I'm interested in axioms that prevent the existence of large cardinals. More precisely:
(Informal definition) $\Phi$ is an anti-large-cardinal axiom iff $V \models \Phi \Rightarrow V \not \models \Psi$ for some garden-variety large cardinal axiom $\Psi$.
Probably the most famous are:
Variety 1 Axioms of constructibility (e.g. $V=L$, $V=L[x]$).
$V=L$ for example implies that there are no measurable cardinals in $V$. Often we don't need the full power of $L$ to get anti-large cardinal features though. For instance:
Variety 2 $\square$-principles
often prevent the existence of large cardinals via their anti-reflection features. For example $\square_\kappa$ prevents the existence of cardinals that reflect stationary sets to $\alpha < \kappa$ (and hence rule out cardinals like supercompacts).
Trivially, we might look at axioms of the following form:
Variety 3 $\neg \Phi$ or $\neg Con(\Phi)$ for some large cardinal axiom $\Phi$.
These obviously and boringly prevents the existence of a large cardinal in the model. Another possible candidate (formalisable in $NBG + \Sigma^1_1$ comprehension) is:
Variety 4 Inner model hypotheses.
For example, the vanilla inner model hypothesis that any parameter-free first-order sentence true in an inner model of an outer model of $V$ is already true in an inner model of $V$. This prevents the existence of inaccessibles in $V$. Other variants rule out cardinals not consistent with $L$.
Final example (if we're being super liberal about what we allow to be revised):
Variety 5 Choice axioms.
For instance $AC$ prevents the existence of Reinhardt (and super-Reinhardt, Berkeley etc.) cardinals in $V$ (assuming that these are, in fact, consistent with $ZF$, which is pretty non-trivial).
My question:
Are there other kinds of axiom with anti-large-cardinal features? Especially non-trivial ones (i.e. not like Variety 3).
Best Answer
Foreman's maximality principle (the statement: any non-trivial forcing either adds a new real or collapses some cardinals) implies there are no inaccessible cardinal.
Also the consistency of Magidor's question (tree property at all regular cardinals above $\aleph_1$) implies the non-existence of inaccessible cardinals.
Another example is the following maximality principle introduced at A new maximality principle and its consequences:
$(MP_*):$ For all uncountable regular cardinals $\kappa, 2^{<\kappa}=\kappa^{+}$ and all trees of height and size $\kappa$ are specialized.