nLab Lindenbaum number

Idea

The Lindenbaum number of a set $X$ is the smallest well-ordered set that is not a quotient of $X$ (as a bare set). It is a ‘surjective analogue’ of the Hartogs number.

Definition

Given a set $X$, denote by $\aleph^*(X)$, the set of ordinals $\alpha$ that admit a surjection $X\to \alpha$. Equivalently, it is the set of order isomorphism classes of well-orderings of quotients of $X$.

Properties

By construction, if there were a surjection function $X \to \aleph^*(X)$, then $\aleph^*(X)$ would be order isomorphic to one of its own elements, and hence to a proper initial segment of itself, which is impossible.

If $\aleph(X)$ denotes the Hartogs number of $X$, then we also have:

• there is an injection $\aleph(X) \to \aleph^*(X)$,

• for well-orderable $X$, then $\aleph(X) \simeq \aleph^*(X)$,

• $X$ and $\aleph^*(X)$ are comparable if and only if $X$ is well-orderable, and this holds for both $\lt$ and $\lt^*$, the surjective preordering on cardinals,

• it is consistent that for any well-orderable cardinal $\kappa$, there is a Dedekind-finite set $X$ (i.e. an infinite set such that $\aleph(X)=\aleph_0$) and $\aleph^*(X)\geq\kappa$ (Monro 1975, so in particular the gap between $\aleph(X)$ and $\aleph^*(X)$ can be arbitrarily large,

• the statement that $\aleph(X)=\aleph^*(X)$ for all $X$ is equivalent to the axiom of choice for well-ordered families (i.e. if $A$ is a well-orderable family of non-empty sets, then $A$ admits a choice function). This was proved by Pincus, and appears in Pelc 1978,

• there is an injection $\aleph^*(X) \to \aleph(P(X))$ (Diener 1992).

Related entries

• Hartogs number
• successor
• diagonal argument

References

The concept appeared as exercise 7.1.6 in

• Karel Hrbacek and Thomas Jech, Introduction to Set Theory, 3rd Edition, Marcel Dekker (1999)

[Note: the exercise is also in chapter 8 in the 1984 2nd edition, not sure about the 1st edition]

And is mentioned in

• Karl‐Heinz Diener, On the transitive hull of a $\kappa$-narrow relation, Math. Logic Quarterly, Volume 38, Issue1 (1992) pp 387-398, doi:10.1002/malq.19920380137

The name appears to be due to

• Asaf Karagila, Embedding Posets Into Cardinals With $DC_\kappa$, arXiv:1212.4396v1 (published as Embedding Orders Into The Cardinals With $DC_\kappa$ Fund. Math. 226 (2014), 143-156. and omitting the Lindenbaum number)

note: version 1 of the arXiv paper!

• G. P. Monro, Independence results concerning Dedekind-finite sets, J. Austral. Math. Soc. 19 (1975), 35–46.

• Andrzej Pelc, On some weak forms of the axiom of choice in set theory. Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 26 (1978), no. 7, 585–589.