Definition of $H_\lambda$ (hereditary cardinality)

Question

It seems to me that the definition of $H_\lambda$ (the set of sets of hereditary cardinality less than $\lambda$) on the web page at Cantor's Attic is not quite correct. From the page:

$H_\lambda=\{x: |\operatorname{trcl}(x)|<\lambda\}$ where $\operatorname{trcl}(x)$ denotes the transitive closure of $x$.

This is correct when $\lambda$ is a regular cardinal, but if $\lambda$ is singular then the definition should include any cofinal subset of $\lambda$ with size less than $\lambda$, which this definition doesn't. Indeed, if $\lambda$ is a singular cardinal, $A \subset \lambda$ is cofinal, and $|A| < \lambda$, then because $A$, its elements, their elements, and so on all have cardinality less than $\lambda$ then we should say that $A$ has hereditary cardinality less than $\lambda$ if we want the terminology to make any sense.

I think the closest definition of $H_\lambda$ that would be correct is

$H_\lambda=\{x: \forall y \in \operatorname{trcl}(\{x\})\, (|y| <\lambda)\}$.

My question for Math.SE is whether I'm correct that there is a mistake, and if so, what the best definition would be (so that I can fix the page.) In my experience $H_\lambda$ is only ever used when $\lambda$ is regular, so the existing definition could be fixed by restricting it to this case, which is what Jech does in the exercises to Chapter 12. On the other hand, the second definition proposed above has the feature that it generalizes better to the case of "hereditarily in $P$" where $P$ is some class other than $\{y : |y| < \lambda\}$, I think. There might be a third definition that is even better. What is standard? What is best?

Answer 1

You are correct. For the reasons you discuss, the appropriate definition of $H_\lambda$ for singular $\lambda$ is the set of $x$ such that $x$ and all sets in its transitive closure have size below $\lambda$. (Though I could not instantly find a reference, this is actually standard among those that consider the set at all, which are not too many, and its use is somewhat limited. For example, I would imagine one may want to consider the class in settings where choice fails, but I have yet to see an application in that context.)

On the other hand, it is also standard (and far more common) to only define $H_\lambda$ for $\lambda$ regular.

---

Mostly as a curiosity: Some people (Forster, for example), present $H_\kappa$ as $\bigcap\{y\mid \mathcal P_\kappa(y)\subseteq y\}$. This presentation generalizes: We can define $$\mathcal P_\phi(x)=\{y\subseteq x\mid \phi(y)\},$$ so $\mathcal P_\kappa(x)$ is $\mathcal P_{|y|<\kappa}(x)$. One can then define $H_\phi$ as $\bigcap\{y\mid\mathcal P_\phi(y)\subseteq y\}$. This notation seems to go back to

Maurice Boffa. Sur l'ensemble des ensembles héréditairement de puissance inférieure à un cardinal infini donné, Bull. Soc. Math. Belg., 22, (1970), 115–118. MR0309732 (46 #8837).

Note this version coincides with the standard one for $\kappa$ regular, but is the problematic one for $\kappa$ singular.

Answer 2

Although the definition that Trevor quoted from Cantor's Attic fails, as he says, to match the usual meaning of "hereditarily", it is, as far as I know, fairly standard. I suspect the reason (or at least a reason) for that is that it works well in Lévy's cardinal boundedness theorem: Anything $\Delta_1$-definable from sets in $H_\lambda$ is itself in $H_\lambda$. If you change the definition so that a small cofinal subset $A$ of a singular $\lambda$ becomes an element of $H_\lambda$, then this theorem fails, because $\lambda$ itself is $\Delta_1$-definable from $A$ as the union of $A$.