Tukey-order with models on Pawlikowski's theorems

Miguel A. Cardona

Published 2021-09-02Version 1

In J. Symbolic Logic,51(4): 957-968, 1986, Pawlikowski proved that, if $r$ is a random real over $\mathbf{N}$, and $c$ is Cohen real over $\mathbf{N}[r]$, then (a) in $\mathbf{N}[r][c]$ there is a Cohen real over $\mathbf{N}[c]$, and (b) $2^\omega\cap\mathbf{N}[c]\notin\mathcal{N}\cap\mathbf{N}[r][c]$, so in $\mathbf{N}[r][c]$ there is no random real over $\mathbf{N}[c]$. To prove this, Pawlikowski proposes the following notion: Given two models $\mathbf{N}\subseteq \mathbf{M}$ of ZFC, we associate with a cardinal characteristic $\mathfrak{x}$ of the continuum, a sentence $\mathfrak{x}_\mathbf{N}^\mathbf{M}$ saying that in $\mathbf{M}$, the reals in $\mathbf{N}$ give an example of a family fulfilling the requirements of the cardinal. So to prove (a) and (b), it suffices to prove that (a') $\mathrm{cov}(\mathcal{M})_{\mathbf{N}[c]}^{\mathbf{M}[c]}\Rightarrow\mathrm{cof}(\mathcal{M})_{\mathbf{N}}^{\mathbf{M}}\Rightarrow\mathrm{cov}(\mathcal{N})_{\mathbf{N}}^{\mathbf{M}}$, and (b') $\mathrm{cov}(\mathcal{M})_\mathbf{N}^\mathbf{M}\Rightarrow\mathrm{add}(\mathcal{M})_{\mathbf{N}}^{\mathbf{M}}\Rightarrow\mathrm{non}(\mathcal{M})_{\mathbf{N}[c]}^{\mathbf{M}[c]}\Rightarrow\mathrm{cov}(\mathcal{N})_{\mathbf{N}[c]}^{\mathbf{M}[c]}$. In this paper, we introduce the notion of Tukey-order with models, which expands the concept of Tukey-order introduced by Vojt\'{a}\v{s} (Israel Math. Conf. Proc. 6: 619-643, 1991) to prove expressions of the form $\mathfrak{x}_\mathbf{N}^\mathbf{M}\Rightarrow\mathfrak{y}_\mathbf{N}^\mathbf{M}$. In particular, we show (a') and (b') using Tukey-order with models.