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Has by well-foundedness every non-empty class an $R$-minimal element? Also if...
I asked this question here on Math.SE but uptil now it was not answered. So I decided to give it a try. Thank you in advance.Working in $\mathbf{ZF}$ let $R$ be a proper class of ordered pairs that is...
View ArticleAnswer by Giraffro for Has by well-foundedness every non-empty class an...
Here's a counterexample: it is consistent with $\mathbf{ZFC}^-$ that $\mathbf{U} := \{x : x = \{x\}\}$ is a proper class with no infinite subsets [1]. Once you have this, consider the class $\mathbf{R}...
View ArticleAnswer by Emil Jeřábek for Has by well-foundedness every non-empty class an...
Let me mention another counterexample. In [1, Thm. 11], we construct a model of $\mathrm{ZFC}^-$ with the collection schema which contains a definable class relation $\langle A,<\rangle$ such...
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