A universe of hierarchies

When doing set theory (or mathematics in general) we’re working inside some universe, usually denoted by V. Since we can’t work with everything there is (in a first-order way), we resort to working with initial segments of V. The confusion then arises, since what do we mean by an initial segment? Some prefer to work with the “rank-hierarchy” V_\alpha and others prefer to work with the “hereditary cardinality hierarchy” H_\kappa. It gets even worse if we’re working in Gödel’s constructible universe L, since then we also got Gödel’s L_\alpha hierarchy and Jensen’s J_\alpha hierarchy. How do we picture these hierarchies? What are their relation to each other?

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Painting by Nikolay Lavetsky.

The “vanilla” hierarchy V_\alpha is also the way we build our universe in the first place. Here we start out with the empty set V_0=\emptyset and then successively apply the power set operation; i.e. we set V_{\alpha+1}=\mathcal P(V_\alpha) and for \eta a limit ordinal we put V_\eta=\bigcup_{\alpha<\eta}V_\alpha.

The foundation axiom is equivalent to saying that this construction reaches all sets. That is, V:=\bigcup_{\alpha<\infty}V_\alpha is the entire universe we’re working in. To every set x we can then associate it’s rank, which is the least ordinal \alpha such that x\in V_{\alpha+1}. Conveniently, the rank of any ordinal is the ordinal itself.

So in some sense, it seems natural to work with the V_\alpha‘s, and indeed, in many case it’s the most convenient choice. Sometimes however, these rank initial segments V_\alpha do have properties which isn’t ideal in every scenario. Firstly, it almost never satisfies the Replacement axiom: only when \alpha is 0, \omega or inaccessible. This is basically because the rank initial segments are incredibly “wide”: the size of V_\alpha is \beth_\alpha, which is way larger than \alpha unless \alpha is a strong limit cardinal. We’re interested in satisfying Replacement to do things like taking ultrapowers.

Another hierarchy, H_\kappa, consists of all sets x such that x has hereditary cardinality strictly less than \kappa. This intuitively means that x has size {<}\kappa, every element of x has size {<}\kappa, and so on. A downside here is that we get a coarser hierarchy: we can only distinguish between two initial segments H_\kappa and H_\lambda when |\kappa|\neq|\lambda|. On the plus side we get that H_\kappa satisfies all of the ZFC axioms except the Power Set axiom, whenever \kappa is regular. Note that the rank initial segments V_\alpha satisfy the Power Set axiom whenever \alpha is a limit ordinal.

These fellas are almost always smaller than the rank initial segments: the size of H_\kappa is \sup_{\alpha<\kappa}2^\alpha. So if \kappa=\lambda^+ then it’s of size 2^\lambda. This means that if GCH holds then |H_\kappa|=\kappa whenever \kappa is regular. I keep talking about regularity of \kappa, which is because if it’s singular then we would get that H_\kappa=H_{\kappa^+}. To make a sensible definition we can then put H_\kappa:=\bigcup\{H_\lambda\mid\lambda<\kappa\text{ regular}\}.

Let’s picture the relationship between the V_\alpha‘s and the H_\kappa‘s. To make a coherent picture we have to agree on how the cardinal exponent function behaves in our universe. I’ve arbitrarily chosen that 2^\kappa=\kappa^{++} here. The picture in general will be similar though, but the larger 2^\kappa is, the “narrower” the H_\kappa‘s become. Also, \kappa in the picture is meant to be an inaccessible cardinal — hopefully this overloading of the use of “\kappa” isn’t too confusing.

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So far so good. Now, let’s have a look at the initial segments of L. We of course still have the relativised hierarchies V_\alpha^L and H_\kappa^L (for L-cardinals \kappa), but L has its own two special hierarchies: the L_\alpha‘s and the J_\alpha‘s.

The L_\alpha‘s look like they’re simply the L version of the rank initial segments. We again set L_0=\emptyset and we then set L_{\alpha+1} to be all definable subsets of L_\alpha, with parameters from L_\alpha. The difference here is that we keep getting new subsets of earlier stages: L_{\omega+10} will contain new subsets of L_\omega. This continues until we reach the next cardinal, so that it isn’t until we reach \omega_1 that we stop adding subsets to L_\omega.

When we are at cardinal stages though, then we’re simply back at the H_\kappa hierarchy: L_\kappa=H_\kappa^L, so we can see the L_\alpha‘s to give the missing fine-grained pieces to the H_\kappa hierarchy. Also, since GCH holds in L we do get that L_\kappa has size \kappa. This holds in general: L_\alpha has size |\alpha|.

A downside to this hierarchy is that the hierarchy is mostly only well-behaved at limit stages. Most formulas we’d like to work with in an L-setting, like checking if a set is of the form L_\alpha, or how two sets are related in the constructibility ordering, or even V=L, are only absolute to L_\alpha when \alpha is a limit ordinal.

This is a first motivation for Jensen’s hierarchy J_\alpha, whose initial segments always have limit height. Here we again start with the empty set, at limit stages we take unions, but we take successor stages J_{\alpha+1} to be the closure of J_\alpha\cup\{J_\alpha\} under all the rudimentary functions. Closing something off under (finitely many) functions takes \omega stages, so here we then get that the height of J_{\alpha+1} is the height of J_\alpha plus \omega, so that in general we get that J_\alpha has height \omega\cdot\alpha.

A notable feature of the Jensen hierarchy is that J_{\alpha+1} agrees with L_{\alpha+1} below height \alpha+1 — all new sets are added on top. The main motivation for working with the Jensen hierarchy rather than Gödel’s is more than the absolute properties however: the Jensen hierarchy allows a fine structural analysis, making it the hierarchy of choice for inner model theorists.

Here’s a picture of the two L-hierarchies, situated inside another universe V (here we assume V\neq L because, why not). Note here that the Jensen hierarchies are really just the Gödel hierarchy “stretched upwards”, and that they agree whenever the index is a cardinal.

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