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Beau Madison Mount
Beau Madison Mount
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Kunen showed that there is no nontrivial $j: V \rightarrow_e V$. One might wonder what happens in $\mathsf{ZFC}$ with atoms.

Let's denote the universe by $U$. We aren't assuming that the atoms form a set, or even that there are no more atoms than pure sets. Of course, if there at least two atoms, there will be nontrivial automorphisms on $U$, so that isn't very interesting. The following seems like a better way of translating Kunen's question into this framework: let's call an elementary embedding atrivial if it is nontrivial but it is the identity mapping on atoms. Can there be an atrivial $j: U \rightarrow_e U$?

Two comments: 1) I've tried to prove thisthat there can be no such $j$ by, in essence, quotienting out distinctions among Urelemente (similar to a Fraenkel-Mostowski model) and building a $j'$ that inherits nontriviality, but without success; 2) note that every object has at most set-many atoms in its transitive closure, so one can find a set of atoms $A$ such that $j$ is nontrivial on the restricted hierarchy $U(A)$ built over those atoms alone; unfortunately, there's no guarantee that $j``U(A)$ is a class built up only over set-many atoms.

Kunen showed that there is no nontrivial $j: V \rightarrow_e V$. One might wonder what happens in $\mathsf{ZFC}$ with atoms.

Let's denote the universe by $U$. We aren't assuming that the atoms form a set, or even that there are no more atoms than pure sets. Of course, if there at least two atoms, there will be nontrivial automorphisms on $U$, so that isn't very interesting. The following seems like a better way of translating Kunen's question into this framework: let's call an elementary embedding atrivial if it is nontrivial but it is the identity mapping on atoms. Can there be an atrivial $j: U \rightarrow_e U$?

Two comments: 1) I've tried to prove this by, in essence, quotienting out distinctions among Urelemente (similar to a Fraenkel-Mostowski model) and building a $j'$ that inherits nontriviality, but without success; 2) note that every object has at most set-many atoms in its transitive closure, so one can find a set of atoms $A$ such that $j$ is nontrivial on the restricted hierarchy $U(A)$ built over those atoms alone; unfortunately, there's no guarantee that $j``U(A)$ is a class built up only over set-many atoms.

Kunen showed that there is no nontrivial $j: V \rightarrow_e V$. One might wonder what happens in $\mathsf{ZFC}$ with atoms.

Let's denote the universe by $U$. We aren't assuming that the atoms form a set, or even that there are no more atoms than pure sets. Of course, if there at least two atoms, there will be nontrivial automorphisms on $U$, so that isn't very interesting. The following seems like a better way of translating Kunen's question into this framework: let's call an elementary embedding atrivial if it is nontrivial but it is the identity mapping on atoms. Can there be an atrivial $j: U \rightarrow_e U$?

Two comments: 1) I've tried to prove that there can be no such $j$ by, in essence, quotienting out distinctions among Urelemente (similar to a Fraenkel-Mostowski model) and building a $j'$ that inherits nontriviality, but without success; 2) note that every object has at most set-many atoms in its transitive closure, so one can find a set of atoms $A$ such that $j$ is nontrivial on the restricted hierarchy $U(A)$ built over those atoms alone; unfortunately, there's no guarantee that $j``U(A)$ is a class built up only over set-many atoms.

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Beau Madison Mount
Beau Madison Mount
  • 1.2k
  • 5
  • 8

Kunen inconsistency with atoms

Kunen showed that there is no nontrivial $j: V \rightarrow_e V$. One might wonder what happens in $\mathsf{ZFC}$ with atoms.

Let's denote the universe by $U$. We aren't assuming that the atoms form a set, or even that there are no more atoms than pure sets. Of course, if there at least two atoms, there will be nontrivial automorphisms on $U$, so that isn't very interesting. The following seems like a better way of translating Kunen's question into this framework: let's call an elementary embedding atrivial if it is nontrivial but it is the identity mapping on atoms. Can there be an atrivial $j: U \rightarrow_e U$?

Two comments: 1) I've tried to prove this by, in essence, quotienting out distinctions among Urelemente (similar to a Fraenkel-Mostowski model) and building a $j'$ that inherits nontriviality, but without success; 2) note that every object has at most set-many atoms in its transitive closure, so one can find a set of atoms $A$ such that $j$ is nontrivial on the restricted hierarchy $U(A)$ built over those atoms alone; unfortunately, there's no guarantee that $j``U(A)$ is a class built up only over set-many atoms.