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A Grothendieck universe provides an easy-to-understand example for a model of ZFC. Because ZFC, if consistent, cannot prove the existence of any model of itself, the existence of universes needs to be assumed by new axioms, which is the approach of TG set theory.

TG set theory, if still consistent, cannot prove the existence of a model of itself. But we can add another "hyper-universe" axiom:

For every set $X$ there exists a universe $U$ with $X\in U$, such that for every set $Y \in U$, there exists a universe $V$ with $Y\in V\in U$.

In other words: Every set is contained as an element in a "hyper-universe", which is a model of TG set theory.

As soon as one gets on that trip, adding more axioms concerning "hyper-hyper-universes" etc. seems natural.

My first question is: How far can we exploit this thought? Is it even transfinitely continuable ("$\omega$-hyper-universes", "$\omega+1$-hyper-universes" etc.)? (Sorry, I dont know how to describe this line of thought properly, I hope you know what I mean.) Are there some amazing consequences? Or inconsistencies?

My second question is: Can you imagine an "ordinary mathematician" needing such "hyper-universes"?

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You're starting up the large cardinal hierarchy.


Iterating universes transfinitely is but the very very very very very beginning of the gigantic large cardinal hierarchy. Universes correspond exactly to (strongly) inaccessible cardinals, which are (more or less) the smallest of these: $\kappa$ is inaccessible iff $\kappa$ is an uncountable regular strong limit cardinal, and a Grothendieck universe is exactly a set of the form $V_\kappa$ for some inaccessible $\kappa$ (see the discussion here). Hyperuniverses, meanwhile, correspond$^1$ to $1$-inaccessibles. A natural limit of this iterative idea is the hyperinaccessible cardinal.

... And that's still extremely tiny by the standards of the subject. There's a general divide between the "small" large cardinal properties and the "large" large cardinal properties - roughly, the latter begin with the measurable cardinals. Sadly, it's the smaller ones that have the cool names, like (in increasing strength/size, and each larger than Mahlos) indescribables, ethereals, and ineffables. Wikipedia has a good list.


So what are the mathematical consequences of these things?

For one thing, large cardinal principles high up enough in this hierarchy provide tameness results: e.g. once you start hitting the level of measurable cardinals (which are incalculably bigger than inaccessibles) you start being able to prove that more and more sets of reals are Lebesgue measurable. As a concrete example, the measurability of every continuous image of a coanalytic set is provable from ZFC + large cardinals but not from ZFC.

This aspect of large cardinal implications is quite well understood at this point, through inner model theory and descriptive set theory. Very briefly, the key ingredient is the idea of determinacy: large cardinals implying that more and more abstract games are determined, and tameness properties often correspond to the determinacy of some associated game. See Larson's essay for more on this.

There are, however, other strange appearances of large cardinals in mathematics. In algebra, they appear in the study of left distributive algebras, and there are still results here which have no known large cardinal proof; this survey of Dehornoy is relevant. There are also appearances in purely finite combinatorics, primarily studied by Harvey Friedman in a variety of forms.


Now what about consistency?

In my opinion, the situation is one of surprising consistency: there aren't any "deep" inconsistencies that have arisen, and the few plausible large cardinal notions which we now know to be inconsistent (e.g. Reinhardt cardinals and Moschovakis cardinals) were quickly quickly discovered to be so.

Indeed, there's a hilarious story here about one particular large cardinal notion - Vopenka's principle. This was originally introduced by Vopenka as a joke; Vopenka was annoyed by the proliferation of large cardinal principles, and whipped one up that seemed plausible at first but which he thought he could prove inconsistent. But his inconsistency argument broke down, and now Vopenka cardinals are actually quite significant in set theory - and a lot of their serious study was kicked of by Jech, Vopenka's own student! See Pudlak's account of the topic.


$^1$Actually they don't - the definition you've given isn't really what you want it to be. For example, under reasonable assumptions there is a countable transitive model of TG set theory. Note that the usual definition of a universe is more than just a transitive model of ZFC - it's also demanded to be closed under powersets and ranges of functions, which are external stipulations.

Instead, the right way to define a hyperuniverse is as follows: $U$ is a hyperuniverse iff $U$ is a universe and for each $x\in U$ there is a universe $U_x$ with $x\in U_x\in U$. Then the hyperuniverses are exactly the sets of the form $V_\kappa$ for $\kappa$ an inaccessible limit of inaccessibles (= $1$-inaccessible).

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  • $\begingroup$ Let me disagree with "A natural limit of this iterative idea is the Mahlo cardinal" for two reasons: (1) I don't really see the sort of limit that could be involved here. Mahlo cardinals seem (to me) to be essentially different from anything I'd dream up starting with the idea of iteration. (2) A plug for Paul Mahlo. He introduced what are now called Mahlo cardinals in 1911. For comparison, the most important property of stationary sets, Fodor's Lemma, was published in 1956. $\endgroup$ Commented Jul 20, 2019 at 22:29
  • $\begingroup$ @AndreasBlass Whoops, I conflated Mahlos with hyperinaccessibles! Fixing ... $\endgroup$ Commented Jul 20, 2019 at 22:32
  • $\begingroup$ Thanks for that extensive answer! I am more interested in the technical realization of hyperuniverse axioms. I can imagine axioms for every set being contained in 1-,2-,3-universes etc. But how do you formalize a transfinite universe axiom? The cool thing about Grothendieck universes is, that their simple definition does not rely on sophisticated set theory concerning the cumulative hierarchy and von Neumann cardinals. $\endgroup$
    – Lucina
    Commented Jul 21, 2019 at 9:23
  • $\begingroup$ I think the coolest-sounding cardinal-name I have seen is "totally indescribable ". $\endgroup$ Commented Jul 21, 2019 at 11:52
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    $\begingroup$ the notion of elementary embedding is a purely model-theoretic one, and one which is of interest to category theorists (at least some of them). Now granted, the "mirror universe principle" isn't expressible in ZFC, but it is expressible in NBG (and that's a more natural framework for category theory anyways). And it's extremely powerful. For example, any $M\prec V$ must satisfy powerset, since $V$ does; by elementarity, whenever $x\in M$ the thing $M$ thinks is the powerset of $x$ actually is the powerset of $x$. More generally, $M$ is a universe. It's also a hyperuniverse - and so on. $\endgroup$ Commented Jul 21, 2019 at 16:23

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