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.



Large cardinal near inconsistencies
I would argue that a "restricting versions" of large cardinals are such.


Starting from the top down, we have the inconsistent Berkeley cardinals:


* $κ$ is Berkeley if for every transitive $M\ni\kappa$ and $ν<κ$ we have an elementary embedding on $M$ with critical point between $\nu$ and $κ$.


A way to make this cardinal consistent with $\sf AC$ is by restricting the sets $M$ can be:


* $κ$ is $\sf HOD$-Berkeley if the above condition holds for $M\in \sf HOD$


This large cardinal is very important, the consistency of $\sf ZFC+HOD$-Berkeley cardinal is implied by the consistency of $\sf ZF$+Berkeley cardinal (±some other large cardinals), and if $\sf ZFC+HOD$-Berkeley is consistent it means that $\sf HOD$ is very far from $V$ (again, ±some other LCA).


Similarly to Berkeley, we can weaken Reinhardt cardinals to smaller classes:


* $κ$ is $\sf HOD$-Reinhardt if there exists an elementary embedding on $\sf HOD$ with critical point $κ$.


Similarly to the Berkeley case, the consistency of $\sf ZF$+Reinhardt implies the consistency of $\sf ZFC+HOD$-Reinhardt (±LCA), and the existence of (sufficiently large) $\sf HOD$-Reinhardt implies that $\sf HOD$ is far from $V$ (±LCA).


The $\sf HOD$-Berkeley/Reinhardt cardinals are $\sf HOD$-analog to $0^\sharp$ of $L$, which brings us to:


* a non-trivial elementary embedding on $L$.
* a non-trivial elementary embedding from $V$ to some transitive class


I would argue that the only reason this is less concerning to be consistent is that we know that $V$ can be *very* far from $L$, and because we know the theory of ultrapowers.


