I am looking for examples of results about large cardinals, large cardinal axioms, or other objects of high (or seemingly high) consistency strength that are almost inconsistencies. I am looking for large cardinal axioms that only remain consistent because they "got off on a technicality".

In particular, I am looking for axioms $C$ along with theorems $T$ proven using $C$ such that the theorem $T$ calls into question the consistency of $C$ but which does not establish a contradiction under $C$.

These examples should fall under the standard large cardinal hierarchy. For example, the partition relation $\kappa\rightarrow(\aleph_0)^{\aleph_0}_2$ (this says that for each function $f:[\kappa]^{\aleph_0}\rightarrow\{0,1\}$, there is an infinite $A$ where $f|_{[A]^{\aleph_0}}$ is constant) is an inconsistent strengthening of the notion of an $\omega$-Erdos cardinal, but I would not consider this axiom since it is not implied by the existence of a non-trivial rank-into-rank embedding (unless non-trivial rank-into-rank embeddings are inconsistent).

Any near inconsistency of an axiom $C$ where $\text{Con}(\text{ZFC}+\text{I0})\rightarrow\text{Con}(C)$ and where $\text{Con}(C)\rightarrow\text{Con}(\text{ZFC})$ should be fine. I am not currently interested in near inconsistencies that are reformulations of the Kunen inconsistency nor am I currently interested in exceedingly strong axioms such as Berkeley cardinals.

When an axiom which we shall call Axiom $C$ (Axiom $C$ could be the existence of a Vopenka or a rank-into-rank cardinal) is originally formulated and it is not immediately obvious that the consistency of Axiom $C$ is implied by stronger large cardinals, then it is natural to be skeptical about the consistency of Axiom $C$. But I am generally not currently interested in these apparent near inconsistencies because the doubt of the consistency of Axiom $C$ seems to wane over time as the theory around Axiom $C$ or about stronger axioms is developed. I am instead looking for a case where after Axiom $C$ is formulated and the theory of Axiom $C$ is developed, a result $T$ derived from Axiom $C$ causes people to doubt the consistency of Axiom $C$.

Perhaps Jack Silver's concerns about the consistency of ZFC or of measurable cardinals should count as a near inconsistency.

**My own example of a near inconsistency**

It is easy to prove that if $j:V_\lambda\rightarrow V_\lambda$ is an elementary embedding and $\alpha<\lambda$, then $(j*j)(\alpha)\leq j(\alpha)$, but there are many finite algebraic structures that appear similar to algebras of elementary embeddings but which violate this inequality.

An algebra $(X,*,1)$ that satisfies the identities $x*(y*z)=(x*y)*(x*z)$ and $x*1=1,1*x=x$ is said to be nilpotent if for all $x\in X$, there is some $n$ with $x^{[n]}=1$ where we define $x^{[n]}$ recursively by letting $x^{[1]}=x,x^{[n+1]}=x*x^{[n]}$. Define $x_{[1]}=x,x_{[n+1]}=x_{[n]}*x$. Then there are 6854 nilpotent self-distributive algebras $(X,*,1,a,b)$ generated by $a,b$ up to a critical equivalence preserving $a,b$ with $a_{[64]}=1,b*b=1$ and where $\text{crit}[X]$ is linearly ordered but only 1145 of these (16.7056%) satisfy the inequality $\text{crit}((x*x)*y)\leq\text{crit}(x*y)$. This looks like a near inconsistency since it seemed feasible to obtain one of those 5709 algebras from rank-into-rank embeddings.