and derivative ranks. Finally, \mathcal P(ω)/\mathcal H is a σ-closed reduced power \mathbb B≅\mathbb Bω/Fin, contains \mathcal P(ω)/Fin regularly, and under CH is forcing-equivalent to (\mathcal P(ω)/Fin)+.