en.wikipedia.org/wiki/Large_cardinal#Motivations_and_epistemic_status
1 correction found
That is, no exception is known to the following: Given two large cardinal axioms A1 and A2, exactly one of three things happens:
This is outdated. A 2025 published paper gives counterexamples, exhibiting large-cardinal set theories with incomparable consistency strengths.
Full reasoning
The article states that there are no known exceptions to a three-way trichotomy for pairs of large-cardinal axioms: either they are equiconsistent, or one proves the consistency of the other.
That is no longer accurate. In 2025, Joel David Hamkins published "Nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength" in Monatshefte für Mathematik. The abstract says the paper presents "counterexamples" and specifically describes "various large cardinal set theories, which exhibit incomparability and illfoundedness in consistency strength."
That directly contradicts the claim that no exception is known. If two theories are incomparable in consistency strength, then none of the article's three listed cases holds: neither proves the other's consistency, and they are not equiconsistent in the required sense.
To be fair, many familiar large-cardinal hypotheses are still often discussed as if they line up in a near-linear way. But the Wikipedia sentence is broader than that and says there are no known exceptions. After Hamkins's 2025 paper, that statement is false as written.
2 sources
- Nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength
Abstract: "In this paper I present counterexamples, natural instances of nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength... I present ... various large cardinal set theories, which exhibit incomparability and illfoundedness in consistency strength."
- [2208.07445] Nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength
Abstract: "In this paper I present counterexamples, natural instances of nonlinearity and illfoundedness in the hierarchy of large cardinal consistency strength... I present diverse cautious enumerations of ZFC and large cardinal set theories, which exhibit incomparability and illfoundedness in consistency strength."