# “Bootstrapping” an unbounded class of inaccessible cardinals

The "richness principle" of set theory asserts roughly that "everything that happens once should happen an unbounded number of times".
An example would be the existence of an unbounded class of inaccessible cardinals.
Now there are many examples of large cardinals $$\kappa$$ whose existence guarantees an unbounded class of inaccessibles below $$\kappa$$.
So, my first question:
Is there any large cardinal such that "$$\kappa$$ exists" $$\implies$$ "There exists an unbounded class of inaccessible cardinals in V" ?

An even stronger example of this would be something like:
"There exists a large cardinal of Type A" $$\implies$$ "There exists an unbounded class of cardinals of Type A". Is there any large cardinal with this property?
I know that rank-into-rank cardinals satisfy an "upward reflection" property, but that only happens $$\omega$$-many times.
Would a Reinhardt cardinal (in ZF rather than ZFC say) reflect upwards unboundedly many times ?

• For your second question, if you consider first order expressibility in ZFC, then the answer is no: otherwise consider the second one of type $A$, and cut the universe at that level. – Mohammad Golshani Jul 22 at 5:31
• If you go to ZF, then the existence of a regular cardinal ($\omega$) does not imply that there is a proper class of regular cardinals. – Asaf Karagila Jul 22 at 6:11

$$\kappa$$ is superhuge if for any $$\gamma$$, there exists $$j: V\to M$$ such that $$crit(j)=\kappa,$$ $$\gamma and $${}^{j(\kappa)}M\subset M.$$ But $$j(\kappa)$$ (inaccessible in M) must be inaccessible in $$V$$ as well.
If $$\kappa$$ is extendible, then there exists a proper class of inaccessible cardinals (and even more).
• One question: Does this hold because $V_\kappa\prec_{\Sigma_3} V$? (So that "even more" could be any $\Pi_1$ notion) – Pedro Sánchez Terraf Jul 22 at 12:39