The title is my question. I'm curious since I can't seem to find any axiomatic set theory that say that large cardinals exist. Another thing I’d like to know is that if there are any axiomatic set theories for larger cardinals like Mahlo cardinals.
2026-03-25 20:06:33.1774469193
What axiomatic set theories say that large cardinals exist
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One usually takes an axiomatic set theory like $\mathrm{ZFC}$ that neither posits their existence nor denies them, and then adjoins a large cardinal axiom $\varphi$ of one's choosing, thereby obtaining $\mathrm{ZFC}+\varphi.$ However there's exceptions, such as TG.
There's also some set theories that posit the existence of collections that "standard" set theories consider "too large" to be sets and consequently disallow, such as MK, and NF.