<jats:p>A model <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {M}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>M</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> of ZF is said to be <jats:italic>condensable </jats:italic>if <jats:inline-formula><jats:alternatives><jats:tex-math>$$ {\mathcal {M}}\cong {\mathcal {M}}(\alpha )\prec _{\mathbb {L}_{{\mathcal {M}}}} {\mathcal {M}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mo>≅</mml:mo>
<mml:mi>M</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>α</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:msub>
<mml:mo>≺</mml:mo>
<mml:msub>
<mml:mi>L</mml:mi>
<mml:mi>M</mml:mi>
</mml:msub>
</mml:msub>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> for some “ordinal” <jats:inline-formula><jats:alternatives><jats:tex-math>$$\alpha \in \mathrm {Ord}^{{\mathcal {M}}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>α</mml:mi>
<mml:mo>∈</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi>Ord</mml:mi>
</mml:mrow>
<mml:mi>M</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathcal {M}(\alpha ):=(\mathrm {V}(\alpha ),\in )^{{\mathcal {M}}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>α</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>:</mml:mo>
<mml:mo>=</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>V</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>α</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mo>,</mml:mo>
<mml:mo>∈</mml:mo>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:mi>M</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> and <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {L}_{{\mathcal {M}}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>L</mml:mi>
<mml:mi>M</mml:mi>
</mml:msub>
</mml:math></jats:alternatives></jats:inline-formula> is the set of formulae of the infinitary logic <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathbb {L}_{\infty ,\omega }$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:msub>
<mml:mi>L</mml:mi>
<mml:mrow>
<mml:mi>∞</mml:mi>
<mml:mo>,</mml:mo>
<mml:mi>ω</mml:mi>
</mml:mrow>
</mml:msub>
</mml:math></jats:alternatives></jats:inline-formula> that appear in the well-founded part of <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {M}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>M</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula>. The work of Barwise and Schlipf in the 1970s revealed the fact that every countable recursively saturated model of ZF is cofinally condensable (i.e., <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {M}}\cong {\mathcal {M}}(\alpha ) \prec _{\mathbb {L}_{{\mathcal {M}}}}{\mathcal {M}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mo>≅</mml:mo>
<mml:mi>M</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>α</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
<mml:msub>
<mml:mo>≺</mml:mo>
<mml:msub>
<mml:mi>L</mml:mi>
<mml:mi>M</mml:mi>
</mml:msub>
</mml:msub>
<mml:mi>M</mml:mi>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula> for an unbounded collection of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\alpha \in \mathrm {Ord}^{{\mathcal {M}}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>α</mml:mi>
<mml:mo>∈</mml:mo>
<mml:msup>
<mml:mrow>
<mml:mi>Ord</mml:mi>
</mml:mrow>
<mml:mi>M</mml:mi>
</mml:msup>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula>). Moreover, it can be readily shown that any <jats:inline-formula><jats:alternatives><jats:tex-math>$$\omega $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>ω</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula>-nonstandard condensable model of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathrm {ZF}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>ZF</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula> is recursively saturated. These considerations provide the context for the following result that answers a question posed to the author by Paul Kindvall Gorbow.</jats:p><jats:p><jats:bold>Theorem A.</jats:bold><jats:italic>Assuming a modest set-theoretic hypothesis, there is a countable model </jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {M}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>M</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula><jats:italic>of </jats:italic>ZFC <jats:italic>that is both</jats:italic><jats:italic>definably well-founded</jats:italic> (<jats:italic>i.e., every first order definable element of </jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {M}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>M</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula><jats:italic> is in the well-founded part of </jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathcal {M)}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mo>)</mml:mo>
</mml:mrow>
</mml:math></jats:alternatives></jats:inline-formula><jats:italic>and</jats:italic><jats:italic>cofinally condensable</jats:italic>. We also provide various equivalents of the notion of condensability, including the result below.</jats:p><jats:p><jats:bold>Theorem B.</jats:bold><jats:italic>The following are equivalent for a countable model</jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {M}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>M</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula><jats:italic>of </jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathrm {ZF}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>ZF</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula>: (a) <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {M}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>M</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula><jats:italic>is condensable</jats:italic>. (b) <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {M}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>M</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula><jats:italic>is cofinally condensable</jats:italic>. (c) <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {M}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mi>M</mml:mi>
</mml:math></jats:alternatives></jats:inline-formula><jats:italic>is nonstandard and </jats:italic><jats:inline-formula><jats:alternatives><jats:tex-math>$$\mathcal {M}(\alpha )\prec _{\mathbb {L}_{{\mathcal {M}}}}{\mathcal {M}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
<mml:mrow>
<mml:mi>M</mml:mi>
<mml:mrow>
<mml:mo>(</mml:mo>
<mml:mi>α</mml:mi>
<mml:m
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