nLab
Lindenbaum number

Contents

Idea

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

Definition

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

Properties

By construction, if there were a surjection function X *(X)X \to \aleph^*(X), then *(X)\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 (X)\aleph(X) denotes the Hartogs number of XX, then we also have:

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

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

  • XX and *(X)\aleph^*(X) are comparable if and only if XX 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 κ\kappa, there is a Dedekind-finite set XX (i.e. an infinite set such that (X)= 0\aleph(X)=\aleph_0) and *(X)κ\aleph^*(X)\geq\kappa (Monro 1975, so in particular the gap between (X)\aleph(X) and *(X)\aleph^*(X) can be arbitrarily large,

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

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

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 κ\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 κDC_\kappa, arXiv:1212.4396v1 (published as Embedding Orders Into The Cardinals With DC κDC_\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.

Last revised on May 11, 2019 at 05:35:42. See the history of this page for a list of all contributions to it.