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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.

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    $\begingroup$ Regarding your third paragraph, the existence of Reinhardt cardinals in $\mathsf{ZF}$ implies infinitary partition relations of the kind you mention (and stronger ones). $\endgroup$ Jun 25, 2023 at 23:35

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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.

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It seems to me there are at least two canonical examples of "large cardinal axioms which are almost inconsistent, but get off on a technicality".

  1. ZFC itself. The theory was carefully designed to have all the advantages of naive set theory, but to avoid Russell's paradox.

  2. Rank-into-rank cardinals, wholeness axiom, (others?): These axioms are carefully designed to get as close to the Kunen inconsistency as possible while carefully avoiding it.

You've stipulated various provisos ruling out certain possible answers; let's see how (1) and (2) fare in the face of these provisos.

  • They fall under the standard large-cardinal hierarchy.

Sure.

  • Their consistency should follow from the consistency of an $I0$ cardinal.

Sure.

  • The relevant inconsistencies should not be reformulations of the Kunen inconsistency.

(1) satisfies this, though (2) does not.

  • They shouldn't be exceedingly strong like Berkeley cardinals

I'm not sure what this means -- maybe this proviso is redundant if you're already asking to be weaker than $I0$? I think (1) and (2) satisfy this.

  • Doubts about the consistency of the axiom in question should come "before" the development of Vopenka or rank-into-rank cardinals.

I'm not sure what "before" means (historically? logically?...). I think that (1) passes here while (2) does not.

  • The inconsistency which the axiom almost hits should be a "consequence of the development of the theory"

I really don't know what this proviso means. What theory are you referring to?

So if my hermeneutical forays into the text of your question are accurate, it seems like (1) is an answer to your question, even though (2) is not (because you explicitly ruled it out). If I'm off-base, perhaps you could clarify your question a bit?

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  • $\begingroup$ I reformulated the question to try to make it more clear what I meant. Do you have a specific theorem proven in ZFC (rank-into-rank) that is close to being a contradiction in ZFC (rank-into-rank)? While the notion of a rank-into-rank cardinal may have originally be defined as a simple axiom that narrowly avoids the Kunen inconsistency, now we should know that the fact that if $j:V_\lambda\rightarrow V_\lambda$, then $\lambda$ has countable cofinality is essential to the theory of rank-into-rank embeddings since the quotient algebras of rank-into-rank embeddings must be locally finite. $\endgroup$ Jun 25, 2023 at 23:13
  • $\begingroup$ The rank-into-rank cardinals I0-I3 have a sophisticated theory behind them, so I am more interested in large cardinal axioms that follow from I0 cardinals than the exceedingly strong axioms. If Berkeley cardinals are found to be inconsistent, not much would change, but some axiom between I3 and I0 were found to be inconsistent, then that would be a big deal. I decided to narrow my question down to the parts of the large cardinal hierarchy that I (and probably others) care about, so I am not that interested in Berkeley cardinals at the moment. $\endgroup$ Jun 25, 2023 at 23:22
  • $\begingroup$ I will leave it up to the reader and answerer whether they want to consider things such as Reinhardt cardinal as notable enough for us to be concerned about an inconsistency. $\endgroup$ Jun 25, 2023 at 23:23
  • $\begingroup$ Instead of "ZFC," maybe Tim Campion means the comprehension axiom, which gets off Russell's paradox on the technicality of being restricted. $\endgroup$ Jun 26, 2023 at 12:18
  • $\begingroup$ @JosephVanName considering that Berkeley cardinals are known to be inconsistent (with AC), the assertion "If Berkeley cardinals are found to be inconsistent, not much would change" is as true as you can get $\endgroup$
    – Holo
    Jun 26, 2023 at 21:01

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