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I've been looking into the idea of large cardinals recently, and I found this question in particular to be interesting.

Large Cardinals Ordered by Cardinality of Least Instance

The most popular answer to this question goes very in depth about the ordering of cardinality for large cardinals, and it made me think of a few questions.

The answer mentions stationary transcendence and measurable transcendence, and talks about how they are two different magnitudes. I don't know about stationary, but measurable transcendence seems to have something to do with measurable cardinals, so I am curious. Is there something akin to "extendable transcendence" (in relation to extendable cardinals), and could there theoretically be larger levels of transcendence, as you get into larger and larger cardinals.

Patrick Malone
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    Tangentially, here's a quick gloss on stationarity (since it's super important and interesting). If $\kappa$ is a regular uncountable cardinal (or really any ordinal of uncountable cofinality), there is an analytic analogy connecting "club subset of $\kappa$" with "measure-one subset of $[0,1]$" and "stationary subset of $\kappa$" with "non-null subset of $[0,1]$." So stationarity, similar to non-null-ness in analysis, is a very strong nonemptiness condition. To see just how strong stationarity can be, consider the following natural type of strengthening of (strong) inaccessibility: (cont'd) – Noah Schweber Mar 06 '21 at 07:58
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    a property of the form "$\kappa$ is inaccessible and there are many inaccessibles $<\kappa$." If we take "many" here to mean "$\kappa$-many" we get the notion of $1$-inaccessibles, whereas if we take "many" to be "stationarily many" we get the vastly stronger notion of a Mahlo cardinal. The gulf between $1$-inaccessibility and Mahloness is gigantic (although of course Mahloness is still vastly weaker than weak compactness which in turn is galactically weaker than measurability), and this reflects the surprising "richness" of stationarity as a nonemptiness property. – Noah Schweber Mar 06 '21 at 07:59
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    I should say that the term "transcendence" is not particularly standard in that context. Even looking at Kanamori's book, the term appears in a more descriptive context than a mathematical context (and mostly to describe a relationship between inner models and sharps, actually). I don't recall ever seeing the term being used in this context until now. – Asaf Karagila Mar 07 '21 at 02:46
  • Is there a better term that you think might accurately describe what the guy was saying? Maybe something akin to indescribability? I know that a something is considered totally indescribable if it is Πn m-indescribable for all positive integers n and m, so I've wondered if it would be possible for something to be indescribable for a large cardinal. Could transcendence and its various levels be something similar to that idea? Or does being α-indescribable stop making a difference once α is infinite? – Patrick Malone Mar 09 '21 at 21:16

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