TITLE: Definition of $H_\lambda$ (hereditary cardinality)
QUESTION [6 upvotes]: 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?

REPLY [2 votes]: In ON THE TRANSITIVE HULL OF A K-NARROW RELATION(DOI: 10.1002/malq.19920380137) 
and On κ-hereditary Sets and Consequences of the
Axiom of Choice(DOI:  10.1002/1521-3870(200010)46:4<563)  
They define:

The Strict Hereditary Set of Cardinality $<κ$

as: $$H_κ=\{x∈{\bf WF}\mid|\mbox{trcl}(x)|<κ\}$$
and define:

The Hereditary Set of Cardinality $<κ$

as: $$HC_κ=\{x∈{\bf WF}\mid y\in\mbox{trcl}(\{x\})⇒|y|<κ\}$$
And the $HC_κ$ definition is exactly what you suggested.