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Given theories $T,U$ extending $\textsf{ZFC}$, Steel (p3) defines the consistency strength hierarchy as follows: $T \leqslant_{\text{Con}} U$ iff $\textsf{ZFC} \vdash \operatorname{Con}{U} \to \operatorname{Con}{T}$, where we get strict inequality iff we also have $\textsf{ZFC}\nvdash \operatorname{Con}{T} \to \operatorname{Con}{U}$.

Hamkins (p3) notes that we may give as a sufficient condition for strict inequality that $U \vdash \operatorname{Con}{T}$, also noting (p4) that this is how Simpson (p2) defines his consistency strength relation outright. However these hierarchies are not the same, since (Hamkins p4), for example, the latter order is not dense on extensions of the base theory.

Following from the above, I've been looking for a pair of theories $T,U$ such that we have $T <_{\text{Con}} U$, but $U \nvdash \operatorname{Con}{T}$. More fully, the three conditions I require to be met are:

  1. $\textsf{ZFC}\vdash \operatorname{Con}{U} \to \operatorname{Con}{T}$,
  2. $\textsf{ZFC}\nvdash \operatorname{Con}{T} \to \operatorname{Con}{U}$,
  3. $U \nvdash \operatorname{Con}{T}$.

It seems like theories witnessing the density of Steel's $\leqslant_{\text{Con}}$ over Simpson's would provide the required counter-example, but I'm not aware of these. A related question from math.SE (with no answer) is here.

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    $\begingroup$ As in Theorem 4 of the paper by Hamkins you cite, you can find a theory $U$ that is strictly between $T:=\mathrm{ZFC}$ and $\mathrm{ZFC}+\mathrm{Con}(\mathrm{ZFC})$ w.r.t. $<_{\mathrm{Con}}$. These $U$ and $T$ should work. $\endgroup$ Feb 24, 2022 at 17:33

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For future reference, Andreas' suggestion does work. Take $\mathsf{ZFC} <_{\text{Con}} \mathsf{ZFC} + \operatorname{Con}{(\mathsf{ZFC})}$ and following Theorem 4 of here let $$S = (\mathsf{ZFC} + \operatorname{Con}{(\mathsf{ZFC})}) \vee (\mathsf{ZFC} + \mathrm{R})$$ where $\mathrm{R}$ is the Rosser sentence of the theory $\mathsf{ZFC} + \operatorname{Con}{(\mathsf{ZFC})} + \neg \operatorname{Con}{(\mathsf{ZFC} + \operatorname{Con}(\mathsf{ZFC}))}$. This is the theory with axioms of the form $\varphi \vee (\psi \wedge \mathrm{R})$, where $\varphi \in \mathsf{ZFC} + \operatorname{Con}{(\mathsf{ZFC})}$, $\psi \in \mathsf{ZFC}$.

Hamkins (among other things) shows that for such a theory we will have $$\mathsf{ZFC} <_{\text{Con}} S <_{\text{Con}} \mathsf{ZFC} + \operatorname{Con}{(\mathsf{ZFC})}.$$

We will also have $S \nvdash \operatorname{Con}{(\mathsf{ZFC})}$: suppose for contradiction that $S \vdash \operatorname{Con}{(\mathsf{ZFC})}$, then since by construction we have $S \vdash \mathsf{ZFC}$, and thus $$S \vdash \mathsf{ZFC} + \operatorname{Con}{(\mathsf{ZFC})}.$$ Thus any proof of a contradiction from $\mathsf{ZFC} + \operatorname{Con}{(\mathsf{ZFC})}$ would also constitute a proof of a contradiction from $S$, thus $S$ must have at least the consistency strength of $\mathsf{ZFC} + \operatorname{Con}{(\mathsf{ZFC})}$, which contradicts the above.

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