Membership Chains: Why Length Is at Most Omega

[Dragonfall]

Why do membership chains ($$a\in b\in c\in ...$$) have length at most $$\omega$$?

 

[CRGreathouse]

They don't. The axiom of infinity gives us a chain $\varnothing\in\{\varnothing\}\in\cdots\in\omega,$, but $\omega\in\{\omega,\cup\omega\}.$ That's a chain of length $\omega+1.$

Perhaps you mean $a\ni b\ni c\ni\cdots$, which has length less than $\omega$ by the axiom of regularity/foundation?

 

[Dragonfall]

All chains in $$\omega$$ are finite.