Timeline for Has by well-foundedness every non-empty class an $R$-minimal element? Also if axiom REG is not assumed?
Current License: CC BY-SA 3.0
5 events
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| Aug 20, 2015 at 14:15 | comment | added | Giraffro | Nice! Need to remember this one. I've been thinking a bit more about permutation models, and if you construct one from ZF(C) you still have collection and every non-empty, well-founded class relation has a minimal element. It'll probably require some thinking to come up with a counterexample with collection... | |
| Aug 20, 2015 at 12:53 | comment | added | Emil Jeřábek | Let $X$ be a class. First, by a straightforward induction on $n$, prove for all $n\in\omega$ that $X$ is a finite set with $< n$ elements, or it contains an $n$-element subset. Thus, assuming $X$ is not a finite set, we have $\forall n\in\omega\,\exists x\subseteq X\,|x|=n$. Applying collection, there is a set $z$ such that $\forall n\in\omega\,\exists x\in z\,(x\subseteq X\land|x|=n)$. Then $X\cap\bigcup z$ is an infinite subset of $X$. | |
| Aug 20, 2015 at 11:54 | comment | added | Giraffro | It certainly satisfies Replacement. Could you provide a proof of that statement? I'm a little slow this morning. | |
| Aug 20, 2015 at 9:13 | comment | added | Emil Jeřábek | That every proper class has an infinite subset is implied by collection, so I take it you don't include the collection schema in the list of axioms of ZFC^-? | |
| Aug 20, 2015 at 0:22 | history | answered | Giraffro | CC BY-SA 3.0 |