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  • $\begingroup$ I actually wonder why we can't index universes with natural numbers rather than having a different type for universe levels... $\endgroup$
    – Couchy
    Commented Feb 20, 2022 at 21:07
  • $\begingroup$ @Couchy my conjecture is that if you can unlift the level of something (which is possible with naturals) you might be able to construct the Girard's paradox $\endgroup$
    – ice1000 ♦
    Commented Feb 20, 2022 at 21:15
  • $\begingroup$ What do you mean by unlifting a level? $\endgroup$
    – Couchy
    Commented Feb 20, 2022 at 21:22
  • $\begingroup$ @Couchy reduce the level of something. Thought about this for a while and I feel like I was probably making no sense. Please don't take it seriously :-) $\endgroup$
    – ice1000 ♦
    Commented Feb 20, 2022 at 21:36
  • $\begingroup$ I was under the impression you needed ordinals/limits for useful universes? $\endgroup$
    – Ms. Molly Stewart-Gallus
    Commented May 2, 2022 at 0:13